Research Article Performance Analysis of Heterogeneous...

Research ArticlePerformance Analysis of Heterogeneous Data Centers inCloud Computing Using a Complex Queuing Model

Wei-Hua Bai12 Jian-Qing Xi1 Jia-Xian Zhu2 and Shao-Wei Huang2

1School of Computer Science and Engineering South China University of Technology Guangzhou 510006 China2School of Computer Science Zhaoqing University Zhaoqing 526061 China

Correspondence should be addressed to Jian-Qing Xi jianqingxi163com

Received 3 March 2015 Revised 28 May 2015 Accepted 1 June 2015

Academic Editor Javier Plaza

Copyright copy 2015 Wei-Hua Bai et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Performance evaluation of modern cloud data centers has attracted considerable research attention among both cloud providersand cloud customers In this paper we investigate the heterogeneity of modern data centers and the service process used in theseheterogeneous data centers Using queuing theory we construct a complex queuing model composed of two concatenated queuingsystems and present this as an analyticalmodel for evaluating the performance of heterogeneous data centers Based on this complexqueuing model we analyze the mean response time the mean waiting time and other important performance indicators We alsoconduct simulation experiments to confirm the validity of the complex queuingmodelWe further conduct numerical experimentsto demonstrate that the traffic intensity (or utilization) of each execution server as well as the configuration of server clusters in aheterogeneous data center will impact the performance of the system Our results indicate that our analytical model is effective inaccurately estimating the performance of the heterogeneous data center

1 Introduction

Cloud computing is a popular paradigm for providing ser-vices to users via three fundamental models Infrastructureas a service (IaaS) Platform as a service (PaaS) and Softwareas a service (SaaS) [1 2] Cloud computing providers offercomputing resources (eg servers storage networks devel-opment platforms and applications) to users either elasticallyor dynamically according to user-demand and form ofpayment (eg pay-as-you-go) [3] A modern data center isconsidered a heterogeneous environment because it containsmany generations of servers with hardware configurationsthat may differ especially in terms of the speed and capacityof the processors Generally these servers have been addedto the data center gradually and provisioned to replace theexisting (or ldquolegacyrdquo) machines already in the data centerrsquosinfrastructure [4 5]Theheterogeneity of thismix ofmachineplatforms affects the performance of data centers and theexecution of cloud computing applications

The ability to ensure the desired Quality of Service (QoS)which is a standard element of the Service Level Agreement(SLA) established between consumers and cloud providers is

one the most important business considerations for a cloudcomputing provider [6 7] A typical QoS outlines a set ofcritical performance indicators mean response time meantask queuing lengthmeanwaiting timemean task number ofthroughput task capacity of the system blocking probabilityand probability of immediate service of the system All ofthese indicators can be analyzed and described using queuingtheory [8]

A task completed by the cloud computing center followsseveral steps (1) a customerrsquos request is transformed into acloud computing task and sent to the task queue (the firstlevel queue) maintained by the main scheduler server (2)

the scheduler allocates computing resources for the task anddispatches it to a node (execution server) to execute (3)

the node takes the task from the second level queue andallocates space in a CPU core (the main computing resource)to complete it (4) the system outputs the result and thetask leaves the system In this process blocking will occurif a task leaves the system without having been executed (orthe task may have been abandoned by the system) becauseof capacity constraint at the first level queue A queuingmodel for a cloud computing heterogeneous data center can

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 980945 15 pageshttpdxdoiorg1011552015980945

2 Mathematical Problems in Engineering

analyze the relationship among three factors the arrival rateof tasks the core utilization and the probability of a node(execution server) being dispatched Cloud providers canutilize this information to analyze their systemsrsquomanagementand optimize their allocation of computing resources [9] Aswe show in Section 2 much existing research has discussedthe problem of how to analyze data center performanceHowever these studies have not considered the followingimportant issues

(i) A cloud center can include a large number of serversof different generations with different CPU speedsand processor capacities Existing research rarelyconsiders cloud center heterogeneity rather studiesgenerally assume that every node operates at the samespeed

(ii) A cloud system has at least two levels of schedulingprocesses (service processes) the main schedulerwhich allocates computing resources and the secondscheduler (a nodeexecution server) which executestasks The arrival rate of a task to the main schedulerversus to a node is different and this must be takeninto account in the queuing analysis

(iii) A node server with more than one speed and proces-sor capacity will have different probabilities of beingdispatched depending on the situation Each nodehas its own task queue to maintain which is trackedby a separate queuing model that is included in theoverall system queuing model

(iv) A state-of-the-art resource allocation scheme basedon the utilization threshold [120575119897 120575ℎ] controls nodesthat are to be added to a cloud system or removedfrom it [10]

To fill these gaps in this paper we employ a complexqueuing model to investigate the performance of a cloudcenter and the relationship among various performanceindicators in a heterogeneous data center We aim to makethe following contributions with this paper

(1) Wemodel the heterogeneous data center as a complexqueuing system to characterize the service processTwo concatenated queuing systems compose thecomplex queuingmodelThe first is the finite capacityscheduler queuing system which is used to study themain scheduling server The second is the executionqueuing system which has multiple servers for eachmulticore server processor

(2) Based on this complex queuing model and usingqueuing theory we analyze the real output rate ofthe main scheduling servermdashthat is to say the realarrival rate of the execution queuing systems themean response time of the system the blockingprobability and other performance indicatorsmdashanduse this analysis to evaluate the performance of andthe relationships among these indicators in a cloudcenter

(3) We conduct extensive discrete event simulations toevaluate and validate the results of using the complex

queuing model to analyze the performance of hetero-geneous data centers in cloud computing

The remainder of this paper is organized as followsSection 2 reviews and discusses the related scholarship onqueuing models for cloud centers Section 3 discusses ouranalytical model in detail All of the performance metricsobtained by using this complex queuing model are presentedin Section 4 In Section 5 we show the simulation results andcompare them with our analytical results to evaluate and val-idate the applicability of the complex queuing model Finallyconclusions and future work are presented in Section 6

2 Related Work

Although much scholarship has been conducted on the per-formance analysis of a cloud center based on queuing models[11ndash16] factors including the loosely coupled architecturesof cloud computing and the heterogeneity and dynamicinfrastructure of a cloud center have not been considered

In [11] the authors used an119872119866119898119898+119903 queuing systemto evaluate a cloud computing center and obtained a completeprobability distribution of response time number of tasks inthe system and other important performance metrics Theyalso considered the different mean waiting times betweenheterogeneous services and homogeneous services under thesame conditions in the system Based on queuing theoryand open Jacksonrsquos networks a combination of 1198721198721 and119872119872119898 was presented to model the cloud platform forQoS requirements in [12] In [13] the authors presented an119872119872119898 queuing system and proposed a synthesis opti-mization of model function and strategy to optimize theperformance of services in the cloud center In [14 15] theauthors introduced a queuing model consisting of three con-catenated queuing systems (the schedule queue 1198721198721 thecomputation queue 119872119872119898 and the transmission queue1198721198721) to characterize the service process in a multimediacloud and to investigate the resource optimization problemsof multimedia cloud computing In [16] a cloud center wasmodeled as a 119866119868

119883119872119878119873 queuing model to discuss cloudservice performance related to fault recovery

In many existing methodologies [17ndash20] queuing theoryhas been used to analyze or optimize factors such as powerallocation load distribution and profit control

In [17] the authors presented an energy proportionalmodelmdashwhich treats a server as an1198721198661 queuing systemmdashas a strategy to improve performance efficiency In [18]an 119872119872119898 queuing model which considers factors suchas the requirement 120574 of a service and the configurationof a multiserver system is used to optimize multiserverconfiguration for profit maximization in cloud computingIn [19] the authors presented an 119872119872119903119896 queuing systemto model a cloud server farm provided with finite capacityfor profit optimization Another study presents a queuingmodel for a group of heterogeneous multicore servers withdifferent sizes and speeds to optimize power allocation andload distribution in a cloud computing environment [20]In [17] the authors used the 119872119872119888119888 queuing system tomodel a cloud center with different priority classes to analyze

Mathematical Problems in Engineering 3

Internet Responsedeparture

Tasks stream

Users

Users

Users

Users

Scheduledispatch

Main scheduler

Node(execution server)

Data center

middot middot middot

Figure 1 Cloud center architecture and the service model

the general problem of resource deployment within cloudcomputing

However none of the above literature considers that themain scheduler and the execution servers have their owntask queuing that an execution server with different speedswill have different processing times and that each executionserver has a different probability of being dispatched givendifferent arrival rates of tasks As a result a queuing modelused to model a heterogeneous data center for performanceanalysis should characterize the service process within thecloud center as well as the infrastructural architecture of adata center

3 Queuing Model for Performance Analysis

In this section we present a complex queuingmodel with twoconcatenated queuing systems to characterize the service pro-cess and the heterogeneity of the infrastructural architecturein order to analyze the performance of a heterogeneous datacenter

31 Cloud Center Architecture and the Service Model In thecloud computing environment a cloud computing providerbuilds a data center as the infrastructure to provide servicefor customers [21] Figure 1 illustrates the architecture ofa heterogeneous data center and its service model Theheterogeneous data center consists of large-scale servers ofdifferent generations It includes the master server whichworks as amain scheduler to allocate computing resources fora task or to dispatch a node to execute a task and large-scalecomputing nodes which are multicore servers with differentspeeds that work as execution servers

All customers submit requests to the data center forservices through the Internet which has been deployed bythe users or supplied by the cloud providers When the tasks(user requests) arrive at the data center the main schedulerallocates resources for the task or dispatches nodes the taskis then distributed to the execution servers to be executedThemain scheduler has a waiting buffer with a finite capacityin which it saves waiting tasks that cannot be scheduledimmediately The bufferrsquos finite capacity means that somerequests cannot enter the waiting buffer andwill instead leave

the system immediately after arriving Each execution serverreceives all requests from the main scheduler and maintainsits own task queue

After a task has been completed it leaves the system andthe result is sent back to the customer

32 Other Queuing Models for Reference

321 Multiple-Server Queue with the Same Service Rate Aqueuing model containing multiple servers with the sameservice rate is shown in Figure 2(a) Using this queuingmodel the authors treated a cloud computing data center asan 119872119866119898119898 + 119903 queuing system in [11] as an 119872119872119903119896

queuing system in [19] and as an 119872119872119898 queuing systemin [12 13 18] Because each server has the same service ratethis queuing model treats each server in the data center thesame and assigns it the same dispatching probability 119901 = 1119899

This queuingmodel simplifies analysis andmakes it easierto deduce an equation for the important performancemetrics(eg the mean response time mean number of tasks inthe system and mean waiting time) However the processpresented in this model does not apply to the service processin a cloud data center It ignores the scheduling activity of themain server and the different speeds of execution servers ofdifferent generations

322 Multiple-Server Queue with Different Service RatesA queuing model including multiple servers with differentservice rates is shown in Figure 2(b)We can use this queuingsystem to model a heterogeneous data center that includesmulticore serverswith different speeds Each execution serverin the data center has a different service rate 120583119894 and dis-patching probability 119901119894 We assume that the conditions are119873 = 2 and task arrival rate 120582The state-transition-probabilitydiagram for 119873 = 2 is shown in Figure 3

When a task arrives and there are two servers that areidle the scheduler will select one of them to execute the taskThe selection probability of the number 1 server is 1199011 andthat of the number 2 server is 1199012 (1199012 = 1 minus 1199011) The stateof 1198780 indicates that there is no task in the queuing system11987810 indicates that the task has been distributed to the number1 server and the number 2 server is idle 11987801 indicates thatthe task has been distributed to the number 2 server andthe number 1 server is idle 1198782 indicates that there are twotasks in the system and each has been distributed to one ofthe two servers 1198783 1198784 119878119899 indicate that there are 3 4 119899

tasks in the system and two of them are in the two executionservers Let 1205831 and 1205832 denote the service rates of the number1 and number 2 servers respectivelyThe systemrsquos service rateis 120583 = sum

2

119894=1120583119894 Figure 3 shows how this model calculates

probability when 120588 = 120582120583 and 120572 = 12058321205831 [22]

12058701 =120588

1 + 2120588sdot1 + 120572

120572(120588 + 1199012) 1205870 (1198641)

12058710 =120588

1 + 2120588sdot (1 + 120572) (120588 + 1199011) 1205870 (1198642)

1205870 =1 minus 120588

1 + 120588 [1 + (1 + 1205722) 120588 minus (1 minus 1205722) 1199011] 120572 (1 + 2120588) (1198643)


Waiting queue

S1

S2

Sn

120583

120583

120583

120582

(a)

Waiting queue

λ

S1

S2

Sn

p1

p2

pn

1205831

1205832

120583n

(b)

Figure 2 Other queuing models for reference

λ

λ λ λ λ λλ

S0

S10

S2

S01

S3 Sn Sn+1

1205831

1205831

1205832 1205832

120583 120583 120583 120583 120583p1120582

p2120582

middot middot middot middot middot middot

Figure 3 State-transition-probability diagram for 119873 = 2

From (1198641)ndash(1198643) we can see that if 119873 ge 3 it can use 120587119899 =

120588120587119899minus1 to calculate the probability of each state in Figure 3Thestate-transition-probability diagram for 119873 ge 3 will be morecomplex

Although it considers the heterogeneity of data centersthis model still does not accurately describe the serviceprocess in the cloud center also if 119873 ge 3 the performanceanalysis will be more difficult

33 Queuing Model for Performance Analysis A queuingmodel of heterogeneous data centers with a large numberof execution servers with multiple cores of different gener-ations is shown in Figure 4 The two concatenated queuingsystemsmdashthe scheduler queuing system and the executionqueuing systemmdashmake up the complex queuing model thatcharacterizes the heterogeneity of the data center and of theservice processes involved in cloud computing This modeluses amonitor to obtain performancemetricsmdashincluding thetask number in the queuing 119871119902 the mean waiting time of atask 119882119902 and the utilization of the execution server 120588 Themodel then sends this information to the balance controllerand optimizes performance by proposing an optimal strategybased on these metrics This is a project we will considerdoing in future

The complex queuing model is presented as followsThe master server works as the main scheduler and

maintains scheduler queuing for all requests from all usersSince the process of allocating resources for tasks in the cloudcomputing environment should consider all resources in thedata center the master server is treated as an 1198721198721119896

queuing system with finite capacity 119896Each node in a data center works as an execution server

which is a multicore server that has 119898119894 identical coreswith the core execution speed 119891119894 (GIPS measured by gigainstructions per second) Since each multicore executionserver can parallel-process multiple tasks it is treated as an119872119872119888 queuing system

The stream of tasks is a Poisson process with arrival rate120582 (the task interarrival times are independent and identicallydistributed exponential random variables with a rate of 1120582)The related notations of the system parameters are listed inthe Notations

Since arrivals are independent of the queuing state andthe scheduler queuing system is a finite capacity queuingmodel the blocking probability of the scheduling system 119901119887will be greater than or equal to 0 (119901119887 ge 0) and 120582 ge 120582119890 to yieldthe following equations

120582119890 = 120582 (1minus 119901119887) (1)

120582119894 = 120582119890119901119894 1 le 119894 le 119899 sum 119901119894 = 1 (2)

120583119894 =1

(119868119891119894)=

119891119894

119868 (3)

120588119894 =120582119894

(119862119894120583119894)=

(120582119894119868)

(119862119894119891119894) 120588119894 isin [1205751 120575ℎ] (4)

4 Performance Metrics of the Queuing Model

In this section we will use the complex queuing modelwith the two concatenated queuing systems proposed inSection 33 to analyze the performance problems of a het-erogeneous data center in cloud computing We consider theperformance metrics of the scheduler queuing system theexecution queuing systems and the entire queuing system

41 The Scheduler Queuing System Since management ofcomputing resources and scheduling tasks across entiredata center and cloud environments are done by a uni-fied resources management platform (eg Mesos [23] andYARN [24]) and the system should be accompanied byfinite capacity the master server is treated as an 1198721198721119896

queuing system with finite capacity 119896 The state-transition-probability diagram for the scheduler queuing model isshown in Figure 5

If the scheduler utilization 120588119904 = 120582120583119904 and 120588119904 lt 1 then theprobability that 119894 tasks are in the queuing system 120587

(119904)

119894is [22]

120587(119904)

0 =1 minus 120588119904

1 minus 120588119896+1119904

120587(119904)

119894=

1 minus 120588119904

1 minus 120588119896+1119904

120588119894

119904 119894 = 1 2 119896

(5)


Mainscheduler

λe

Monitor

Balancecontroller

Departure

System capacitySystem response time

hellip

Scheduler queuing system Execution queuing systems

Masterserver

Executionserver

λ

S1

S2

Sn

c0

c0c1

c0

c1

c1

cm1

cm2

cmn

(1205831)

(1205832)120583s

(120583n)

f1

fn

f2

p1

p2

pn

Lq Wq 120588

1205821

1205822

120582n

k





Figure 4 Queuing model of heterogeneous data centers

1 20

λ λλλλ

k minus 1 k

120583s 120583s 120583s 120583s 120583s


Figure 5 State-transition-probability diagram for the schedulerqueuing model

If 119894 ge 119896 the task request could not enter the queuingsystem The blocking probability 119901119887 is

119875119887 = 120587(119904)

119896=

1 minus 120588119904

1 minus 120588119896+1119904

120588119896

119904 (6)

From (1) and (6) the master server throughput rate (theeffective arrival rate of the execution queuing systems) 120582119890 is

120582119890 = 120582 (1minus 119901119887) = 1205821 minus 120588119896119904

1 minus 120588119896+1119904

(7)

The tasks waiting in the queue 119862(119904)

119908and the tasks sched-

uled by the main scheduler 119862(119904)

sch are respectively

119862(119904)

119908=

119896minus1sum119894=0

119896120587(119904)

119894+1 = 1205882119904120587(119904)

0

119896minus1sum119894=1

119896120588119896minus1119904

119862(119904)

sch = (1minus 120587(119904)

0 ) =120588119904 minus 120588119896+1

119904

1 minus 120588119896+1119904

(8)

From (8) the mean task number (including the taskswaiting in the queue and the tasks scheduled by the mainscheduler) in the scheduler queuing system 119862

(119904)

total is

119862(119904)

total = 119862(119904)

119908+ 119862(119904)

sch =120588119904

1 minus 120588119904minus

(119896 + 1) 120588119896+1119904

1 minus 120588119896+1119904

(9)

Using (7) and (9) 120588119904 = 120582120583119904 applying Littlersquos formulas(the response time 119905 = the tasks number in the system

the tasks arrival rate) the mean task response time of thescheduler queuing system 119879

(119904)

is

119879(119904)

=119862(119904)

total120582119890

=1 minus (119896 + 1) 120588119896

119904+ 119896120588119896+1119904

120583119904 (1 minus 120588119904) (1 minus 120588119896119904)

=1

120583119904 minus 120582minus

119896120582119896

120583119896+1119904

minus 120582119896

(10)

42 The Execution Queuing Systems Assume that there are 119899

execution servers in the data center Each execution serverhas 119862119894 cores and can be treated as an 119872119872119862119894 queuingsystem

The utilization of one core in the 119894th execution server 1205881198940is

1205881198940 =120582119894

120583119894=

120582119894119868

119891119894 (11)

From the state-transition-probability diagram for an119872119872119862119894 queuing system [22] let120587(119890)

119894119898indicate the probability

that there are 119898 task requests (including waiting in the queueand being executed) in the execution queuing system of theexecution server 119878119894

120587(119890)

119894119898=

120587(119890)

1198940 sdot1205881198981198940

119898= 120587(119890)

1198940 sdot119862119898119894

119898120588119898119894

0 le 119898 lt 119862119894

120587(119890)

1198940 sdot1205881198981198940

119862119894119862119898minus119862119894

119894

= 120587(119890)

1198940 sdot119862119862119894

119894

119862119894120588119898

119894 119898 ge 119862119894

(12)

Under the stability condition there are constraints asfollows

(1) suminfin

119898=0 120587(119890)

119894119898= sum119862119894minus1119898=0 (1119898)120588119898

1198940 + (119862119894119862119894)120588119862119894minus11198940 (120588119894(1 minus

120588119894))1205871198940 = 1(2) 120588119894 lt 1


We can derive the probability that there are 0 task requestsin the execution queuing system 120587

(119890)

1198940

120587(119890)

1198940 = (

119862119894minus1

sum119898=0

120588119898

1198940119898

+120588119862119894

1198940119862119894

sdot1

1 minus 120588119894)

minus1

(13)

The probability that a newly arrived task request from themain scheduler to the 119894th execution server will be executedimmediately 119902119894 is

119902119894 = 1minus

infin

sum119898=119862119894

120587(119890)

119894119898= 1minus

infin

sum119898=119862119894

119862119862119894

119894

119862119894120588119898

119894120587(119890)

1198940

=119862119894 minus 1205881198940 minus 119862119894120587

(119890)

119894119862119894

119862119894 minus 1205881198940

(14)

The mean probability of a newly arrived task requestentering the execution queuing systems 119902

(119890) is

119902(119890)

=

119899

sum119894=1

119901119894119902119894 (15)

In the 119894th execution server the tasks waiting in the queue119862(119890)

119908 119894and the tasks executed by the 119894th execution server 119862

(119890)

exc 119894are respectively

119862(119890)

119908 119894=

infin

sum119898=119862119894

(119898 minus 119862119894) 120587(119890)

119894119898

119862(119890)

exc 119894 =

119862119894

sum119898=0

119898120587(119890)

119894119898+ 119862119894

infin

sum119898=119862119894+1

120587(119890)

119894119898

(16)

From (16) the mean task number (including the taskswaiting in the queue and the tasks executed by the executionserver) in the 119894th execution server 119862

(119890)

total 119894 is

119862(119890)

total 119894 = 119862(119890)

119908 119894+ 119862(119890)

exc 119894

= 120587(119890)

1198940 sdot120588119862119894+11198940

(119862119894 minus 1) (119862119894 minus 1205881198940)2 + 1205881198940

(17)

Applying Littlersquos formulas themean task response time ofthe 119894th execution queuing systems 119879

(119890)

119894and the mean waiting

time for execution of a task in the 119894th execution queuingsystems 119882

(119890)

119894are respectively

119879(119890)

119894=

119862(119890)

total 119894120582119894

=119868

119891119894(1+ 120587

(119890)

1198940 sdot120588119862119894

1198940

(119862119894 minus 1) (119862119894 minus 1205881198940)2 )

119882(119890)

119894=

119862(119890)

119908 119894

120582119894=

1120582119894

infin

sum119898=119862119894

(119898 minus 119862119894) 120587(119890)

119894119898

= 120587(119890)

1198940 sdot119868 sdot 120588119862119894

1198940

119891119894 (119862119894 minus 1) (119862119894 minus 1205881198940)2

(18)

Themean response time of the execution queuing systems119879(119890)

for a group of 119899 execution servers in the data center is

119879(119890)

=

119899

sum119894=1

119875119894119879(119890)

119894 (19)

The mean waiting time of the execution queuing systems119882(119890) in a group of 119899 execution servers in the data center is

119882(119890)

=

119899

sum119894=1

119875119894119882(119890)

119894 (20)

43The PerformanceMetrics of the Entire Queuing System Inorder to analyze the performance of heterogeneous data cen-ters in cloud computing we consider the main performancemetrics of a complex queuing system which consists of twoconcatenated queuing systems as follows

(i) Response Time Based on analyzing the two concatenatedqueuing systems the equilibrium response time in heteroge-neous data centers is the sum of the response time of the twophases The response time equation can be formulated as

119879 = 119879(119904)

+ 119879(119890)

= 119879(119904)

+

119899

sum119894=1

119875119894119879(119890)

119894 (21)

(ii) Waiting Time The mean waiting time for a task request is

119882 = 119882(119904)

+ 119882(119890)

=119862(119904)

119882

120582119890+

119899

sum119894=1

119875119894119882(119890)

119894

=120588119904

120583119904(

11 minus 120588119904

minus119896120588119896minus1119904

1 minus 120588119896119904

) +

119899

sum119894=1

119875119894119882(119890)

119894

(22)

(iii) Loss (Blocking) Probability Since the scheduler queuingsystem is a finite capacity queuingmodel blockingwill occur

1198751 = 119875119887 (23)

From (6) the loss (blocking) probability of the system isrelated to the finite capacity of the scheduler queuing system119896 and the scheduling rate of themain scheduler server 120583119904Theparameters of 119896 or 120583119904 can be adjusted to obtain different loss(blocking) probabilities

(iv) Probability of Immediate Execution Here the probabilityof immediate execution indicates the probability of a taskrequest being executed immediately by an execution serverafter being scheduled by the scheduler server and sent to theexecution server

119902 = 119902(119890)

(24)

(v) Execution Server Utilization Optimization Model Basedon the Arrival Rate 120582119890 Our analytical model can be used to


optimize the performance for a data center This is one of thesignificant applications of the model

Based on the utilization threshold [120575119897 120575ℎ] of the executionservers we can determine the number of execution serversand the utilization of each execution server in the system byusing the calculation model which is based on our analyticalmodel to configure the execution servers in the data center

to deal with tasks with the arrival rate 120582119890 It can minimize themean response time of the system

Assume the use of FCFS as the scheduling strategy and setthe heterogeneous data center as a group of 119899 heterogeneousexecution servers 1198781 1198782 119878119899 withmulticores1198621 1198622 119862119899and execution speeds of 1198911 1198912 119891119899 The execution serverutilization optimization model based on arrival rate 120582119890 canbe formulated as follows

Min120582119890 1198681198911 1198911198991198621119862119899

119899

sum119894=1

119885119894119875119894119879(119890)

119894

ST (1) 120588119894 =

0 119885119894 = 0

isin [120575119897 120575ℎ] 119885119894 = 11 le 119894 le 119899

(2) 120582119894 =120588119894119862119894119891119894

119868 1 le 119894 le 119899

(3)

119899

sum119894=1

120582119894 = 120582119890

(4) 119875119894 =120588119894119862119894119891119894

119868120582119890 1 le 119894 le 119899

(5)

119899

sum119894=1

119875119894 = 1

(6) 119879(119890)

119894=

119868

119891119894

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

1 le 119894 le 119899

(7) 119885119894 isin 0 1 1 le 119894 le 119899

(25)

The utilization of the execution servers will determinemean response time occupied time and resource capac-ity therefore the execution server utilization optimizationmodel based on arrival rate 120582119890 can be used to analyze theproblem of optimal resource cost or the problem of energyefficiency We will accomplish this research in future work

In this paper we just consider the special casemdashthecondition of 120582119890 isin [(120575119897119868) sum

119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898

119894=1119862119894119891119894] In a

small case we can use a numerical method to gain the valuesof 120588119894 for optimal performance Under the condition of 120582119890 isin

[(120575119897119868) sum119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898

119894=1119862119894119891119894] the parameter of119885119894 in (25)

is 119885119894 = 1 (25) can be rewritten as follows

Min120582119890 11986811989111198911198991198621119862119899

119899

sum119894=1

120588119894119862119894

120582119890

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

ST (1) 120575119897 le 120588119894le 120575ℎ 1 le 119894 le 119899

(2)

119899

sum119894=1

120588119894119862119894119891119894

119868= 120582119890

(3)

119899

sum119894=1

120588119894119862119894119891119894

119868120582119890= 1

(26)


Table 1 Hosts and VMs resources setting

HostID (PM) Setting ST1 Setting ST2CoreNum

(VM Num 119862119894)

Computingpower (GIPS 119891

119894) VMsrsquo setting CoreNum

(VM Num 119862119894)


119894) VMsrsquo setting

Host1 2 2 1 Core2G 2 2 1 Core2GHost2 4 2 1 Core2G 2 2 1 Core2GHost3 6 2 1 Core2G 2 2 1 Core2GHost4 8 2 1 Core2G 4 2 1 Core2GHost5 10 2 1 Core2G 4 2 1 Core2GHost6 12 2 1 Core2G 4 2 1 Core2GHost7 14 2 1 Core2G 8 4 1 Core4GHost8 16 2 1 Core2G 8 4 1 Core4GHost9 18 2 1 Core2G 12 5 1 Core5GHost10 20 2 1 Core2G 12 5 1 Core5G

5 Numerical Validation

In order to validate the performance analysis equationspresented above we built a heterogeneous cloud computingdata center farm in the CloudSim platform and a tasksgenerator with using the discrete event tool MATLAB Inthe numerical examples and the simulation experiments wepresent the parameters are illustrative and can be changed tosuit the cloud computing environment

51 Performance Simulation In this section we describe thesimulation experiments that we conducted to validate theperformance analysis equations of the performance metricsIn all cases the mean number of instructions of a task to beexecuted by the execution server was 119868 = 1 (giga instruc-tions) In the CloudSim we considered a heterogeneous datacenter with a group of 119899 = 10 multicore execution serversHost1 Host2 Host10 and a main scheduler serverMs The traffic intensity of the main scheduler server was120588119904 = 080 The finite capacity of the scheduler 119896 = lceil01120582rceilThe different settings of the Host119894 (1 le 119894 le 10) areshown in Table 1 The allocation policy of the VM-scheduleris space-shared which makes one task in one VM The totalcomputing ability (computing power) of these two groups ofexecution servers with different settings was the same

In Table 1 the heterogeneous data centers of ST1 and ST2are configured with 10 PMs and 110VMs and 10 PMs and58VMs respectively The total computing powers of ST1 andST2 are all 220 (GIPS) The hardware environment is 2timesDellPowerEdge T720 (2timesCPUs Xeon 6-core E5-2630 23G 12cores in total)

The interarrival times of task requests were independentand followed an exponential distribution with arrival rate120582 (the number of task requests per second) The cloudletinformation is created by tasks generator which generatestask arrival interval time task scheduling time and tasklength

In the experiments we assigned eachmulticore executionservermdashwhich had a dispatched probability according to

its core speeds or set traffic intensitymdashto each multicoreexecution server to control a task The rule for setting thedispatched probability and the traffic intensity which can bechanged by administrators was as follows

(i) Setting the Dispatched Probability Consider

119875119894 =119862119894119891119894

sum10119894=1 119862119894119891119894

1 le 119894 le 10 (27)

Under this pattern the traffic intensity of each executionserver was the same and may have exceeded the scope ofthe utilization threshold [120575119897 120575ℎ] Using these two groups ofexecution servers (ST1 and ST2) the dispatched probabilityof each execution server was set using (27)

Simulation Experiments 1We have the following

(1) Tasks number cloudletsNo = 100000 tasks length(GIPS) minus log(rand(1 cloudletNo))(119891119894119868) and 1205821575 1605 1635 1665 1695 1725 1755 17851815 1845 1875

(2) The main scheduler computing power 120583119904 = 120582120588119904hosts (PMs) and VMs setting ST1 and St2 (shown inTable 1) and the probability of host (PM) dispatched119875119894 (27)

(3) Process simulation experiment times 30 In everytime we recorded the loss tasks number in the mainscheduler the immediate execution tasks number (ifthe arrival time equates to start time) the responsetime and thewaiting time of each host After 30 timesrsquoexperiments we calculated the average values of theloss (blocking) probability 119875119897(119875119887) the probability ofimmediate execution 119902 the mean response time 119879and themeanwaiting time119882Then we calculated the95 confidence interval (95 CI) for each metric tovalidate these metrics in the analysis model

(4) The analytical results are calculated by using theanalytical model with (21)ndash(24) The settings of each


Table 2 The simulations and analytical results of 119875119897(119875119887)

120582 Host set Simulation 95 CI for 119875119897(119875119887) Analytical results

Mean Lower Upper

1575 ST1 00058 0005683 0005917 0005759ST2 000573 0005491 0005969 0005759

1725 ST1 000452 0004404 0004636 0004586ST2 000462 0004504 0004736 0004586

1875 ST1 000295 0002853 0003047 0002916ST2 000296 0002842 0003078 0002916

Table 3 The simulations and analytical results of 119902

120582 Host set Simulation 95 CI for 119902 Analytical resultsMean Lower Upper

1575 ST1 080593 0803737 0808123 0806097ST2 072404 0723025 0725055 0724773

1725 ST1 06867 0683968 0689432 0688503ST2 060165 0599881 0603419 060225

1875 ST1 0526 0524375 0527625 0525132ST2 044677 0445745 0447795 0447063



1575 ST1 005987 0059035 0060705 0060115ST2 008227 0081676 0082864 0082617

1725 ST1 009591 0094468 0097352 0096286ST2 01249 0123706 0126094 012491

1875 ST1 017872 0174778 0182662 0178457ST2 021112 0207124 0215116 0213743



1575 ST1 056496 0565194 056374 056618ST2 035094 0350361 0351519 0351333

1725 ST1 060029 0598638 0601942 0600923ST2 039317 0391785 0394555 0393184

1875 ST1 06829 0678949 0686851 0682724ST2 047903 0474915 0483145 0481646

parameter are120582 1575 1605 1635 1665 1695 17251755 1785 1815 1845 1875 120588119904 = 080 119896 = lceil01120582rceil119868 = 1 119875119894 (27) and 119862119894 and 119891119894 shown in Table 1

The simulations and the analytical results (120582 1575

1725 1875) are shown in Tables 2ndash5The comparisons between the analytical and the simula-

tions results (120582 1575 1605 1635 1665 1695 1725 17551785 1815 1845 1875) are shown in Figures 6ndash9

By comparing the simulation results obtained fromCloudSim and the analytical results calculated by using themodel we can observe that the analytical results of 119875119897(119875119887)

00025

0003

00035

0004

00045

0005

00055

0006

00065

155 160 165 170 175 180 185 190

Loss

(blo

ckin

g) p

rob

()

95 CI of ST195 CI of ST2

Analy of ST1Analy of ST2

00034

00036

00038

181 1815 182

00044

00046

00048

166 1665 167

Task arrival rate (120582)

Figure 6 Simulation 95 CI for Pl versus analytical result

044

048

052

056

06

064

068

072

076

08

Prob

of i

mm

edia

te ex

ecut

ion

()

0735

074

0745

166 1665 167

0679

068

0681

163 1635 164

155 160 165 170 175 180 185 190




Figure 7 Simulation 95 CI for 119902 versus analytical result

119902 119882 and 119879 are all in the range of 95 CI which areshown in Tables 2ndash5 and Figures 6ndash9 It demonstrates thatthe analytical results are in agreement with the CloudSimsimulation results It also confirms that the metrics of 119875119897(119875119887)119902 119882 and 119879 in the analytical model can be trusted under theconfidence level of 95

Simulation Experiments 2 Use the same dataset (task num-ber 100000 ST1 and ST2) as the simulation experiments 1to complete 30 timesrsquo experiments under the arrival rate 120582 =

1875 From this experiment we recorded the response andwaiting times of each host in ST1 and ST2 (119879

(119890)

119894and 119882

(119890)

119894 1 le

119894 le 10) Then we can get the 95 confidence interval (95CI) of 119879

(119890)

119894and 119882

(119890)

119894to validate these metrics in the model

The analytical results are gotten by using (18) and (19)In Tables 6 and 7 we demonstrate the simulation results

and the 95 CI of 119879(119890)

119894and 119882

(119890)

119894(1 le 119894 le 10) of every host

in ST1 and ST2 with a task request arrival rate 120582 = 1875


Table 6 The simulations and analytical results of each host in ST1

HostID Simulation 95 CI for 119879(119890)

119894 Analytical results of 119879(119890)

119894

Simulation 95 CI for 119882(119890)


119894Mean Lower Upper Mean Lower UpperHost1 1808109 1730226 1885993 179946 1308227 1231192 1385263 129946Host2 1090659 1064057 1117261 1073279 0590882 0565038 0616725 0573279Host3 0840732 082654 0854924 0845942 0341209 0327489 0354929 0345942Host4 0737109 0730784 0743434 0738017 0237314 0231432 0243195 0238017Host5 0674368 066885 0679887 0676187 0174809 0169671 0179947 0176187Host6 0634818 0630513 0639123 0636685 0135545 013165 0139441 0136685Host7 0607036 0603298 0610775 0609575 0107477 0104036 0110919 0109575Host8 0589295 058681 059178 059 0089395 0087273 0091518 009Host9 0576559 0572962 0580156 057532 0076586 0073578 0079594 007532Host10 0565364 0563253 0567474 0563981 0065318 0063482 0067154 0063981




119894




004

006

008

01

012

014

016

018

02

022

Mea

n w

aitin

g tim

e (s)

015

0155

016

184 1845 185

0094

0096

0098

163 1635 164

155 160 165 170 175 180 185 190





The comparisons between the analytical and the simula-tions results of the hosts are shown in Figures 10-11

As shown in Tables 6 and 7 and Figures 10 and 11 we canobserve that the analytical results of 119879

(119890)

119894and 119882

(119890)

119894for all

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)

0656

066

184 1845 185

0356

0358

036

160 1605 161

0392

0394

172 1725 173

155 160 165 170 175 180 185 190





the hosts in ST1 and ST2 are all in the range of 95 CIIt demonstrates that the analytical results are in agreementwith theCloudSim simulation results and the hostsrsquo analyticalmodel can be trusted under the confidence level of 95


02

04

06

08

1

12

14

16

18

2

0 1 2 3 4 5 6 7 8 9 10 11

Mea

n re

spon

se ti

me (

s)

HostID

104

108

3 4 5 6 70576

05805

8 9 10



Figure 10 Simulation 95 CI for 119879(119890)

119894versus analytical result

After comparing the results we also conclude that

(1) the relative errors of the simulation and the analyticsare less than 35

(2) under the same arrival rate 120582119890 the performancemetrics will be different with different settings despitehaving the same computing abilities

(ii) Setting the Traffic IntensityThe traffic intensityutilization120588119894 (1 le 119894 le 10) was set in the scope of [120575119897 120575ℎ] (setting120575119897 = 065 and 120575ℎ = 085) Under this pattern the scope ofthe task request arrival rate of the 119894th execution server will be120582119894 isin [120575119897119862119894119891119894119868 120575ℎ119862119894119891119894119868] (1 le 119894 le 10) according to (4) Agroup of 119898 (119898 le 119899) multicore execution servers can handlethe effective arrival rate 120582

1015840

119890

1205821015840

119890=

119898

sum119894=1

120582119894 997904rArr120575119897

119868

119898

sum119894=1

119862119894119891119894 le 1205821015840

119890le

120575ℎ

119868

119898

sum119894=1

119862119894119891119894 (28)

Simulation Experiments 3 Let us consider using the samehosts set to deal with the same arrival rate 120582 = 172 ((28)172 isin [220 times 065 220 times 085]) at designated rates of trafficintensityutilization 120588119894 (1 le 119894 le 10) setting The different120588119894 (1 le 119894 le 10) sets are shown in Table 8 as TIs1 and TIs2respectively

We completed 30 times with these settings to get themean valuesThemean response time results of the executionqueuing systems 119879

(119890)

119894(1 le 119894 le 10) are shown in Table 8

FromTable 8 we can see that the maximum relative errorbetween the simulation and the analysis is 051 By settingthe traffic intensity of TIs1 and TIs2 for each host in thegroup of ST1 we find different mean response times for theexecution queuing systems The 119879

(119890)

in TIs1 has fallen from0571203 (second) to 05607632 (second) in TIs2mdashan increaseof 18 It shows that the same hardware resource will have

0

Mea

n w

aitin

g tim

e (s)

056

0595

3 4 5 6 7

00530054005500560057

8 9 10 1102

04

06

08

1

12

14

16

0 1 2 3 4 5 6 7 8 9 10 11HostID





different performance with different traffic intensity settingfor each host

We can compare the results of our analytical calculationwith the results of simulation data shown in Tables 2ndash8 andFigures 6ndash11 and all of the analytical resultsrsquo relative errorsare less than 35 The results also showed that all of theanalytical results are in the range of 95 CI for the metricsThis shows that our analysis agrees with our simulationresults which confirms the validity of our analytical model

52 Numerical Examples We demonstrate some numericalexamples to analyze the relationship among performancemetrics with using the analytical model

(i) Numerical Example 1 We use the two groups of executionservers (hosts in ST1 and ST2) shown in Table 1 with the sametotal computing ability to handle an arrival rate 120582 = 155 Theexecution servers in ST1 and ST2 are sorted by computingability (computing power) In the experiments we adjustthe traffic intensity of each execution server so that we cananalyze the mean response time119879 and the mean waiting time119882 of the systemWe select 16 sets of traffic intensity and eachset has 10 120588119894 (1 le 119894 le 10) to fit 10 execution servers

The results of the experiments are shown in Figures 12 and13 In the 16 experiments we set similar traffic intensity forthe 119894th execution server in ST1 and ST2 Although ST1 andST2 have the same total computing ability themean responsetime 119879 and the mean waiting time 119882 of the system differsignificantly from each other

In Figures 12 and 13 we can see that adjusting thetraffic intensity (utilization) of each execution server canenhance the response and waiting times In ST1 the meanresponse time 119879 is decreased from 0707159 (second) to0554670 (second) and response time has been enhanced by2156Themean waiting time119882 is decreased from 0201998(second) to 0049509 (second) and response time has been


Table 8 The results of each host with different traffic intensity in ST1

HostID Traffic intensity set 1 (TIs1) Traffic intensity set 2 (TIs2)120588119894 Simulation Analytical results Relative error 120588119894 Simulation Analytical results Relative error

Host1 0751 1143158 1146792 032 0651 087184 0867756 047Host2 0755 076765 0764214 045 0665 0642525 0640293 035Host3 0756 0648089 0647759 005 0686 0580317 0583304 051Host4 0759 0595157 0597003 031 0735 0578816 0577729 019Host5 0761 0571153 0572848 030 0749 0560845 0560706 002Host6 0764 0554577 0555183 011 0782 056216 0562954 014Host7 0766 0541873 0543499 030 0786 0551055 0550656 007Host8 0771 0534157 0535349 022 0807 0551097 0551994 016Host9 0815 054695 0547189 004 0811 0542748 0544765 037Host10 0798 052541 0525306 002 0813 0538181 0538257 001

03

035

04

045

05

055

06

065

07

075

0 2 4 6 8 10 12 14 16

Mea

n re

spon

se ti

me (

s)

ith setting of the traffic intensity

120582 = 155 k = 17 120588s = 08

ST1_TST2_T

Figure 12 The mean response times of ST1 and ST2

enhanced by 7549 In ST2 the mean response time 119879

is decreased from 0523104 (second) to 0333339 (second)and response time has been enhanced by 3627 The meanwaiting time 119882 is decreased from 0240814 (second) to0067228 (second) and response time has been enhancedby 7208 We can see that the traffic intensity (utilization)of each execution server has a significant impact on theperformance of the heterogeneous data center

The configuration of a server cluster in a heterogeneousdata center is an important factor that greatly impacts thesystemrsquos performance Again Figures 12 and 13 show theresults of our 16 experiments and we can see that the meanresponse time 119879 of ST2 is better than ST1 under a similartraffic intensity (utilization) setting Mean response timeimproves by 3518 on average and the maximum is 399However the mean waiting time 119882 of ST2 is worse than ST1Mean waiting time is decreased by 2495 on average andthe maximum is 29 It is important for a cloud providerto configure the server cluster into a reasonable structure toprovide services for the customer This conclusion will beconfirmed in numerical example 2

004006008

01012014016018

02022024026

0 2 4 6 8 10 12 14 16

Mea

n w

aitin

g tim

e (s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T

Figure 13 The mean waiting times of ST1 and ST2

(ii) Numerical Example 2 Let us consider the third group ofexecution servers named ST3 that includes 10 servers eachof which is configured as119862119894 = 4 and119891119894 = 55 (1 le 119894 le 10)Theother conditions are the same as the conditions in simulationexperiments 1 Assume that the arrival rate 120582 is set from 150 to1875 which means that the traffic intensity of each executionserver would be in the threshold [120575119897 120575ℎ]

The performance results (119879119882 and 119902) of the three groups(ST1 ST2 and ST3) of execution servers are shown in Figures14ndash16 Different configurations of server clusters will havedifferent performance results The best performance of themean response time is group ST3 and the worst is ST1 asshown in Figure 14 The best performance of mean waitingtime is group ST1 and the worst is ST3 as shown in Figure 15In Figure 16 ST1 has the maximum immediate executionprobability while the ST3 has theminimumvalueWe can usethis queuing model to estimate the performance results of aserver cluster with a certain configuration under different 120582In order to enhance the performance of mean response timeconfiguring the server cluster in a reasonable structure is anefficient method


025

03

150

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)

Task arrival rate (120582)155 160 165 170 175 180 185 190

ST1_TST2_TST3_T

Figure 14 The mean response times of ST1 ST2 and ST3

In practice users will present some constraints when theyask cloud computing providers for servicesThey will presentconstraints as follows

(1) The task request arrival rate 120582119894119899 isin [120582119897 120582ℎ](2) The mean response time 120591119903 le 119879119897(3) The mean wasting time 120591119908 le 119882119897 or the immediate

execution probability 120577 ge 119902119897

Cloud computing providers could configure the servercluster to serve customers under certain constraints by usingour queuing model In our future work we will use thecomplex queuing model to further research dynamic clustertechnology in a heterogeneous data center to learn howit handles different tasks with different arrival rates undervarious constraints

(iii) Numerical Example 3 One application of our analyticalmodel is to use this model for optimal system performanceby finding a reasonable traffic intensity setting for eachexecution server Numerical example 3 shows the optimiza-tion under the condition of (120575119897119868) sum

119898

119894=1119862119894119891119894 le 120582119890 le

(120575ℎ119868) sum119898

119894=1119862119894119891119894 to get the values of 120588119894

Assume using three execution servers 1198781 (1198621 = 41198911 = 4)1198782 (1198622 = 4 1198912 = 3) and 1198783 (1198623 = 6 1198913 = 4) to handlethe arrival rate 120582119890 = 383 ([(16 + 12 + 24) times 065 (16 +

12 + 24) times 085]) Equations (25) and (26)mdashthe executionserver utilization optimization model based on the arrivalrate 120582119890mdashcan use a numerical method to get the minimummean response time of this system under the condition of120582119890 = 383 As Figure 17 shows we can see that the minimummean response time 119879min = 036317 when 1205881 = 0727121205882 = 067627 and 1205883 = 077295 Controlling the trafficintensity (or utilization) of each execution server allows forthe control of the dispatched probability of each executionserver and the optimization of system performance

004

006

008

01

012

014

016

018

02

022

024

Mea

n w

aitin

g tim

e (s)

150Task arrival rate (120582)

155 160 165 170 175 180 185 190

ST1_TST2_TST3_T

Figure 15 The mean waiting times of ST1 ST2 and ST3

03

04

05

06

07

08

09Pr

ob o

f im

med

iate

exec

utio

n (

)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T

Figure 16 Immediate execution probability of ST1 ST2 and ST3

The traffic intensity (or utilization) of each executionserver will impact the response time computing resourceallocation and energy usage of the system therefore we willfurther optimize the execution server utilization optimiza-tion model in future work

6 Conclusions and Future Work

Performance analysis of heterogeneous data centers is acrucial aspect of cloud computing for both cloud providersand cloud service customers Based on an analysis of thecharacteristics of heterogeneous data centers and their serviceprocesses we propose a complex queuing model composedof two concatenated queuing systemsmdashthe main schedulequeue and the execution queuemdashto evaluate the performance


06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)

Figure 17 The mean response time with different traffic intensitysettings

of heterogeneous data centers We theoretically analyzedmean response time mean waiting time and other impor-tant performance indicators We also conducted simulationexperiments to validate the complex queuing model Thesimulation results and the calculated values demonstrate thatthe complex queuing model provides results with a highdegree of accuracy for each performance metric and allowsfor a sophisticated analysis of heterogeneous data centers

We have further conducted some numerical examples toanalyze factors such as the traffic intensity (or utilization)of each execution server and the configuration of serverclusters in a heterogeneous data center these are factors thatsignificantly impact the performance of the system Based onthis complex queuingmodel we plan to extend our analyticalmodel to the study of dynamic cluster technology in aheterogeneous data center In doing so we aim to help serviceproviders optimize resource allocation using the executionserver utilization optimization model

Notations

119896 The finite capacity of the scheduler queuing system120583119904 The scheduling rate of the master server120582119890 The master server throughput ratethe effective

arrival rate of the execution queuing systems119901119894 The probability that execution server 119878119894 has been

dispatched to execute a taskthe probability of a taskrequest sent to the execution server 119878119894

120582119894 The arrival rate of the task request distributed to theexecution server 119878119894

119878119894 The 119894th nodethe 119894th execution server119862119894 The number of cores in the 119894th execution server 119878119894

120583119894 The execution rate of the core in the 119894th executionserver 119878119894

119891119894 The core execution speed of the 119894th execution server119878119894 (measured by giga instructions per second (GIPS))

119868 The mean number of instructions in a task tobe executed in the execution server (giga)

120588119894 The utilizationtraffic intensity of the 119894thexecution server 119878119894

119899 The number of execution servers in the datacenter

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by the Funds of Core Technologyand Emerging Industry Strategic Project of GuangdongProvince (Project nos 2011A010801008 2012A010701011and 2012A010701003) the Guangdong Provincial TreasuryProject (Project no 503-503054010110) and the Technologyand Emerging Industry Strategic Project of Guangzhou(Project no 201200000034)

References

[1] B Furht ldquoCloud computing fundamentalsrdquo in Handbook ofCloud Computing pp 3ndash19 Springer New York NY USA 2010

[2] W Voorsluys J Broberg and R Buyya ldquoIntroduction to cloudcomputingrdquo in Cloud Computing pp 1ndash41 2011

[3] M Armbrust A Fox R Griffith et al ldquoA view of cloudcomputingrdquo Communications of the ACM vol 53 no 4 pp 50ndash58 2010

[4] C Delimitrou and C Kozyrakis ldquoParagon QoS-aware schedul-ing for heterogeneous datacentersrdquo ACM SIGARCH ComputerArchitecture News vol 41 no 1 pp 77ndash88 2013

[5] J Mars L Tang and R Hundt ldquoHeterogeneity in lsquohomoge-neousrsquo warehouse-scale computers a performance opportu-nityrdquo IEEE Computer Architecture Letters vol 10 no 2 pp 29ndash32 2011

[6] C Vecchiola S Pandey and R Buyya ldquoHigh-performancecloud computing a view of scientific applicationsrdquo in Proceed-ings of the 10th International Symposium on Pervasive SystemsAlgorithms and Networks (I-SPAN rsquo09) pp 4ndash16 IEEE Kaohsi-ung Taiwan December 2009

[7] I Foster Y Zhao I Raicu and S Lu ldquoCloud computing and gridcomputing 360-degree comparedrdquo in Proceedings of the GridComputing Environments Workshop (GCE rsquo08) pp 1ndash10 IEEENovember 2008

[8] D Gross J F Shortle J M Thompson and C M HarrisFundamentals of Queueing Theory John Wiley amp Sons 2013

[9] R Buyya C S Yeo and S Venugopal ldquoMarket-orientedcloud computing vision hype and reality for delivering ITservices as computing utilitiesrdquo in Proceedings of the 10th IEEEInternational Conference on High Performance Computing andCommunications (HPCC rsquo08) pp 5ndash13 IEEE September 2008

[10] A Wolke and G Meixner ldquoTwospot a cloud platform forscaling out web applications dynamicallyrdquo in Towards a Service-Based Internet vol 6481 of Lecture Notes in Computer Sciencepp 13ndash24 Springer Berlin Germany 2010

[11] H Khazaei J Misic and V B Misic ldquoPerformance analysis ofcloud computing centers using MGmm+r queuing systemsrdquo


IEEE Transactions on Parallel and Distributed Systems vol 23no 5 pp 936ndash943 2012

[12] J Vilaplana F Solsona I Teixido JMateo F Abella and J RiusldquoA queuing theory model for cloud computingrdquo The Journal ofSupercomputing vol 69 no 1 pp 492ndash507 2014

[13] L Guo T Yan S Zhao and C Jiang ldquoDynamic performanceoptimization for cloud computing using mmm queueingsystemrdquo Journal of Applied Mathematics vol 2014 Article ID756592 8 pages 2014

[14] X Nan Y He and L Guan ldquoOptimal resource allocation formultimedia cloud based on queuing modelrdquo in Proceedingsof the 3rd IEEE International Workshop on Multimedia SignalProcessing (MMSP rsquo11) pp 1ndash6 November 2011

[15] X Nan Y He and L Guan ldquoQueueing model based resourceoptimization for multimedia cloudrdquo Journal of Visual Commu-nication and Image Representation vol 25 no 5 pp 928ndash9422014

[16] B Yang F Tan and Y-S Dai ldquoPerformance evaluation of cloudservice considering fault recoveryrdquoThe Journal of Supercomput-ing vol 65 no 1 pp 426ndash444 2013

[17] X Zheng and Y Cai ldquoAchieving energy proportionality inserver clustersrdquo International Journal of Computer Networksvol 1 no 1 pp 21ndash35 2009

[18] J Cao KHwang K Li andA Y Zomaya ldquoOptimalmultiserverconfiguration for profit maximization in cloud computingrdquoIEEE Transactions on Parallel and Distributed Systems vol 24no 6 pp 1087ndash1096 2013

[19] Y-J Chiang and Y-C Ouyang ldquoProfit optimization in SLA-aware cloud services with a finite capacity queuing modelrdquoMathematical Problems in Engineering vol 2014 Article ID534510 11 pages 2014

[20] J Cao K Li and I Stojmenovic ldquoOptimal power allocation andload distribution for multiple heterogeneous multicore serverprocessors across clouds and data centersrdquo IEEE Transactionson Computers vol 63 no 1 pp 45ndash58 2014

[21] Q Zhang L Cheng and R Boutaba ldquoCloud computing state-of-the-art and research challengesrdquo Journal of Internet Servicesand Applications vol 1 no 1 pp 7ndash18 2010

[22] C L Zhao Queuing Theory Buptprss Beijing China 2ndedition 2004

[23] B Hindman A Konwinski M Zaharia et al ldquoMesos a plat-form for fine-grained resource sharing in the data centerrdquo inProceedings of the 8thUSENIXConference onNetworked SystemsDesign and Implementation (NSDI rsquo11) vol 11 pp 295ndash308 2011

[24] V K Vavilapalli A C Murthy C Douglas et al ldquoApachehadoop YARN yet another resource negotiatorrdquo in Proceedingsof the 4th Annual Symposium on Cloud Computing (SoCC rsquo13)ACM October 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


analyze the relationship among three factors the arrival rateof tasks the core utilization and the probability of a node(execution server) being dispatched Cloud providers canutilize this information to analyze their systemsrsquomanagementand optimize their allocation of computing resources [9] Aswe show in Section 2 much existing research has discussedthe problem of how to analyze data center performanceHowever these studies have not considered the followingimportant issues

(i) A cloud center can include a large number of serversof different generations with different CPU speedsand processor capacities Existing research rarelyconsiders cloud center heterogeneity rather studiesgenerally assume that every node operates at the samespeed

(ii) A cloud system has at least two levels of schedulingprocesses (service processes) the main schedulerwhich allocates computing resources and the secondscheduler (a nodeexecution server) which executestasks The arrival rate of a task to the main schedulerversus to a node is different and this must be takeninto account in the queuing analysis

(iii) A node server with more than one speed and proces-sor capacity will have different probabilities of beingdispatched depending on the situation Each nodehas its own task queue to maintain which is trackedby a separate queuing model that is included in theoverall system queuing model

(iv) A state-of-the-art resource allocation scheme basedon the utilization threshold [120575119897 120575ℎ] controls nodesthat are to be added to a cloud system or removedfrom it [10]

To fill these gaps in this paper we employ a complexqueuing model to investigate the performance of a cloudcenter and the relationship among various performanceindicators in a heterogeneous data center We aim to makethe following contributions with this paper

(1) Wemodel the heterogeneous data center as a complexqueuing system to characterize the service processTwo concatenated queuing systems compose thecomplex queuingmodelThe first is the finite capacityscheduler queuing system which is used to study themain scheduling server The second is the executionqueuing system which has multiple servers for eachmulticore server processor

(2) Based on this complex queuing model and usingqueuing theory we analyze the real output rate ofthe main scheduling servermdashthat is to say the realarrival rate of the execution queuing systems themean response time of the system the blockingprobability and other performance indicatorsmdashanduse this analysis to evaluate the performance of andthe relationships among these indicators in a cloudcenter

(3) We conduct extensive discrete event simulations toevaluate and validate the results of using the complex

queuing model to analyze the performance of hetero-geneous data centers in cloud computing

The remainder of this paper is organized as followsSection 2 reviews and discusses the related scholarship onqueuing models for cloud centers Section 3 discusses ouranalytical model in detail All of the performance metricsobtained by using this complex queuing model are presentedin Section 4 In Section 5 we show the simulation results andcompare them with our analytical results to evaluate and val-idate the applicability of the complex queuing model Finallyconclusions and future work are presented in Section 6

2 Related Work

Although much scholarship has been conducted on the per-formance analysis of a cloud center based on queuing models[11ndash16] factors including the loosely coupled architecturesof cloud computing and the heterogeneity and dynamicinfrastructure of a cloud center have not been considered

In [11] the authors used an119872119866119898119898+119903 queuing systemto evaluate a cloud computing center and obtained a completeprobability distribution of response time number of tasks inthe system and other important performance metrics Theyalso considered the different mean waiting times betweenheterogeneous services and homogeneous services under thesame conditions in the system Based on queuing theoryand open Jacksonrsquos networks a combination of 1198721198721 and119872119872119898 was presented to model the cloud platform forQoS requirements in [12] In [13] the authors presented an119872119872119898 queuing system and proposed a synthesis opti-mization of model function and strategy to optimize theperformance of services in the cloud center In [14 15] theauthors introduced a queuing model consisting of three con-catenated queuing systems (the schedule queue 1198721198721 thecomputation queue 119872119872119898 and the transmission queue1198721198721) to characterize the service process in a multimediacloud and to investigate the resource optimization problemsof multimedia cloud computing In [16] a cloud center wasmodeled as a 119866119868

119883119872119878119873 queuing model to discuss cloudservice performance related to fault recovery

In many existing methodologies [17ndash20] queuing theoryhas been used to analyze or optimize factors such as powerallocation load distribution and profit control

In [17] the authors presented an energy proportionalmodelmdashwhich treats a server as an1198721198661 queuing systemmdashas a strategy to improve performance efficiency In [18]an 119872119872119898 queuing model which considers factors suchas the requirement 120574 of a service and the configurationof a multiserver system is used to optimize multiserverconfiguration for profit maximization in cloud computingIn [19] the authors presented an 119872119872119903119896 queuing systemto model a cloud server farm provided with finite capacityfor profit optimization Another study presents a queuingmodel for a group of heterogeneous multicore servers withdifferent sizes and speeds to optimize power allocation andload distribution in a cloud computing environment [20]In [17] the authors used the 119872119872119888119888 queuing system tomodel a cloud center with different priority classes to analyze



Tasks stream

Users

Users

Users

Users

Scheduledispatch

Main scheduler


Data center


















2



12058701 =120588

1 + 2120588sdot1 + 120572

120572(120588 + 1199012) 1205870 (1198641)

12058710 =120588

1 + 2120588sdot (1 + 120572) (120588 + 1199011) 1205870 (1198642)

1205870 =1 minus 120588

1 + 120588 [1 + (1 + 1205722) 120588 minus (1 minus 1205722) 1199011] 120572 (1 + 2120588) (1198643)


Waiting queue

S1

S2

Sn

120583

120583

120583

120582

(a)

Waiting queue

λ

S1

S2

Sn

p1

p2

pn

1205831

1205832

120583n

(b)


λ

λ λ λ λ λλ

S0

S10

S2

S01

S3 Sn Sn+1

1205831

1205831

1205832 1205832

120583 120583 120583 120583 120583p1120582

p2120582













120582119890 = 120582 (1minus 119901119887) (1)

120582119894 = 120582119890119901119894 1 le 119894 le 119899 sum 119901119894 = 1 (2)

120583119894 =1

(119868119891119894)=

119891119894

119868 (3)

120588119894 =120582119894

(119862119894120583119894)=

(120582119894119868)

(119862119894119891119894) 120588119894 isin [1205751 120575ℎ] (4)






(119904)

119894is [22]

120587(119904)

0 =1 minus 120588119904

1 minus 120588119896+1119904

120587(119904)

119894=

1 minus 120588119904

1 minus 120588119896+1119904

120588119894

119904 119894 = 1 2 119896

(5)


Mainscheduler

λe

Monitor

Balancecontroller

Departure


hellip


Masterserver

Executionserver

λ

S1

S2

Sn

c0

c0c1

c0

c1

c1

cm1

cm2

cmn

(1205831)

(1205832)120583s

(120583n)

f1

fn

f2

p1

p2

pn

Lq Wq 120588

1205821

1205822

120582n

k






1 20

λ λλλλ

k minus 1 k

120583s 120583s 120583s 120583s 120583s




119875119887 = 120587(119904)

119896=

1 minus 120588119904

1 minus 120588119896+1119904

120588119896

119904 (6)


120582119890 = 120582 (1minus 119901119887) = 1205821 minus 120588119896119904

1 minus 120588119896+1119904

(7)





119862(119904)

119908=

119896minus1sum119894=0

119896120587(119904)

119894+1 = 1205882119904120587(119904)

0

119896minus1sum119894=1

119896120588119896minus1119904

119862(119904)

sch = (1minus 120587(119904)

0 ) =120588119904 minus 120588119896+1

119904

1 minus 120588119896+1119904

(8)


(119904)

total is

119862(119904)

total = 119862(119904)

119908+ 119862(119904)

sch =120588119904


(119896 + 1) 120588119896+1119904

1 minus 120588119896+1119904

(9)



(119904)

is

119879(119904)

=119862(119904)

total120582119890

=1 minus (119896 + 1) 120588119896

119904+ 119896120588119896+1119904

120583119904 (1 minus 120588119904) (1 minus 120588119896119904)

=1

120583119904 minus 120582minus

119896120582119896

120583119896+1119904

minus 120582119896

(10)




1205881198940 =120582119894

120583119894=

120582119894119868

119891119894 (11)




120587(119890)

119894119898=

120587(119890)

1198940 sdot1205881198981198940

119898= 120587(119890)

1198940 sdot119862119898119894

119898120588119898119894

0 le 119898 lt 119862119894

120587(119890)

1198940 sdot1205881198981198940

119862119894119862119898minus119862119894

119894

= 120587(119890)

1198940 sdot119862119862119894

119894

119862119894120588119898

119894 119898 ge 119862119894

(12)


(1) suminfin

119898=0 120587(119890)

119894119898= sum119862119894minus1119898=0 (1119898)120588119898

1198940 + (119862119894119862119894)120588119862119894minus11198940 (120588119894(1 minus

120588119894))1205871198940 = 1(2) 120588119894 lt 1



(119890)

1198940

120587(119890)

1198940 = (

119862119894minus1

sum119898=0

120588119898

1198940119898

+120588119862119894

1198940119862119894

sdot1

1 minus 120588119894)

minus1

(13)


119902119894 = 1minus

infin

sum119898=119862119894

120587(119890)

119894119898= 1minus

infin

sum119898=119862119894

119862119862119894

119894

119862119894120588119898

119894120587(119890)

1198940

=119862119894 minus 1205881198940 minus 119862119894120587

(119890)

119894119862119894

119862119894 minus 1205881198940

(14)


(119890) is

119902(119890)

=

119899

sum119894=1

119901119894119902119894 (15)



(119890)


119862(119890)

119908 119894=

infin

sum119898=119862119894

(119898 minus 119862119894) 120587(119890)

119894119898

119862(119890)

exc 119894 =

119862119894

sum119898=0

119898120587(119890)

119894119898+ 119862119894

infin

sum119898=119862119894+1

120587(119890)

119894119898

(16)


(119890)

total 119894 is

119862(119890)

total 119894 = 119862(119890)

119908 119894+ 119862(119890)

exc 119894

= 120587(119890)

1198940 sdot120588119862119894+11198940

(119862119894 minus 1) (119862119894 minus 1205881198940)2 + 1205881198940

(17)


(119890)



(119890)


119879(119890)

119894=

119862(119890)

total 119894120582119894

=119868

119891119894(1+ 120587

(119890)

1198940 sdot120588119862119894

1198940

(119862119894 minus 1) (119862119894 minus 1205881198940)2 )

119882(119890)

119894=

119862(119890)

119908 119894

120582119894=

1120582119894

infin

sum119898=119862119894

(119898 minus 119862119894) 120587(119890)

119894119898

= 120587(119890)

1198940 sdot119868 sdot 120588119862119894

1198940

119891119894 (119862119894 minus 1) (119862119894 minus 1205881198940)2

(18)



119879(119890)

=

119899

sum119894=1

119875119894119879(119890)

119894 (19)


119882(119890)

=

119899

sum119894=1

119875119894119882(119890)

119894 (20)



119879 = 119879(119904)

+ 119879(119890)

= 119879(119904)

+

119899

sum119894=1

119875119894119879(119890)

119894 (21)


119882 = 119882(119904)

+ 119882(119890)

=119862(119904)

119882

120582119890+

119899

sum119894=1

119875119894119882(119890)

119894

=120588119904

120583119904(

11 minus 120588119904

minus119896120588119896minus1119904

1 minus 120588119896119904

) +

119899

sum119894=1

119875119894119882(119890)

119894

(22)


1198751 = 119875119887 (23)



119902 = 119902(119890)

(24)







Min120582119890 1198681198911 1198911198991198621119862119899

119899

sum119894=1

119885119894119875119894119879(119890)

119894

ST (1) 120588119894 =

0 119885119894 = 0

isin [120575119897 120575ℎ] 119885119894 = 11 le 119894 le 119899

(2) 120582119894 =120588119894119862119894119891119894

119868 1 le 119894 le 119899

(3)

119899

sum119894=1

120582119894 = 120582119890

(4) 119875119894 =120588119894119862119894119891119894

119868120582119890 1 le 119894 le 119899

(5)

119899

sum119894=1

119875119894 = 1

(6) 119879(119890)

119894=

119868

119891119894

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

1 le 119894 le 119899

(7) 119885119894 isin 0 1 1 le 119894 le 119899

(25)



119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898

119894=1119862119894119891119894] In a


[(120575119897119868) sum119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898



Min120582119890 11986811989111198911198991198621119862119899

119899

sum119894=1

120588119894119862119894

120582119890

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

ST (1) 120575119897 le 120588119894le 120575ℎ 1 le 119894 le 119899

(2)

119899

sum119894=1

120588119894119862119894119891119894

119868= 120582119890

(3)

119899

sum119894=1

120588119894119862119894119891119894

119868120582119890= 1

(26)




(VM Num 119862119894)



(VM Num 119862119894)












119875119894 =119862119894119891119894

sum10119894=1 119862119894119891119894

1 le 119894 le 10 (27)










Mean Lower Upper

1575 ST1 00058 0005683 0005917 0005759ST2 000573 0005491 0005969 0005759

1725 ST1 000452 0004404 0004636 0004586ST2 000462 0004504 0004736 0004586

1875 ST1 000295 0002853 0003047 0002916ST2 000296 0002842 0003078 0002916



1575 ST1 080593 0803737 0808123 0806097ST2 072404 0723025 0725055 0724773

1725 ST1 06867 0683968 0689432 0688503ST2 060165 0599881 0603419 060225

1875 ST1 0526 0524375 0527625 0525132ST2 044677 0445745 0447795 0447063



1575 ST1 005987 0059035 0060705 0060115ST2 008227 0081676 0082864 0082617

1725 ST1 009591 0094468 0097352 0096286ST2 01249 0123706 0126094 012491

1875 ST1 017872 0174778 0182662 0178457ST2 021112 0207124 0215116 0213743



1575 ST1 056496 0565194 056374 056618ST2 035094 0350361 0351519 0351333

1725 ST1 060029 0598638 0601942 0600923ST2 039317 0391785 0394555 0393184

1875 ST1 06829 0678949 0686851 0682724ST2 047903 0474915 0483145 0481646






00025

0003

00035

0004

00045

0005

00055

0006

00065

155 160 165 170 175 180 185 190

Loss

(blo

ckin

g) p

rob

()



00034

00036

00038

181 1815 182

00044

00046

00048

166 1665 167



044

048

052

056

06

064

068

072

076

08

Prob

of i

mm

edia

te ex

ecut

ion

()

0735

074

0745

166 1665 167

0679

068

0681

163 1635 164

155 160 165 170 175 180 185 190








(119890)

119894and 119882

(119890)

119894 1 le


(119890)

119894and 119882

(119890)



and the 95 CI of 119879(119890)

119894and 119882

(119890)







119894







119894




004

006

008

01

012

014

016

018

02

022

Mea

n w

aitin

g tim

e (s)

015

0155

016

184 1845 185

0094

0096

0098

163 1635 164

155 160 165 170 175 180 185 190







(119890)

119894and 119882

(119890)

119894for all

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)

0656

066

184 1845 185

0356

0358

036

160 1605 161

0392

0394

172 1725 173

155 160 165 170 175 180 185 190







02

04

06

08

1

12

14

16

18

2

0 1 2 3 4 5 6 7 8 9 10 11

Mea

n re

spon

se ti

me (

s)

HostID

104

108

3 4 5 6 70576

05805

8 9 10









1015840

119890

1205821015840

119890=

119898

sum119894=1

120582119894 997904rArr120575119897

119868

119898

sum119894=1

119862119894119891119894 le 1205821015840

119890le

120575ℎ

119868

119898

sum119894=1

119862119894119891119894 (28)



(119890)



(119890)


0

Mea

n w

aitin

g tim

e (s)

056

0595

3 4 5 6 7

00530054005500560057

8 9 10 1102

04

06

08

1

12

14

16

0 1 2 3 4 5 6 7 8 9 10 11HostID















03

035

04

045

05

055

06

065

07

075

0 2 4 6 8 10 12 14 16

Mea

n re

spon

se ti

me (

s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





004006008

01012014016018

02022024026

0 2 4 6 8 10 12 14 16

Mea

n w

aitin

g tim

e (s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





025

03

150

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)


ST1_TST2_TST3_T







119898

119894=1119862119894119891119894 le 120582119890 le

(120575ℎ119868) sum119898




004

006

008

01

012

014

016

018

02

022

024

Mea

n w

aitin

g tim

e (s)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T


03

04

05

06

07

08

09Pr

ob o

f im

med

iate

exec

utio

n (

)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T






06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)




Notations













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Tasks stream

Users

Users

Users

Users

Scheduledispatch

Main scheduler


Data center


















2



12058701 =120588

1 + 2120588sdot1 + 120572

120572(120588 + 1199012) 1205870 (1198641)

12058710 =120588

1 + 2120588sdot (1 + 120572) (120588 + 1199011) 1205870 (1198642)

1205870 =1 minus 120588

1 + 120588 [1 + (1 + 1205722) 120588 minus (1 minus 1205722) 1199011] 120572 (1 + 2120588) (1198643)


Waiting queue

S1

S2

Sn

120583

120583

120583

120582

(a)

Waiting queue

λ

S1

S2

Sn

p1

p2

pn

1205831

1205832

120583n

(b)


λ

λ λ λ λ λλ

S0

S10

S2

S01

S3 Sn Sn+1

1205831

1205831

1205832 1205832

120583 120583 120583 120583 120583p1120582

p2120582













120582119890 = 120582 (1minus 119901119887) (1)

120582119894 = 120582119890119901119894 1 le 119894 le 119899 sum 119901119894 = 1 (2)

120583119894 =1

(119868119891119894)=

119891119894

119868 (3)

120588119894 =120582119894

(119862119894120583119894)=

(120582119894119868)

(119862119894119891119894) 120588119894 isin [1205751 120575ℎ] (4)






(119904)

119894is [22]

120587(119904)

0 =1 minus 120588119904

1 minus 120588119896+1119904

120587(119904)

119894=

1 minus 120588119904

1 minus 120588119896+1119904

120588119894

119904 119894 = 1 2 119896

(5)


Mainscheduler

λe

Monitor

Balancecontroller

Departure


hellip


Masterserver

Executionserver

λ

S1

S2

Sn

c0

c0c1

c0

c1

c1

cm1

cm2

cmn

(1205831)

(1205832)120583s

(120583n)

f1

fn

f2

p1

p2

pn

Lq Wq 120588

1205821

1205822

120582n

k






1 20

λ λλλλ

k minus 1 k

120583s 120583s 120583s 120583s 120583s




119875119887 = 120587(119904)

119896=

1 minus 120588119904

1 minus 120588119896+1119904

120588119896

119904 (6)


120582119890 = 120582 (1minus 119901119887) = 1205821 minus 120588119896119904

1 minus 120588119896+1119904

(7)





119862(119904)

119908=

119896minus1sum119894=0

119896120587(119904)

119894+1 = 1205882119904120587(119904)

0

119896minus1sum119894=1

119896120588119896minus1119904

119862(119904)

sch = (1minus 120587(119904)

0 ) =120588119904 minus 120588119896+1

119904

1 minus 120588119896+1119904

(8)


(119904)

total is

119862(119904)

total = 119862(119904)

119908+ 119862(119904)

sch =120588119904


(119896 + 1) 120588119896+1119904

1 minus 120588119896+1119904

(9)



(119904)

is

119879(119904)

=119862(119904)

total120582119890

=1 minus (119896 + 1) 120588119896

119904+ 119896120588119896+1119904

120583119904 (1 minus 120588119904) (1 minus 120588119896119904)

=1

120583119904 minus 120582minus

119896120582119896

120583119896+1119904

minus 120582119896

(10)




1205881198940 =120582119894

120583119894=

120582119894119868

119891119894 (11)




120587(119890)

119894119898=

120587(119890)

1198940 sdot1205881198981198940

119898= 120587(119890)

1198940 sdot119862119898119894

119898120588119898119894

0 le 119898 lt 119862119894

120587(119890)

1198940 sdot1205881198981198940

119862119894119862119898minus119862119894

119894

= 120587(119890)

1198940 sdot119862119862119894

119894

119862119894120588119898

119894 119898 ge 119862119894

(12)


(1) suminfin

119898=0 120587(119890)

119894119898= sum119862119894minus1119898=0 (1119898)120588119898

1198940 + (119862119894119862119894)120588119862119894minus11198940 (120588119894(1 minus

120588119894))1205871198940 = 1(2) 120588119894 lt 1



(119890)

1198940

120587(119890)

1198940 = (

119862119894minus1

sum119898=0

120588119898

1198940119898

+120588119862119894

1198940119862119894

sdot1

1 minus 120588119894)

minus1

(13)


119902119894 = 1minus

infin

sum119898=119862119894

120587(119890)

119894119898= 1minus

infin

sum119898=119862119894

119862119862119894

119894

119862119894120588119898

119894120587(119890)

1198940

=119862119894 minus 1205881198940 minus 119862119894120587

(119890)

119894119862119894

119862119894 minus 1205881198940

(14)


(119890) is

119902(119890)

=

119899

sum119894=1

119901119894119902119894 (15)



(119890)


119862(119890)

119908 119894=

infin

sum119898=119862119894

(119898 minus 119862119894) 120587(119890)

119894119898

119862(119890)

exc 119894 =

119862119894

sum119898=0

119898120587(119890)

119894119898+ 119862119894

infin

sum119898=119862119894+1

120587(119890)

119894119898

(16)


(119890)

total 119894 is

119862(119890)

total 119894 = 119862(119890)

119908 119894+ 119862(119890)

exc 119894

= 120587(119890)

1198940 sdot120588119862119894+11198940

(119862119894 minus 1) (119862119894 minus 1205881198940)2 + 1205881198940

(17)


(119890)



(119890)


119879(119890)

119894=

119862(119890)

total 119894120582119894

=119868

119891119894(1+ 120587

(119890)

1198940 sdot120588119862119894

1198940

(119862119894 minus 1) (119862119894 minus 1205881198940)2 )

119882(119890)

119894=

119862(119890)

119908 119894

120582119894=

1120582119894

infin

sum119898=119862119894

(119898 minus 119862119894) 120587(119890)

119894119898

= 120587(119890)

1198940 sdot119868 sdot 120588119862119894

1198940

119891119894 (119862119894 minus 1) (119862119894 minus 1205881198940)2

(18)



119879(119890)

=

119899

sum119894=1

119875119894119879(119890)

119894 (19)


119882(119890)

=

119899

sum119894=1

119875119894119882(119890)

119894 (20)



119879 = 119879(119904)

+ 119879(119890)

= 119879(119904)

+

119899

sum119894=1

119875119894119879(119890)

119894 (21)


119882 = 119882(119904)

+ 119882(119890)

=119862(119904)

119882

120582119890+

119899

sum119894=1

119875119894119882(119890)

119894

=120588119904

120583119904(

11 minus 120588119904

minus119896120588119896minus1119904

1 minus 120588119896119904

) +

119899

sum119894=1

119875119894119882(119890)

119894

(22)


1198751 = 119875119887 (23)



119902 = 119902(119890)

(24)







Min120582119890 1198681198911 1198911198991198621119862119899

119899

sum119894=1

119885119894119875119894119879(119890)

119894

ST (1) 120588119894 =

0 119885119894 = 0

isin [120575119897 120575ℎ] 119885119894 = 11 le 119894 le 119899

(2) 120582119894 =120588119894119862119894119891119894

119868 1 le 119894 le 119899

(3)

119899

sum119894=1

120582119894 = 120582119890

(4) 119875119894 =120588119894119862119894119891119894

119868120582119890 1 le 119894 le 119899

(5)

119899

sum119894=1

119875119894 = 1

(6) 119879(119890)

119894=

119868

119891119894

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

1 le 119894 le 119899

(7) 119885119894 isin 0 1 1 le 119894 le 119899

(25)



119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898

119894=1119862119894119891119894] In a


[(120575119897119868) sum119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898



Min120582119890 11986811989111198911198991198621119862119899

119899

sum119894=1

120588119894119862119894

120582119890

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

ST (1) 120575119897 le 120588119894le 120575ℎ 1 le 119894 le 119899

(2)

119899

sum119894=1

120588119894119862119894119891119894

119868= 120582119890

(3)

119899

sum119894=1

120588119894119862119894119891119894

119868120582119890= 1

(26)




(VM Num 119862119894)



(VM Num 119862119894)












119875119894 =119862119894119891119894

sum10119894=1 119862119894119891119894

1 le 119894 le 10 (27)










Mean Lower Upper

1575 ST1 00058 0005683 0005917 0005759ST2 000573 0005491 0005969 0005759

1725 ST1 000452 0004404 0004636 0004586ST2 000462 0004504 0004736 0004586

1875 ST1 000295 0002853 0003047 0002916ST2 000296 0002842 0003078 0002916



1575 ST1 080593 0803737 0808123 0806097ST2 072404 0723025 0725055 0724773

1725 ST1 06867 0683968 0689432 0688503ST2 060165 0599881 0603419 060225

1875 ST1 0526 0524375 0527625 0525132ST2 044677 0445745 0447795 0447063



1575 ST1 005987 0059035 0060705 0060115ST2 008227 0081676 0082864 0082617

1725 ST1 009591 0094468 0097352 0096286ST2 01249 0123706 0126094 012491

1875 ST1 017872 0174778 0182662 0178457ST2 021112 0207124 0215116 0213743



1575 ST1 056496 0565194 056374 056618ST2 035094 0350361 0351519 0351333

1725 ST1 060029 0598638 0601942 0600923ST2 039317 0391785 0394555 0393184

1875 ST1 06829 0678949 0686851 0682724ST2 047903 0474915 0483145 0481646






00025

0003

00035

0004

00045

0005

00055

0006

00065

155 160 165 170 175 180 185 190

Loss

(blo

ckin

g) p

rob

()



00034

00036

00038

181 1815 182

00044

00046

00048

166 1665 167



044

048

052

056

06

064

068

072

076

08

Prob

of i

mm

edia

te ex

ecut

ion

()

0735

074

0745

166 1665 167

0679

068

0681

163 1635 164

155 160 165 170 175 180 185 190








(119890)

119894and 119882

(119890)

119894 1 le


(119890)

119894and 119882

(119890)



and the 95 CI of 119879(119890)

119894and 119882

(119890)







119894







119894




004

006

008

01

012

014

016

018

02

022

Mea

n w

aitin

g tim

e (s)

015

0155

016

184 1845 185

0094

0096

0098

163 1635 164

155 160 165 170 175 180 185 190







(119890)

119894and 119882

(119890)

119894for all

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)

0656

066

184 1845 185

0356

0358

036

160 1605 161

0392

0394

172 1725 173

155 160 165 170 175 180 185 190







02

04

06

08

1

12

14

16

18

2

0 1 2 3 4 5 6 7 8 9 10 11

Mea

n re

spon

se ti

me (

s)

HostID

104

108

3 4 5 6 70576

05805

8 9 10









1015840

119890

1205821015840

119890=

119898

sum119894=1

120582119894 997904rArr120575119897

119868

119898

sum119894=1

119862119894119891119894 le 1205821015840

119890le

120575ℎ

119868

119898

sum119894=1

119862119894119891119894 (28)



(119890)



(119890)


0

Mea

n w

aitin

g tim

e (s)

056

0595

3 4 5 6 7

00530054005500560057

8 9 10 1102

04

06

08

1

12

14

16

0 1 2 3 4 5 6 7 8 9 10 11HostID















03

035

04

045

05

055

06

065

07

075

0 2 4 6 8 10 12 14 16

Mea

n re

spon

se ti

me (

s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





004006008

01012014016018

02022024026

0 2 4 6 8 10 12 14 16

Mea

n w

aitin

g tim

e (s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





025

03

150

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)


ST1_TST2_TST3_T







119898

119894=1119862119894119891119894 le 120582119890 le

(120575ℎ119868) sum119898




004

006

008

01

012

014

016

018

02

022

024

Mea

n w

aitin

g tim

e (s)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T


03

04

05

06

07

08

09Pr

ob o

f im

med

iate

exec

utio

n (

)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T






06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)




Notations













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Waiting queue

S1

S2

Sn

120583

120583

120583

120582

(a)

Waiting queue

λ

S1

S2

Sn

p1

p2

pn

1205831

1205832

120583n

(b)


λ

λ λ λ λ λλ

S0

S10

S2

S01

S3 Sn Sn+1

1205831

1205831

1205832 1205832

120583 120583 120583 120583 120583p1120582

p2120582













120582119890 = 120582 (1minus 119901119887) (1)

120582119894 = 120582119890119901119894 1 le 119894 le 119899 sum 119901119894 = 1 (2)

120583119894 =1

(119868119891119894)=

119891119894

119868 (3)

120588119894 =120582119894

(119862119894120583119894)=

(120582119894119868)

(119862119894119891119894) 120588119894 isin [1205751 120575ℎ] (4)






(119904)

119894is [22]

120587(119904)

0 =1 minus 120588119904

1 minus 120588119896+1119904

120587(119904)

119894=

1 minus 120588119904

1 minus 120588119896+1119904

120588119894

119904 119894 = 1 2 119896

(5)


Mainscheduler

λe

Monitor

Balancecontroller

Departure


hellip


Masterserver

Executionserver

λ

S1

S2

Sn

c0

c0c1

c0

c1

c1

cm1

cm2

cmn

(1205831)

(1205832)120583s

(120583n)

f1

fn

f2

p1

p2

pn

Lq Wq 120588

1205821

1205822

120582n

k






1 20

λ λλλλ

k minus 1 k

120583s 120583s 120583s 120583s 120583s




119875119887 = 120587(119904)

119896=

1 minus 120588119904

1 minus 120588119896+1119904

120588119896

119904 (6)


120582119890 = 120582 (1minus 119901119887) = 1205821 minus 120588119896119904

1 minus 120588119896+1119904

(7)





119862(119904)

119908=

119896minus1sum119894=0

119896120587(119904)

119894+1 = 1205882119904120587(119904)

0

119896minus1sum119894=1

119896120588119896minus1119904

119862(119904)

sch = (1minus 120587(119904)

0 ) =120588119904 minus 120588119896+1

119904

1 minus 120588119896+1119904

(8)


(119904)

total is

119862(119904)

total = 119862(119904)

119908+ 119862(119904)

sch =120588119904


(119896 + 1) 120588119896+1119904

1 minus 120588119896+1119904

(9)



(119904)

is

119879(119904)

=119862(119904)

total120582119890

=1 minus (119896 + 1) 120588119896

119904+ 119896120588119896+1119904

120583119904 (1 minus 120588119904) (1 minus 120588119896119904)

=1

120583119904 minus 120582minus

119896120582119896

120583119896+1119904

minus 120582119896

(10)




1205881198940 =120582119894

120583119894=

120582119894119868

119891119894 (11)




120587(119890)

119894119898=

120587(119890)

1198940 sdot1205881198981198940

119898= 120587(119890)

1198940 sdot119862119898119894

119898120588119898119894

0 le 119898 lt 119862119894

120587(119890)

1198940 sdot1205881198981198940

119862119894119862119898minus119862119894

119894

= 120587(119890)

1198940 sdot119862119862119894

119894

119862119894120588119898

119894 119898 ge 119862119894

(12)


(1) suminfin

119898=0 120587(119890)

119894119898= sum119862119894minus1119898=0 (1119898)120588119898

1198940 + (119862119894119862119894)120588119862119894minus11198940 (120588119894(1 minus

120588119894))1205871198940 = 1(2) 120588119894 lt 1



(119890)

1198940

120587(119890)

1198940 = (

119862119894minus1

sum119898=0

120588119898

1198940119898

+120588119862119894

1198940119862119894

sdot1

1 minus 120588119894)

minus1

(13)


119902119894 = 1minus

infin

sum119898=119862119894

120587(119890)

119894119898= 1minus

infin

sum119898=119862119894

119862119862119894

119894

119862119894120588119898

119894120587(119890)

1198940

=119862119894 minus 1205881198940 minus 119862119894120587

(119890)

119894119862119894

119862119894 minus 1205881198940

(14)


(119890) is

119902(119890)

=

119899

sum119894=1

119901119894119902119894 (15)



(119890)


119862(119890)

119908 119894=

infin

sum119898=119862119894

(119898 minus 119862119894) 120587(119890)

119894119898

119862(119890)

exc 119894 =

119862119894

sum119898=0

119898120587(119890)

119894119898+ 119862119894

infin

sum119898=119862119894+1

120587(119890)

119894119898

(16)


(119890)

total 119894 is

119862(119890)

total 119894 = 119862(119890)

119908 119894+ 119862(119890)

exc 119894

= 120587(119890)

1198940 sdot120588119862119894+11198940

(119862119894 minus 1) (119862119894 minus 1205881198940)2 + 1205881198940

(17)


(119890)



(119890)


119879(119890)

119894=

119862(119890)

total 119894120582119894

=119868

119891119894(1+ 120587

(119890)

1198940 sdot120588119862119894

1198940

(119862119894 minus 1) (119862119894 minus 1205881198940)2 )

119882(119890)

119894=

119862(119890)

119908 119894

120582119894=

1120582119894

infin

sum119898=119862119894

(119898 minus 119862119894) 120587(119890)

119894119898

= 120587(119890)

1198940 sdot119868 sdot 120588119862119894

1198940

119891119894 (119862119894 minus 1) (119862119894 minus 1205881198940)2

(18)



119879(119890)

=

119899

sum119894=1

119875119894119879(119890)

119894 (19)


119882(119890)

=

119899

sum119894=1

119875119894119882(119890)

119894 (20)



119879 = 119879(119904)

+ 119879(119890)

= 119879(119904)

+

119899

sum119894=1

119875119894119879(119890)

119894 (21)


119882 = 119882(119904)

+ 119882(119890)

=119862(119904)

119882

120582119890+

119899

sum119894=1

119875119894119882(119890)

119894

=120588119904

120583119904(

11 minus 120588119904

minus119896120588119896minus1119904

1 minus 120588119896119904

) +

119899

sum119894=1

119875119894119882(119890)

119894

(22)


1198751 = 119875119887 (23)



119902 = 119902(119890)

(24)







Min120582119890 1198681198911 1198911198991198621119862119899

119899

sum119894=1

119885119894119875119894119879(119890)

119894

ST (1) 120588119894 =

0 119885119894 = 0

isin [120575119897 120575ℎ] 119885119894 = 11 le 119894 le 119899

(2) 120582119894 =120588119894119862119894119891119894

119868 1 le 119894 le 119899

(3)

119899

sum119894=1

120582119894 = 120582119890

(4) 119875119894 =120588119894119862119894119891119894

119868120582119890 1 le 119894 le 119899

(5)

119899

sum119894=1

119875119894 = 1

(6) 119879(119890)

119894=

119868

119891119894

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

1 le 119894 le 119899

(7) 119885119894 isin 0 1 1 le 119894 le 119899

(25)



119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898

119894=1119862119894119891119894] In a


[(120575119897119868) sum119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898



Min120582119890 11986811989111198911198991198621119862119899

119899

sum119894=1

120588119894119862119894

120582119890

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

ST (1) 120575119897 le 120588119894le 120575ℎ 1 le 119894 le 119899

(2)

119899

sum119894=1

120588119894119862119894119891119894

119868= 120582119890

(3)

119899

sum119894=1

120588119894119862119894119891119894

119868120582119890= 1

(26)




(VM Num 119862119894)



(VM Num 119862119894)












119875119894 =119862119894119891119894

sum10119894=1 119862119894119891119894

1 le 119894 le 10 (27)










Mean Lower Upper

1575 ST1 00058 0005683 0005917 0005759ST2 000573 0005491 0005969 0005759

1725 ST1 000452 0004404 0004636 0004586ST2 000462 0004504 0004736 0004586

1875 ST1 000295 0002853 0003047 0002916ST2 000296 0002842 0003078 0002916



1575 ST1 080593 0803737 0808123 0806097ST2 072404 0723025 0725055 0724773

1725 ST1 06867 0683968 0689432 0688503ST2 060165 0599881 0603419 060225

1875 ST1 0526 0524375 0527625 0525132ST2 044677 0445745 0447795 0447063



1575 ST1 005987 0059035 0060705 0060115ST2 008227 0081676 0082864 0082617

1725 ST1 009591 0094468 0097352 0096286ST2 01249 0123706 0126094 012491

1875 ST1 017872 0174778 0182662 0178457ST2 021112 0207124 0215116 0213743



1575 ST1 056496 0565194 056374 056618ST2 035094 0350361 0351519 0351333

1725 ST1 060029 0598638 0601942 0600923ST2 039317 0391785 0394555 0393184

1875 ST1 06829 0678949 0686851 0682724ST2 047903 0474915 0483145 0481646






00025

0003

00035

0004

00045

0005

00055

0006

00065

155 160 165 170 175 180 185 190

Loss

(blo

ckin

g) p

rob

()



00034

00036

00038

181 1815 182

00044

00046

00048

166 1665 167



044

048

052

056

06

064

068

072

076

08

Prob

of i

mm

edia

te ex

ecut

ion

()

0735

074

0745

166 1665 167

0679

068

0681

163 1635 164

155 160 165 170 175 180 185 190








(119890)

119894and 119882

(119890)

119894 1 le


(119890)

119894and 119882

(119890)



and the 95 CI of 119879(119890)

119894and 119882

(119890)







119894







119894




004

006

008

01

012

014

016

018

02

022

Mea

n w

aitin

g tim

e (s)

015

0155

016

184 1845 185

0094

0096

0098

163 1635 164

155 160 165 170 175 180 185 190







(119890)

119894and 119882

(119890)

119894for all

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)

0656

066

184 1845 185

0356

0358

036

160 1605 161

0392

0394

172 1725 173

155 160 165 170 175 180 185 190







02

04

06

08

1

12

14

16

18

2

0 1 2 3 4 5 6 7 8 9 10 11

Mea

n re

spon

se ti

me (

s)

HostID

104

108

3 4 5 6 70576

05805

8 9 10









1015840

119890

1205821015840

119890=

119898

sum119894=1

120582119894 997904rArr120575119897

119868

119898

sum119894=1

119862119894119891119894 le 1205821015840

119890le

120575ℎ

119868

119898

sum119894=1

119862119894119891119894 (28)



(119890)



(119890)


0

Mea

n w

aitin

g tim

e (s)

056

0595

3 4 5 6 7

00530054005500560057

8 9 10 1102

04

06

08

1

12

14

16

0 1 2 3 4 5 6 7 8 9 10 11HostID















03

035

04

045

05

055

06

065

07

075

0 2 4 6 8 10 12 14 16

Mea

n re

spon

se ti

me (

s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





004006008

01012014016018

02022024026

0 2 4 6 8 10 12 14 16

Mea

n w

aitin

g tim

e (s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





025

03

150

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)


ST1_TST2_TST3_T







119898

119894=1119862119894119891119894 le 120582119890 le

(120575ℎ119868) sum119898




004

006

008

01

012

014

016

018

02

022

024

Mea

n w

aitin

g tim

e (s)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T


03

04

05

06

07

08

09Pr

ob o

f im

med

iate

exec

utio

n (

)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T






06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)




Notations













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Mainscheduler

λe

Monitor

Balancecontroller

Departure


hellip


Masterserver

Executionserver

λ

S1

S2

Sn

c0

c0c1

c0

c1

c1

cm1

cm2

cmn

(1205831)

(1205832)120583s

(120583n)

f1

fn

f2

p1

p2

pn

Lq Wq 120588

1205821

1205822

120582n

k






1 20

λ λλλλ

k minus 1 k

120583s 120583s 120583s 120583s 120583s




119875119887 = 120587(119904)

119896=

1 minus 120588119904

1 minus 120588119896+1119904

120588119896

119904 (6)


120582119890 = 120582 (1minus 119901119887) = 1205821 minus 120588119896119904

1 minus 120588119896+1119904

(7)





119862(119904)

119908=

119896minus1sum119894=0

119896120587(119904)

119894+1 = 1205882119904120587(119904)

0

119896minus1sum119894=1

119896120588119896minus1119904

119862(119904)

sch = (1minus 120587(119904)

0 ) =120588119904 minus 120588119896+1

119904

1 minus 120588119896+1119904

(8)


(119904)

total is

119862(119904)

total = 119862(119904)

119908+ 119862(119904)

sch =120588119904


(119896 + 1) 120588119896+1119904

1 minus 120588119896+1119904

(9)



(119904)

is

119879(119904)

=119862(119904)

total120582119890

=1 minus (119896 + 1) 120588119896

119904+ 119896120588119896+1119904

120583119904 (1 minus 120588119904) (1 minus 120588119896119904)

=1

120583119904 minus 120582minus

119896120582119896

120583119896+1119904

minus 120582119896

(10)




1205881198940 =120582119894

120583119894=

120582119894119868

119891119894 (11)




120587(119890)

119894119898=

120587(119890)

1198940 sdot1205881198981198940

119898= 120587(119890)

1198940 sdot119862119898119894

119898120588119898119894

0 le 119898 lt 119862119894

120587(119890)

1198940 sdot1205881198981198940

119862119894119862119898minus119862119894

119894

= 120587(119890)

1198940 sdot119862119862119894

119894

119862119894120588119898

119894 119898 ge 119862119894

(12)


(1) suminfin

119898=0 120587(119890)

119894119898= sum119862119894minus1119898=0 (1119898)120588119898

1198940 + (119862119894119862119894)120588119862119894minus11198940 (120588119894(1 minus

120588119894))1205871198940 = 1(2) 120588119894 lt 1



(119890)

1198940

120587(119890)

1198940 = (

119862119894minus1

sum119898=0

120588119898

1198940119898

+120588119862119894

1198940119862119894

sdot1

1 minus 120588119894)

minus1

(13)


119902119894 = 1minus

infin

sum119898=119862119894

120587(119890)

119894119898= 1minus

infin

sum119898=119862119894

119862119862119894

119894

119862119894120588119898

119894120587(119890)

1198940

=119862119894 minus 1205881198940 minus 119862119894120587

(119890)

119894119862119894

119862119894 minus 1205881198940

(14)


(119890) is

119902(119890)

=

119899

sum119894=1

119901119894119902119894 (15)



(119890)


119862(119890)

119908 119894=

infin

sum119898=119862119894

(119898 minus 119862119894) 120587(119890)

119894119898

119862(119890)

exc 119894 =

119862119894

sum119898=0

119898120587(119890)

119894119898+ 119862119894

infin

sum119898=119862119894+1

120587(119890)

119894119898

(16)


(119890)

total 119894 is

119862(119890)

total 119894 = 119862(119890)

119908 119894+ 119862(119890)

exc 119894

= 120587(119890)

1198940 sdot120588119862119894+11198940

(119862119894 minus 1) (119862119894 minus 1205881198940)2 + 1205881198940

(17)


(119890)



(119890)


119879(119890)

119894=

119862(119890)

total 119894120582119894

=119868

119891119894(1+ 120587

(119890)

1198940 sdot120588119862119894

1198940

(119862119894 minus 1) (119862119894 minus 1205881198940)2 )

119882(119890)

119894=

119862(119890)

119908 119894

120582119894=

1120582119894

infin

sum119898=119862119894

(119898 minus 119862119894) 120587(119890)

119894119898

= 120587(119890)

1198940 sdot119868 sdot 120588119862119894

1198940

119891119894 (119862119894 minus 1) (119862119894 minus 1205881198940)2

(18)



119879(119890)

=

119899

sum119894=1

119875119894119879(119890)

119894 (19)


119882(119890)

=

119899

sum119894=1

119875119894119882(119890)

119894 (20)



119879 = 119879(119904)

+ 119879(119890)

= 119879(119904)

+

119899

sum119894=1

119875119894119879(119890)

119894 (21)


119882 = 119882(119904)

+ 119882(119890)

=119862(119904)

119882

120582119890+

119899

sum119894=1

119875119894119882(119890)

119894

=120588119904

120583119904(

11 minus 120588119904

minus119896120588119896minus1119904

1 minus 120588119896119904

) +

119899

sum119894=1

119875119894119882(119890)

119894

(22)


1198751 = 119875119887 (23)



119902 = 119902(119890)

(24)







Min120582119890 1198681198911 1198911198991198621119862119899

119899

sum119894=1

119885119894119875119894119879(119890)

119894

ST (1) 120588119894 =

0 119885119894 = 0

isin [120575119897 120575ℎ] 119885119894 = 11 le 119894 le 119899

(2) 120582119894 =120588119894119862119894119891119894

119868 1 le 119894 le 119899

(3)

119899

sum119894=1

120582119894 = 120582119890

(4) 119875119894 =120588119894119862119894119891119894

119868120582119890 1 le 119894 le 119899

(5)

119899

sum119894=1

119875119894 = 1

(6) 119879(119890)

119894=

119868

119891119894

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

1 le 119894 le 119899

(7) 119885119894 isin 0 1 1 le 119894 le 119899

(25)



119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898

119894=1119862119894119891119894] In a


[(120575119897119868) sum119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898



Min120582119890 11986811989111198911198991198621119862119899

119899

sum119894=1

120588119894119862119894

120582119890

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

ST (1) 120575119897 le 120588119894le 120575ℎ 1 le 119894 le 119899

(2)

119899

sum119894=1

120588119894119862119894119891119894

119868= 120582119890

(3)

119899

sum119894=1

120588119894119862119894119891119894

119868120582119890= 1

(26)




(VM Num 119862119894)



(VM Num 119862119894)












119875119894 =119862119894119891119894

sum10119894=1 119862119894119891119894

1 le 119894 le 10 (27)










Mean Lower Upper

1575 ST1 00058 0005683 0005917 0005759ST2 000573 0005491 0005969 0005759

1725 ST1 000452 0004404 0004636 0004586ST2 000462 0004504 0004736 0004586

1875 ST1 000295 0002853 0003047 0002916ST2 000296 0002842 0003078 0002916



1575 ST1 080593 0803737 0808123 0806097ST2 072404 0723025 0725055 0724773

1725 ST1 06867 0683968 0689432 0688503ST2 060165 0599881 0603419 060225

1875 ST1 0526 0524375 0527625 0525132ST2 044677 0445745 0447795 0447063



1575 ST1 005987 0059035 0060705 0060115ST2 008227 0081676 0082864 0082617

1725 ST1 009591 0094468 0097352 0096286ST2 01249 0123706 0126094 012491

1875 ST1 017872 0174778 0182662 0178457ST2 021112 0207124 0215116 0213743



1575 ST1 056496 0565194 056374 056618ST2 035094 0350361 0351519 0351333

1725 ST1 060029 0598638 0601942 0600923ST2 039317 0391785 0394555 0393184

1875 ST1 06829 0678949 0686851 0682724ST2 047903 0474915 0483145 0481646






00025

0003

00035

0004

00045

0005

00055

0006

00065

155 160 165 170 175 180 185 190

Loss

(blo

ckin

g) p

rob

()



00034

00036

00038

181 1815 182

00044

00046

00048

166 1665 167



044

048

052

056

06

064

068

072

076

08

Prob

of i

mm

edia

te ex

ecut

ion

()

0735

074

0745

166 1665 167

0679

068

0681

163 1635 164

155 160 165 170 175 180 185 190








(119890)

119894and 119882

(119890)

119894 1 le


(119890)

119894and 119882

(119890)



and the 95 CI of 119879(119890)

119894and 119882

(119890)







119894







119894




004

006

008

01

012

014

016

018

02

022

Mea

n w

aitin

g tim

e (s)

015

0155

016

184 1845 185

0094

0096

0098

163 1635 164

155 160 165 170 175 180 185 190







(119890)

119894and 119882

(119890)

119894for all

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)

0656

066

184 1845 185

0356

0358

036

160 1605 161

0392

0394

172 1725 173

155 160 165 170 175 180 185 190







02

04

06

08

1

12

14

16

18

2

0 1 2 3 4 5 6 7 8 9 10 11

Mea

n re

spon

se ti

me (

s)

HostID

104

108

3 4 5 6 70576

05805

8 9 10









1015840

119890

1205821015840

119890=

119898

sum119894=1

120582119894 997904rArr120575119897

119868

119898

sum119894=1

119862119894119891119894 le 1205821015840

119890le

120575ℎ

119868

119898

sum119894=1

119862119894119891119894 (28)



(119890)



(119890)


0

Mea

n w

aitin

g tim

e (s)

056

0595

3 4 5 6 7

00530054005500560057

8 9 10 1102

04

06

08

1

12

14

16

0 1 2 3 4 5 6 7 8 9 10 11HostID















03

035

04

045

05

055

06

065

07

075

0 2 4 6 8 10 12 14 16

Mea

n re

spon

se ti

me (

s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





004006008

01012014016018

02022024026

0 2 4 6 8 10 12 14 16

Mea

n w

aitin

g tim

e (s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





025

03

150

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)


ST1_TST2_TST3_T







119898

119894=1119862119894119891119894 le 120582119890 le

(120575ℎ119868) sum119898




004

006

008

01

012

014

016

018

02

022

024

Mea

n w

aitin

g tim

e (s)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T


03

04

05

06

07

08

09Pr

ob o

f im

med

iate

exec

utio

n (

)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T






06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)




Notations













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119890)

1198940

120587(119890)

1198940 = (

119862119894minus1

sum119898=0

120588119898

1198940119898

+120588119862119894

1198940119862119894

sdot1

1 minus 120588119894)

minus1

(13)


119902119894 = 1minus

infin

sum119898=119862119894

120587(119890)

119894119898= 1minus

infin

sum119898=119862119894

119862119862119894

119894

119862119894120588119898

119894120587(119890)

1198940

=119862119894 minus 1205881198940 minus 119862119894120587

(119890)

119894119862119894

119862119894 minus 1205881198940

(14)


(119890) is

119902(119890)

=

119899

sum119894=1

119901119894119902119894 (15)



(119890)


119862(119890)

119908 119894=

infin

sum119898=119862119894

(119898 minus 119862119894) 120587(119890)

119894119898

119862(119890)

exc 119894 =

119862119894

sum119898=0

119898120587(119890)

119894119898+ 119862119894

infin

sum119898=119862119894+1

120587(119890)

119894119898

(16)


(119890)

total 119894 is

119862(119890)

total 119894 = 119862(119890)

119908 119894+ 119862(119890)

exc 119894

= 120587(119890)

1198940 sdot120588119862119894+11198940

(119862119894 minus 1) (119862119894 minus 1205881198940)2 + 1205881198940

(17)


(119890)



(119890)


119879(119890)

119894=

119862(119890)

total 119894120582119894

=119868

119891119894(1+ 120587

(119890)

1198940 sdot120588119862119894

1198940

(119862119894 minus 1) (119862119894 minus 1205881198940)2 )

119882(119890)

119894=

119862(119890)

119908 119894

120582119894=

1120582119894

infin

sum119898=119862119894

(119898 minus 119862119894) 120587(119890)

119894119898

= 120587(119890)

1198940 sdot119868 sdot 120588119862119894

1198940

119891119894 (119862119894 minus 1) (119862119894 minus 1205881198940)2

(18)



119879(119890)

=

119899

sum119894=1

119875119894119879(119890)

119894 (19)


119882(119890)

=

119899

sum119894=1

119875119894119882(119890)

119894 (20)



119879 = 119879(119904)

+ 119879(119890)

= 119879(119904)

+

119899

sum119894=1

119875119894119879(119890)

119894 (21)


119882 = 119882(119904)

+ 119882(119890)

=119862(119904)

119882

120582119890+

119899

sum119894=1

119875119894119882(119890)

119894

=120588119904

120583119904(

11 minus 120588119904

minus119896120588119896minus1119904

1 minus 120588119896119904

) +

119899

sum119894=1

119875119894119882(119890)

119894

(22)


1198751 = 119875119887 (23)



119902 = 119902(119890)

(24)







Min120582119890 1198681198911 1198911198991198621119862119899

119899

sum119894=1

119885119894119875119894119879(119890)

119894

ST (1) 120588119894 =

0 119885119894 = 0

isin [120575119897 120575ℎ] 119885119894 = 11 le 119894 le 119899

(2) 120582119894 =120588119894119862119894119891119894

119868 1 le 119894 le 119899

(3)

119899

sum119894=1

120582119894 = 120582119890

(4) 119875119894 =120588119894119862119894119891119894

119868120582119890 1 le 119894 le 119899

(5)

119899

sum119894=1

119875119894 = 1

(6) 119879(119890)

119894=

119868

119891119894

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

1 le 119894 le 119899

(7) 119885119894 isin 0 1 1 le 119894 le 119899

(25)



119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898

119894=1119862119894119891119894] In a


[(120575119897119868) sum119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898



Min120582119890 11986811989111198911198991198621119862119899

119899

sum119894=1

120588119894119862119894

120582119890

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

ST (1) 120575119897 le 120588119894le 120575ℎ 1 le 119894 le 119899

(2)

119899

sum119894=1

120588119894119862119894119891119894

119868= 120582119890

(3)

119899

sum119894=1

120588119894119862119894119891119894

119868120582119890= 1

(26)




(VM Num 119862119894)



(VM Num 119862119894)












119875119894 =119862119894119891119894

sum10119894=1 119862119894119891119894

1 le 119894 le 10 (27)










Mean Lower Upper

1575 ST1 00058 0005683 0005917 0005759ST2 000573 0005491 0005969 0005759

1725 ST1 000452 0004404 0004636 0004586ST2 000462 0004504 0004736 0004586

1875 ST1 000295 0002853 0003047 0002916ST2 000296 0002842 0003078 0002916



1575 ST1 080593 0803737 0808123 0806097ST2 072404 0723025 0725055 0724773

1725 ST1 06867 0683968 0689432 0688503ST2 060165 0599881 0603419 060225

1875 ST1 0526 0524375 0527625 0525132ST2 044677 0445745 0447795 0447063



1575 ST1 005987 0059035 0060705 0060115ST2 008227 0081676 0082864 0082617

1725 ST1 009591 0094468 0097352 0096286ST2 01249 0123706 0126094 012491

1875 ST1 017872 0174778 0182662 0178457ST2 021112 0207124 0215116 0213743



1575 ST1 056496 0565194 056374 056618ST2 035094 0350361 0351519 0351333

1725 ST1 060029 0598638 0601942 0600923ST2 039317 0391785 0394555 0393184

1875 ST1 06829 0678949 0686851 0682724ST2 047903 0474915 0483145 0481646






00025

0003

00035

0004

00045

0005

00055

0006

00065

155 160 165 170 175 180 185 190

Loss

(blo

ckin

g) p

rob

()



00034

00036

00038

181 1815 182

00044

00046

00048

166 1665 167



044

048

052

056

06

064

068

072

076

08

Prob

of i

mm

edia

te ex

ecut

ion

()

0735

074

0745

166 1665 167

0679

068

0681

163 1635 164

155 160 165 170 175 180 185 190








(119890)

119894and 119882

(119890)

119894 1 le


(119890)

119894and 119882

(119890)



and the 95 CI of 119879(119890)

119894and 119882

(119890)







119894







119894




004

006

008

01

012

014

016

018

02

022

Mea

n w

aitin

g tim

e (s)

015

0155

016

184 1845 185

0094

0096

0098

163 1635 164

155 160 165 170 175 180 185 190







(119890)

119894and 119882

(119890)

119894for all

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)

0656

066

184 1845 185

0356

0358

036

160 1605 161

0392

0394

172 1725 173

155 160 165 170 175 180 185 190







02

04

06

08

1

12

14

16

18

2

0 1 2 3 4 5 6 7 8 9 10 11

Mea

n re

spon

se ti

me (

s)

HostID

104

108

3 4 5 6 70576

05805

8 9 10









1015840

119890

1205821015840

119890=

119898

sum119894=1

120582119894 997904rArr120575119897

119868

119898

sum119894=1

119862119894119891119894 le 1205821015840

119890le

120575ℎ

119868

119898

sum119894=1

119862119894119891119894 (28)



(119890)



(119890)


0

Mea

n w

aitin

g tim

e (s)

056

0595

3 4 5 6 7

00530054005500560057

8 9 10 1102

04

06

08

1

12

14

16

0 1 2 3 4 5 6 7 8 9 10 11HostID















03

035

04

045

05

055

06

065

07

075

0 2 4 6 8 10 12 14 16

Mea

n re

spon

se ti

me (

s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





004006008

01012014016018

02022024026

0 2 4 6 8 10 12 14 16

Mea

n w

aitin

g tim

e (s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





025

03

150

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)


ST1_TST2_TST3_T







119898

119894=1119862119894119891119894 le 120582119890 le

(120575ℎ119868) sum119898




004

006

008

01

012

014

016

018

02

022

024

Mea

n w

aitin

g tim

e (s)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T


03

04

05

06

07

08

09Pr

ob o

f im

med

iate

exec

utio

n (

)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T






06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)




Notations













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Min120582119890 1198681198911 1198911198991198621119862119899

119899

sum119894=1

119885119894119875119894119879(119890)

119894

ST (1) 120588119894 =

0 119885119894 = 0

isin [120575119897 120575ℎ] 119885119894 = 11 le 119894 le 119899

(2) 120582119894 =120588119894119862119894119891119894

119868 1 le 119894 le 119899

(3)

119899

sum119894=1

120582119894 = 120582119890

(4) 119875119894 =120588119894119862119894119891119894

119868120582119890 1 le 119894 le 119899

(5)

119899

sum119894=1

119875119894 = 1

(6) 119879(119890)

119894=

119868

119891119894

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

1 le 119894 le 119899

(7) 119885119894 isin 0 1 1 le 119894 le 119899

(25)



119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898

119894=1119862119894119891119894] In a


[(120575119897119868) sum119898

119894=1119862119894119891119894 (120575ℎ119868) sum

119898



Min120582119890 11986811989111198911198991198621119862119899

119899

sum119894=1

120588119894119862119894

120582119890

[

[

1+ (

119862119894minus1

sum119898=0

(119862119894120588119894)119898

(119862119894 minus 1)+

1(119862119894 minus 1)

sdot(119862119894120588119894)

119862119894

1 minus 120588119894)

minus1

sdot(119862119894120588119894)

119862119894minus1 120588119894

119862119894 (1 minus 120588119894)]

]

ST (1) 120575119897 le 120588119894le 120575ℎ 1 le 119894 le 119899

(2)

119899

sum119894=1

120588119894119862119894119891119894

119868= 120582119890

(3)

119899

sum119894=1

120588119894119862119894119891119894

119868120582119890= 1

(26)




(VM Num 119862119894)



(VM Num 119862119894)












119875119894 =119862119894119891119894

sum10119894=1 119862119894119891119894

1 le 119894 le 10 (27)










Mean Lower Upper

1575 ST1 00058 0005683 0005917 0005759ST2 000573 0005491 0005969 0005759

1725 ST1 000452 0004404 0004636 0004586ST2 000462 0004504 0004736 0004586

1875 ST1 000295 0002853 0003047 0002916ST2 000296 0002842 0003078 0002916



1575 ST1 080593 0803737 0808123 0806097ST2 072404 0723025 0725055 0724773

1725 ST1 06867 0683968 0689432 0688503ST2 060165 0599881 0603419 060225

1875 ST1 0526 0524375 0527625 0525132ST2 044677 0445745 0447795 0447063



1575 ST1 005987 0059035 0060705 0060115ST2 008227 0081676 0082864 0082617

1725 ST1 009591 0094468 0097352 0096286ST2 01249 0123706 0126094 012491

1875 ST1 017872 0174778 0182662 0178457ST2 021112 0207124 0215116 0213743



1575 ST1 056496 0565194 056374 056618ST2 035094 0350361 0351519 0351333

1725 ST1 060029 0598638 0601942 0600923ST2 039317 0391785 0394555 0393184

1875 ST1 06829 0678949 0686851 0682724ST2 047903 0474915 0483145 0481646






00025

0003

00035

0004

00045

0005

00055

0006

00065

155 160 165 170 175 180 185 190

Loss

(blo

ckin

g) p

rob

()



00034

00036

00038

181 1815 182

00044

00046

00048

166 1665 167



044

048

052

056

06

064

068

072

076

08

Prob

of i

mm

edia

te ex

ecut

ion

()

0735

074

0745

166 1665 167

0679

068

0681

163 1635 164

155 160 165 170 175 180 185 190








(119890)

119894and 119882

(119890)

119894 1 le


(119890)

119894and 119882

(119890)



and the 95 CI of 119879(119890)

119894and 119882

(119890)







119894







119894




004

006

008

01

012

014

016

018

02

022

Mea

n w

aitin

g tim

e (s)

015

0155

016

184 1845 185

0094

0096

0098

163 1635 164

155 160 165 170 175 180 185 190







(119890)

119894and 119882

(119890)

119894for all

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)

0656

066

184 1845 185

0356

0358

036

160 1605 161

0392

0394

172 1725 173

155 160 165 170 175 180 185 190







02

04

06

08

1

12

14

16

18

2

0 1 2 3 4 5 6 7 8 9 10 11

Mea

n re

spon

se ti

me (

s)

HostID

104

108

3 4 5 6 70576

05805

8 9 10









1015840

119890

1205821015840

119890=

119898

sum119894=1

120582119894 997904rArr120575119897

119868

119898

sum119894=1

119862119894119891119894 le 1205821015840

119890le

120575ℎ

119868

119898

sum119894=1

119862119894119891119894 (28)



(119890)



(119890)


0

Mea

n w

aitin

g tim

e (s)

056

0595

3 4 5 6 7

00530054005500560057

8 9 10 1102

04

06

08

1

12

14

16

0 1 2 3 4 5 6 7 8 9 10 11HostID















03

035

04

045

05

055

06

065

07

075

0 2 4 6 8 10 12 14 16

Mea

n re

spon

se ti

me (

s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





004006008

01012014016018

02022024026

0 2 4 6 8 10 12 14 16

Mea

n w

aitin

g tim

e (s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





025

03

150

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)


ST1_TST2_TST3_T







119898

119894=1119862119894119891119894 le 120582119890 le

(120575ℎ119868) sum119898




004

006

008

01

012

014

016

018

02

022

024

Mea

n w

aitin

g tim

e (s)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T


03

04

05

06

07

08

09Pr

ob o

f im

med

iate

exec

utio

n (

)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T






06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)




Notations













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(VM Num 119862119894)



(VM Num 119862119894)












119875119894 =119862119894119891119894

sum10119894=1 119862119894119891119894

1 le 119894 le 10 (27)










Mean Lower Upper

1575 ST1 00058 0005683 0005917 0005759ST2 000573 0005491 0005969 0005759

1725 ST1 000452 0004404 0004636 0004586ST2 000462 0004504 0004736 0004586

1875 ST1 000295 0002853 0003047 0002916ST2 000296 0002842 0003078 0002916



1575 ST1 080593 0803737 0808123 0806097ST2 072404 0723025 0725055 0724773

1725 ST1 06867 0683968 0689432 0688503ST2 060165 0599881 0603419 060225

1875 ST1 0526 0524375 0527625 0525132ST2 044677 0445745 0447795 0447063



1575 ST1 005987 0059035 0060705 0060115ST2 008227 0081676 0082864 0082617

1725 ST1 009591 0094468 0097352 0096286ST2 01249 0123706 0126094 012491

1875 ST1 017872 0174778 0182662 0178457ST2 021112 0207124 0215116 0213743



1575 ST1 056496 0565194 056374 056618ST2 035094 0350361 0351519 0351333

1725 ST1 060029 0598638 0601942 0600923ST2 039317 0391785 0394555 0393184

1875 ST1 06829 0678949 0686851 0682724ST2 047903 0474915 0483145 0481646






00025

0003

00035

0004

00045

0005

00055

0006

00065

155 160 165 170 175 180 185 190

Loss

(blo

ckin

g) p

rob

()



00034

00036

00038

181 1815 182

00044

00046

00048

166 1665 167



044

048

052

056

06

064

068

072

076

08

Prob

of i

mm

edia

te ex

ecut

ion

()

0735

074

0745

166 1665 167

0679

068

0681

163 1635 164

155 160 165 170 175 180 185 190








(119890)

119894and 119882

(119890)

119894 1 le


(119890)

119894and 119882

(119890)



and the 95 CI of 119879(119890)

119894and 119882

(119890)







119894







119894




004

006

008

01

012

014

016

018

02

022

Mea

n w

aitin

g tim

e (s)

015

0155

016

184 1845 185

0094

0096

0098

163 1635 164

155 160 165 170 175 180 185 190







(119890)

119894and 119882

(119890)

119894for all

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)

0656

066

184 1845 185

0356

0358

036

160 1605 161

0392

0394

172 1725 173

155 160 165 170 175 180 185 190







02

04

06

08

1

12

14

16

18

2

0 1 2 3 4 5 6 7 8 9 10 11

Mea

n re

spon

se ti

me (

s)

HostID

104

108

3 4 5 6 70576

05805

8 9 10









1015840

119890

1205821015840

119890=

119898

sum119894=1

120582119894 997904rArr120575119897

119868

119898

sum119894=1

119862119894119891119894 le 1205821015840

119890le

120575ℎ

119868

119898

sum119894=1

119862119894119891119894 (28)



(119890)



(119890)


0

Mea

n w

aitin

g tim

e (s)

056

0595

3 4 5 6 7

00530054005500560057

8 9 10 1102

04

06

08

1

12

14

16

0 1 2 3 4 5 6 7 8 9 10 11HostID















03

035

04

045

05

055

06

065

07

075

0 2 4 6 8 10 12 14 16

Mea

n re

spon

se ti

me (

s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





004006008

01012014016018

02022024026

0 2 4 6 8 10 12 14 16

Mea

n w

aitin

g tim

e (s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





025

03

150

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)


ST1_TST2_TST3_T







119898

119894=1119862119894119891119894 le 120582119890 le

(120575ℎ119868) sum119898




004

006

008

01

012

014

016

018

02

022

024

Mea

n w

aitin

g tim

e (s)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T


03

04

05

06

07

08

09Pr

ob o

f im

med

iate

exec

utio

n (

)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T






06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)




Notations













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Mean Lower Upper

1575 ST1 00058 0005683 0005917 0005759ST2 000573 0005491 0005969 0005759

1725 ST1 000452 0004404 0004636 0004586ST2 000462 0004504 0004736 0004586

1875 ST1 000295 0002853 0003047 0002916ST2 000296 0002842 0003078 0002916



1575 ST1 080593 0803737 0808123 0806097ST2 072404 0723025 0725055 0724773

1725 ST1 06867 0683968 0689432 0688503ST2 060165 0599881 0603419 060225

1875 ST1 0526 0524375 0527625 0525132ST2 044677 0445745 0447795 0447063



1575 ST1 005987 0059035 0060705 0060115ST2 008227 0081676 0082864 0082617

1725 ST1 009591 0094468 0097352 0096286ST2 01249 0123706 0126094 012491

1875 ST1 017872 0174778 0182662 0178457ST2 021112 0207124 0215116 0213743



1575 ST1 056496 0565194 056374 056618ST2 035094 0350361 0351519 0351333

1725 ST1 060029 0598638 0601942 0600923ST2 039317 0391785 0394555 0393184

1875 ST1 06829 0678949 0686851 0682724ST2 047903 0474915 0483145 0481646






00025

0003

00035

0004

00045

0005

00055

0006

00065

155 160 165 170 175 180 185 190

Loss

(blo

ckin

g) p

rob

()



00034

00036

00038

181 1815 182

00044

00046

00048

166 1665 167



044

048

052

056

06

064

068

072

076

08

Prob

of i

mm

edia

te ex

ecut

ion

()

0735

074

0745

166 1665 167

0679

068

0681

163 1635 164

155 160 165 170 175 180 185 190








(119890)

119894and 119882

(119890)

119894 1 le


(119890)

119894and 119882

(119890)



and the 95 CI of 119879(119890)

119894and 119882

(119890)







119894







119894




004

006

008

01

012

014

016

018

02

022

Mea

n w

aitin

g tim

e (s)

015

0155

016

184 1845 185

0094

0096

0098

163 1635 164

155 160 165 170 175 180 185 190







(119890)

119894and 119882

(119890)

119894for all

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)

0656

066

184 1845 185

0356

0358

036

160 1605 161

0392

0394

172 1725 173

155 160 165 170 175 180 185 190







02

04

06

08

1

12

14

16

18

2

0 1 2 3 4 5 6 7 8 9 10 11

Mea

n re

spon

se ti

me (

s)

HostID

104

108

3 4 5 6 70576

05805

8 9 10









1015840

119890

1205821015840

119890=

119898

sum119894=1

120582119894 997904rArr120575119897

119868

119898

sum119894=1

119862119894119891119894 le 1205821015840

119890le

120575ℎ

119868

119898

sum119894=1

119862119894119891119894 (28)



(119890)



(119890)


0

Mea

n w

aitin

g tim

e (s)

056

0595

3 4 5 6 7

00530054005500560057

8 9 10 1102

04

06

08

1

12

14

16

0 1 2 3 4 5 6 7 8 9 10 11HostID















03

035

04

045

05

055

06

065

07

075

0 2 4 6 8 10 12 14 16

Mea

n re

spon

se ti

me (

s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





004006008

01012014016018

02022024026

0 2 4 6 8 10 12 14 16

Mea

n w

aitin

g tim

e (s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





025

03

150

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)


ST1_TST2_TST3_T







119898

119894=1119862119894119891119894 le 120582119890 le

(120575ℎ119868) sum119898




004

006

008

01

012

014

016

018

02

022

024

Mea

n w

aitin

g tim

e (s)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T


03

04

05

06

07

08

09Pr

ob o

f im

med

iate

exec

utio

n (

)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T






06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)




Notations













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119894







119894




004

006

008

01

012

014

016

018

02

022

Mea

n w

aitin

g tim

e (s)

015

0155

016

184 1845 185

0094

0096

0098

163 1635 164

155 160 165 170 175 180 185 190







(119890)

119894and 119882

(119890)

119894for all

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)

0656

066

184 1845 185

0356

0358

036

160 1605 161

0392

0394

172 1725 173

155 160 165 170 175 180 185 190







02

04

06

08

1

12

14

16

18

2

0 1 2 3 4 5 6 7 8 9 10 11

Mea

n re

spon

se ti

me (

s)

HostID

104

108

3 4 5 6 70576

05805

8 9 10









1015840

119890

1205821015840

119890=

119898

sum119894=1

120582119894 997904rArr120575119897

119868

119898

sum119894=1

119862119894119891119894 le 1205821015840

119890le

120575ℎ

119868

119898

sum119894=1

119862119894119891119894 (28)



(119890)



(119890)


0

Mea

n w

aitin

g tim

e (s)

056

0595

3 4 5 6 7

00530054005500560057

8 9 10 1102

04

06

08

1

12

14

16

0 1 2 3 4 5 6 7 8 9 10 11HostID















03

035

04

045

05

055

06

065

07

075

0 2 4 6 8 10 12 14 16

Mea

n re

spon

se ti

me (

s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





004006008

01012014016018

02022024026

0 2 4 6 8 10 12 14 16

Mea

n w

aitin

g tim

e (s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





025

03

150

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)


ST1_TST2_TST3_T







119898

119894=1119862119894119891119894 le 120582119890 le

(120575ℎ119868) sum119898




004

006

008

01

012

014

016

018

02

022

024

Mea

n w

aitin

g tim

e (s)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T


03

04

05

06

07

08

09Pr

ob o

f im

med

iate

exec

utio

n (

)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T






06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)




Notations













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02

04

06

08

1

12

14

16

18

2

0 1 2 3 4 5 6 7 8 9 10 11

Mea

n re

spon

se ti

me (

s)

HostID

104

108

3 4 5 6 70576

05805

8 9 10









1015840

119890

1205821015840

119890=

119898

sum119894=1

120582119894 997904rArr120575119897

119868

119898

sum119894=1

119862119894119891119894 le 1205821015840

119890le

120575ℎ

119868

119898

sum119894=1

119862119894119891119894 (28)



(119890)



(119890)


0

Mea

n w

aitin

g tim

e (s)

056

0595

3 4 5 6 7

00530054005500560057

8 9 10 1102

04

06

08

1

12

14

16

0 1 2 3 4 5 6 7 8 9 10 11HostID















03

035

04

045

05

055

06

065

07

075

0 2 4 6 8 10 12 14 16

Mea

n re

spon

se ti

me (

s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





004006008

01012014016018

02022024026

0 2 4 6 8 10 12 14 16

Mea

n w

aitin

g tim

e (s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





025

03

150

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)


ST1_TST2_TST3_T







119898

119894=1119862119894119891119894 le 120582119890 le

(120575ℎ119868) sum119898




004

006

008

01

012

014

016

018

02

022

024

Mea

n w

aitin

g tim

e (s)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T


03

04

05

06

07

08

09Pr

ob o

f im

med

iate

exec

utio

n (

)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T






06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)




Notations













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













03

035

04

045

05

055

06

065

07

075

0 2 4 6 8 10 12 14 16

Mea

n re

spon

se ti

me (

s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





004006008

01012014016018

02022024026

0 2 4 6 8 10 12 14 16

Mea

n w

aitin

g tim

e (s)


120582 = 155 k = 17 120588s = 08

ST1_TST2_T





025

03

150

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)


ST1_TST2_TST3_T







119898

119894=1119862119894119891119894 le 120582119890 le

(120575ℎ119868) sum119898




004

006

008

01

012

014

016

018

02

022

024

Mea

n w

aitin

g tim

e (s)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T


03

04

05

06

07

08

09Pr

ob o

f im

med

iate

exec

utio

n (

)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T






06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)




Notations













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










025

03

150

035

04

045

05

055

06

065

07

Mea

n re

spon

se ti

me (

s)


ST1_TST2_TST3_T







119898

119894=1119862119894119891119894 le 120582119890 le

(120575ℎ119868) sum119898




004

006

008

01

012

014

016

018

02

022

024

Mea

n w

aitin

g tim

e (s)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T


03

04

05

06

07

08

09Pr

ob o

f im

med

iate

exec

utio

n (

)


155 160 165 170 175 180 185 190

ST1_TST2_TST3_T






06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)




Notations













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










06065

07075

08085

06065

07075

08085

035

04

045

05

Resp

onse

tim

e

12058821205881

120582e = 383

S1 (C1 = 4 f1 = 4)

S2 (C2 = 4 f2 = 3)

S3 (C3 = 6 f3 = 4)

(Tmin 1205881 1205882 1205883) = (036317 072712 067627 077295)




Notations













Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Performance Analysis of Heterogeneous...

Documents

Transcript of Research Article Performance Analysis of Heterogeneous...