CS 3150 Presentation 1 Ihsan Ayyub Qazi “Achievable sojourn times in M/M/1 and GI/GI/1 systems: A...

CS 3150 PresentationCS 3150 Presentation 11

Ihsan Ayyub Qazi

“Achievable sojourn times in M/M/1 and GI/GI/1 systems: A

Comparison between SRPT and Blind Policies”

A(t),

Q(t)

D(t)


Plan for the Presentation

Some basics Some basics Motivation for minimizing sojourn timesMotivation for minimizing sojourn times Intuition for the Optimality of SRPTIntuition for the Optimality of SRPT Paper 1: “Paper 1: “On the average sojourn time

under M/M/1/SRPT” by Nikhil Bansal Paper 2: “Paper 2: “Achievable sojourn times by

non-size based policies in a GI/GI/1 queue” by Nikhil Bansal


Some Basics…The The sojourn timesojourn time of a job is the time of a job is the time

since a job arrives until the time it since a job arrives until the time it completes its service requirement.completes its service requirement.


Why minimize sojourn time?

Sojourn time is one of the most Sojourn time is one of the most useful measures of useful measures of user satisfaction and user satisfaction and performance in a scheduling system.performance in a scheduling system.

Keeping the Keeping the average sojourn timeaverage sojourn time low is often an important criteria in low is often an important criteria in the choice of a scheduling policy.the choice of a scheduling policy.


Optimizing the average sojourn time

SRPT policy that at any works on the SRPT policy that at any works on the job with the smallest remaining job with the smallest remaining service requirement, achieves the service requirement, achieves the optimum possible average sojourn optimum possible average sojourn time for all possible instances of the time for all possible instances of the problem [1], [2] problem [1], [2]


Optimality of SRPT

Shortest Job First (SJF) is a non-preemptive policy that schedules the jobs in increasing order of their sizes.

It is optimal for all non-preemptive polices

However, the SJF policy is optimal only when all the jobs are present at the server at once.

Total Completion Time = 20+25 = 45Average Completion Time = 45/2 = 22.5

Scheduler

Scheduler

Total Completion Time = 5+25 = 30Average Completion Time = 30/2 = 15

20

5

20

5


Optimality of SRPT

One can view SRPT as a preemptive version of SJF.

Hence one can get an idea about the optimality of SRPT

Scheduler

20

5

15 5



However, one can consider the remaining processing time (i.e. 15)

as a separate job and hence swap it with 5


Disadvantages of SRPT

SRPT requires exact knowledge of the SRPT requires exact knowledge of the service requirement for each jobservice requirement for each job

It is unfair to Jobs with large service It is unfair to Jobs with large service requirements and may even starve requirements and may even starve them.them.

Blind policies have many attractive Blind policies have many attractive properties such as properties such as they are stateless andthey are stateless and easier to implement in a real system.easier to implement in a real system.


“On the average sojourn time under M/M/1/SRPT” by

Nikhil Bansal


Plan

ProblemProblem ResultsResults BackgroundBackground AnalysisAnalysis


Problem

““How much more can the sojourn How much more can the sojourn time improve if the knowledge of time improve if the knowledge of job sizes is used while scheduling?”job sizes is used while scheduling?”


Classical Result

It is a well-known result that the It is a well-known result that the average sojourn time in an M/M/1 average sojourn time in an M/M/1 system issystem is

This holds for all scheduling policies This holds for all scheduling policies that do not make use of the job sizes that do not make use of the job sizes while scheduling [3].while scheduling [3].

)1(

1


Problem Definition (revisited)

““How much more can the sojourn time improve if How much more can the sojourn time improve if the knowledge of job sizes is used while the knowledge of job sizes is used while scheduling?”scheduling?”

In essence, this question reduces to finding the expression for the average sojourn time in a M/M/1/SRPT system


Plan



Main result of the Paper…

The average sojourn time [under M/M/1/SRPT] The average sojourn time [under M/M/1/SRPT] with exponentially distributed job sizes varies as with exponentially distributed job sizes varies as

))

1ln()1(

1(

e

Thus SRPT offers a Thus SRPT offers a factor factor improvement over policies that ignore improvement over policies that ignore knowledge of job sizes while schedulingknowledge of job sizes while scheduling

))1

(ln(

e


Improvement Factor

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Utilization (p)

Impr

ovem

ent F

acto

r


Contribution of the paper

This result places a lower bound on This result places a lower bound on the achievable average sojourn time the achievable average sojourn time under any scheduling policy in an under any scheduling policy in an M/M/1 systemM/M/1 system

The result provides an analytical The result provides an analytical justification for the empirically justification for the empirically observed improvement under SRPT observed improvement under SRPT at high loads.at high loads.


Plan



Some Previous work…

Schrage and Miller first obtained the Schrage and Miller first obtained the expression for the expected sojourn time for a expression for the expected sojourn time for a job of size x under SRPT in the more general job of size x under SRPT in the more general M/G/1 queuing system. They showed that, M/G/1 queuing system. They showed that,

x

x

t

dt

x

dttFt

xT0

20

)(1))(1(

)(

)(

Mean Waiting Time

Mean Residence Time


Mean Waiting and Residence times for a job of size (Intuition)

2

0

))(1(

)(

)(x

dttFt

xW

x

x

t

dtxR

0 )(1)(

Average time a job of Average time a job of size x takes from when size x takes from when it arrives to when it it arrives to when it receives service for receives service for the first timethe first time

Average time it Average time it takes for a job of takes for a job of size x to complete size x to complete once it begins once it begins execution. execution.

Expected Waiting Time of a job of size x corresponds to waiting for all jobs of sizes

less than or equal to x to complete


Plan



Analysis: Expressions with exponentially distributed job sizes

2

0

)))1(1(1(

))1(1()]([

,

))1(1()(

)(

)(

x

x

xtx

t

t

ex

exxWE

Thus

exdtetx

etF

etf


Expression for the sojourn time under SRPT for exponential distributed job sizes

0)]([][ dxexTETE x

The average sojourn time under SRPT is The average sojourn time under SRPT is given bygiven by


Results

Theorem 1: For all Theorem 1: For all p p between 0 and 1between 0 and 1

Theorem 2 (Heavy Traffic Case):Theorem 2 (Heavy Traffic Case):

)1

ln(

1

)1(

1.5][

)1

ln(

1

)1(

1

9

1

e

TEee

1)))1(

1ln()1(].([lim

1

TE


Sketch of the proof An upper bound on E[R] is derived. An upper bound on E[R] is derived. Then an upper bound for E[W] is derived for Then an upper bound for E[W] is derived for

the case when the utilization is more than the case when the utilization is more than 75%. 75%.

For other values of utilization, the result is For other values of utilization, the result is shown to be true by considering the average shown to be true by considering the average sojourn time under FCFS as the upper bound.sojourn time under FCFS as the upper bound.

It is shown that that the contribution of E[R] It is shown that that the contribution of E[R] to E[T] is not significant.to E[T] is not significant.

Hence a good tight bound can be obtained Hence a good tight bound can be obtained for E[T] by lower bounding it by E[W]for E[T] by lower bounding it by E[W]


Upper Bounding the Sojourn Time

Lemma 1: For any load Lemma 1: For any load p, p, such that it is such that it is between 0 and 1between 0 and 1

For any load For any load pp, such that it is between 2/3 , such that it is between 2/3 and 1and 1

)1

1ln(.

1][

ppRE

)1

ln()1(2

7][

pe

pWE

0

)]([][ dxexRERE x

0

)]([][ dxexWEWE x


Comparison between the bounds on E[R] and E[W]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50

Utilization (p)

Upp

er B

ound

s

Expected Waiting Time

Expected Residence Time

)1

1ln(.

1][

ppRE

)1

ln()1(2

7][

pe

pWE


Lower Bounding the Sojourn Time

Lemma 3:Lemma 3:

))1

)(ln(1(9

1)]([][

ee

dxexWETE x


Sketch of the proof (contd) For any load, the average sojourn time under SRPT For any load, the average sojourn time under SRPT

is atleast equal to the average job size. Since SRPT is atleast equal to the average job size. Since SRPT is optimal E[T] is upper bounded by the average is optimal E[T] is upper bounded by the average sojourn time under the policy FCFS. Thussojourn time under the policy FCFS. Thus

And henceAnd hence for for pp < 2/3. The < 2/3. The theorem equation is seen to be true in this case.theorem equation is seen to be true in this case.

)1(

1][

1

TE

)1(

3][

1

TE


Upper Bound for Theorem 1

)1

ln()1(2

3][

e

RE)

1ln()1(2

7][

e

WE

)1

ln()1(

5][

)1

ln()1(2

3

)1

ln()1(2

7][][

eTE

eeWERE

Lemma 3 gives the required lower bound, hence theorem 1 is proved


Thankyou


“Achievable sojourn times by non-size based policies in a GI/GI/1 queuing

system” by Nikhil Bansal


Plan



Problem

How much worse can the How much worse can the average time under the best average time under the best blind policy be as compared to blind policy be as compared to SRPT (in a GI/GI/1 system)?SRPT (in a GI/GI/1 system)?


Plan



Main result of the paper

““For a GI/GI/1 system, the average For a GI/GI/1 system, the average sojourn time under the best blind sojourn time under the best blind policy is atmost policy is atmost time time worse (upto a constant factors) worse (upto a constant factors) then the best average sojourn time then the best average sojourn time possible under any arbitrary policy”possible under any arbitrary policy”

Thus in a sense, the lack of knowledge of Thus in a sense, the lack of knowledge of actual job-sizes does not pose a serious actual job-sizes does not pose a serious problem if the blind policy is chosen carefully.problem if the blind policy is chosen carefully.

))1/(log( pe


Improvement Factor of SRPT against the Best Blind Scheduling Policy

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Utilization (p)

Impr

ovem

ent F

acto

r


Background and Motivation A GI/GI/1 queuing system A GI/GI/1 queuing system

Inter-arrival times are i.i.d rvs, A, taken from a General Inter-arrival times are i.i.d rvs, A, taken from a General distribution Gdistribution Gaa

Job Sizes are i.i.d rvs, S, taken from a General distribution GJob Sizes are i.i.d rvs, S, taken from a General distribution Gss

Different from a G/G/1 system. Different from a G/G/1 system. GGss and G and Gaa specify a GI/GI/1 system completely. specify a GI/GI/1 system completely.

Therefore, the optimum blind policy Therefore, the optimum blind policy A (A (which which minimizes the average sojourn time) only depends minimizes the average sojourn time) only depends on the respective distributions.on the respective distributions.

OptOpt(G(Gaa, G, Gss) = Average sojourn time under the ) = Average sojourn time under the optimum blind policy. optimum blind policy.


Difference in the analyses of M/M/1 and G1/G1/1 queuing systems

In a M/M/1 system all blind policies are In a M/M/1 system all blind policies are identical as far as the average sojourn time is identical as far as the average sojourn time is concerned. In particular any blind scheduling concerned. In particular any blind scheduling policy has average sojourn equal topolicy has average sojourn equal to

So the question reduced to determining the So the question reduced to determining the average sojourn time under SRPT. Bansal average sojourn time under SRPT. Bansal showed that for exponential job sizes the showed that for exponential job sizes the average sojourn time under M/M/1/SRPT isaverage sojourn time under M/M/1/SRPT is

)1(

1

))1

ln()1(

1((

e

p


So why is the analysis of a G1/G1/1 queuing system different and more

difficult? Unlike the case in M/M/1, all blind policies are Unlike the case in M/M/1, all blind policies are

no more identicalno more identical !!! !!! No single blind policy is optimal for e.g.No single blind policy is optimal for e.g.

In an M/G/1 system FCFS is optimum for job size In an M/G/1 system FCFS is optimum for job size distributions with increasing failure rates.distributions with increasing failure rates.

Foreground Background (FB) [that at time works on Foreground Background (FB) [that at time works on the job with the least attained service] is optimum the job with the least attained service] is optimum for job size distributions with for job size distributions with decreasing failure ratesdecreasing failure rates


So why is the analysis of a G1/G1/1 queuing system different and more

difficult?Same blind policy can have very different Same blind policy can have very different

behaviours for different job sizes.behaviours for different job sizes.While FB has average sojourn time While FB has average sojourn time

when job sizes are exponentially distributed, when job sizes are exponentially distributed, thethe

Average sojourn time under FB varies as Average sojourn time under FB varies as when the job sizes have the pareto when the job sizes have the pareto distribution withdistribution with

)1

log(e

21

)1(

1


What to do? So why is the analysis of a G1/G1/1 queuing system so

difficult?Furthermore, As the analytic Furthermore, As the analytic

expression for average sojourn time expression for average sojourn time under an arbitrary GI/GI/1 system is under an arbitrary GI/GI/1 system is not known and moreover no analytic not known and moreover no analytic expression for expression for OptOpt(G(Gaa, G, Gss) for general ) for general GGaa, G, Gss is known, therefore we cannot is known, therefore we cannot adopt the approach that we used in adopt the approach that we used in the case of M/M/1the case of M/M/1


Hmmm…


Plan



Competitive Analysis Let A(I): Total sojourn time when the instance is executed Let A(I): Total sojourn time when the instance is executed

according to the algorithm A.according to the algorithm A. We say that a deterministic algorithm has a competitive ratio We say that a deterministic algorithm has a competitive ratio

c(n) if c(n) if

The definition of competitive ratio is quite strict, The definition of competitive ratio is quite strict, A more useful notion is that of a randomized algorithm.A more useful notion is that of a randomized algorithm.

)()(

)(max

|:|nc

ISRPT

IAnII

Worst Case Ratio over all input instances of size atmost n achieved by A and the optimum cost on that instance


Competitive Analysis

The competitive ratio of a randomized algorithm is The competitive ratio of a randomized algorithm is defined asdefined as

The crucial thing to observe is that there is no The crucial thing to observe is that there is no probabilistic assumption on the input instance.probabilistic assumption on the input instance.

The input instance is still chosen adversarially to The input instance is still chosen adversarially to maximize the ratio. maximize the ratio.

)(

)]([max

|:| IOpt

IAEnII


Competitive Analysis Randomized algorithms can have Randomized algorithms can have

significantly better competitive ratios than significantly better competitive ratios than deterministic algorithms.deterministic algorithms.

NOTE: The notion of the performance of a NOTE: The notion of the performance of a randomized algorithm is dual to the notion of randomized algorithm is dual to the notion of the average case performance of a system the average case performance of a system like GI/GI/1like GI/GI/1

While the former deals with the performance While the former deals with the performance over a distribution over algorithms on a fixed over a distribution over algorithms on a fixed input instance, the latter deals with the input instance, the latter deals with the performance of a fixed algorithm on a performance of a fixed algorithm on a distribution over input instances. Can we distribution over input instances. Can we relate the two?relate the two?


Yao’s Minimax Theorem (Contrapositive) [Minimization

Problem] Suppose a cost minimization problem P has a c(R,n) Suppose a cost minimization problem P has a c(R,n)

competitive randomized online algorithm R for competitive randomized online algorithm R for request sequences of length atmost n. Let request sequences of length atmost n. Let distribution y(j) be any distribution over request distribution y(j) be any distribution over request sequences of length atmost n. Then, sequences of length atmost n. Then,

In particular, for any distribution over the input In particular, for any distribution over the input instances, if we consider the algorithm Ainstances, if we consider the algorithm Aii that has that has the best average case performance on this the best average case performance on this distribution, then this performance is no worse than distribution, then this performance is no worse than c(R,n) times the performance of the optimum c(R,n) times the performance of the optimum algorithm.algorithm.

)]([)()]([min )()( jjyjijyi

OptEncAE


Plan



Some known results for competitive ratio of blind scheduling algorithm For the problem of minimizing the average For the problem of minimizing the average

sojourn time:sojourn time:Motwani, Phillips and Torng showed that no blind Motwani, Phillips and Torng showed that no blind

deterministic scheduling algorithm can have a deterministic scheduling algorithm can have a competitive ratio better than competitive ratio better than

The same people showed that any randomized The same people showed that any randomized algorithm has a competitive ratio atleast algorithm has a competitive ratio atleast

)( 3/1n

)(logn


BreakThrough…. In a significant breakthrough, Pruhs and In a significant breakthrough, Pruhs and

Kalayanasundaram gave a non-trivial randomized Kalayanasundaram gave a non-trivial randomized algorithm that they called RMLF [5] and proved that algorithm that they called RMLF [5] and proved that it has a competitive ratio ofit has a competitive ratio of

Later it was shown that RMLF is infact anLater it was shown that RMLF is infact an

competitive randomized algorithm and hence the competitive randomized algorithm and hence the best possible upto constant factors. In other words,best possible upto constant factors. In other words,

“ “ There is a universal constant r, such that There is a universal constant r, such that for any scheduling instance with atmost n for any scheduling instance with atmost n jobs, the expected total sojourn time under jobs, the expected total sojourn time under RMLF is atmost rlogn times that under SRPT” RMLF is atmost rlogn times that under SRPT”

)loglog(log nn)(logn


Analysis

SRPTsa TEC

dGGOpt ][)1

2log(.),(

6/1

6

6

6

6

][

][

][

][

SE

SE

AE

AEC

Theorem 1: GI/GI/1 system with inter-arrival Theorem 1: GI/GI/1 system with inter-arrival distribution Gdistribution Gaa and service distribution G and service distribution Gss, , there exists a universal constant d such thatthere exists a universal constant d such that

Where C is a bound on the sixth coefficient of Where C is a bound on the sixth coefficient of variation of Gvariation of Gaa and G and Gss, that is, that is


Analysis Lemma 1: Given any probability distribution on the Lemma 1: Given any probability distribution on the

input instances with atmost n jobs, there is some input instances with atmost n jobs, there is some deterministic algorithm the expected total sojourn deterministic algorithm the expected total sojourn time of which is atmost r(logn) times that under time of which is atmost r(logn) times that under SRPTSRPT

If we take each busy period in a GI/GI/1 system as a If we take each busy period in a GI/GI/1 system as a separate instance then we can think of a GI/GI/1 separate instance then we can think of a GI/GI/1 system as defining a probability distribution on input system as defining a probability distribution on input instances.instances.

Can we apply Lemma1 ?Can we apply Lemma1 ? No!No! Because we cannot get a bound on the number of Because we cannot get a bound on the number of

jobs a busy period might contain !!jobs a busy period might contain !!


Analysis (Basic Idea)

For every n, there is a non-zero fraction of busy For every n, there is a non-zero fraction of busy periods that contain more than n jobs.periods that contain more than n jobs.

Can we come up with a Can we come up with a n n such that most of the such that most of the action happens with those many jobs in a busy action happens with those many jobs in a busy period?period?

The analysis is based on this premise.The analysis is based on this premise. A busy period has about 1/1-p jobs on the average, A busy period has about 1/1-p jobs on the average,

hence the hence the lognlogn factor in lemma 1 should essentially factor in lemma 1 should essentially be be log(1/1-p)log(1/1-p)

Most of the action essentially happens in busy Most of the action essentially happens in busy periods with atmost Cperiods with atmost C66/(1-p)/(1-p)66 jobs and hence lemma jobs and hence lemma 1 can be replaced by this number.1 can be replaced by this number.


AnalysisThe busy periods are independent of the

actual scheduling policy involved.Let

B = set of all possible busy periodsM = measure induced by GI/GI/1 system on B

TA(B) = Total sojourn time incurred when A is executed during the busy period B

n(B)= number of jobs in BAverage sojourn time of a job under an

algorithm A can be expressed as

)]([

)]([][

BnE

BTETE

M

AMA


Analysis For any GI/GI/1 system, E[n(B)] is identical for every

work-conserving scheduling policy. Thus it suffices to compare the expected total sojourn time of a busy period for two scheduling policies, in order to compare the average sojourn time under them.

A busy period is called bad if it contains more than N0=24C6/(1-p)6.

Define a process P as follows: Whenever there is a bad busy period of length B, we replace it with another busy period with n(B) “dummy” jobs of length 0.

Let M’ be the measure induced by the new process on the busy periods. By construction: EM’[n(B)] = EM[n(B)].

Now lemma 1 can be applied..


AnalysisKey point: Key point: The only contribution to the The only contribution to the

expected total sojourn time under the measure M’ expected total sojourn time under the measure M’ is due to the busy periods that contain atmost Nis due to the busy periods that contain atmost N00 jobs.jobs.

Let Opt’ = blind algorithm A that Let Opt’ = blind algorithm A that minimizes minimizes EM’[TA(B)]


Few Lemmas…

)]([log)]([ '0'' BTENrBTE SRPTMOptM

)]([2)]([)]([ '' BTEBTEBTE AMAMAM

Lemma 2 (follows easily from lemma 1):

Lemma 3 [For any work-conserving algorithm A, uses lemma 4,5]

Lemma 4: Let Sn=X1+X2+….+Xn where Xi are independent and identically distributed according to the random variable X. Then, for epsilon > 0,

366

6

][

]])[[(]][|][Pr[|

nXE

XEXEXnEXnESn


Few Lemmas…

6/16

6

6

6

)][

][

][

][(

SE

SE

AE

AEC

)1( p

Lemma 5: Given a GI/GI/1 system, let A and S be the random variables representing arrived time and service time respectively. Let and

36

6

36

6

)(

4])(Pr[

3])(Pr[

t

CtBl

n

CnBn


Proof of lemma 3 Since M and M’ differ only on bad busy periods, Since M and M’ differ only on bad busy periods,

therefore for any work-conserving algorithmtherefore for any work-conserving algorithm

To show the other side of the inequality we need to To show the other side of the inequality we need to upper bound the contribution due to bad busy upper bound the contribution due to bad busy periods in periods in EM[TA(B)]

In a busy period of length l and consisting of n jobs, the total sojourn time of the jobs involved can be atmost nl, irrespective of the choice of the algorithm.

Thus we need to upper bound:

)]([)]([' BTEBTE AMAM

0 0

])( and )(Pr[Nn t

dtnBntBltn0

6

63

N

C


Proof of Theorem 1

)]([2)]([ ''' BTEBTE OptMOptM

Consider the optimum blind policy Opt’ that Consider the optimum blind policy Opt’ that minimizes Eminimizes EM’M’[T[TOptOpt’(B)]. It follows by lemma 3’(B)]. It follows by lemma 3

By definition:By definition:

By lemma 2:By lemma 2: Which by lemma 3 impliesWhich by lemma 3 implies

Combining the above we getCombining the above we get

)]([),( ' BTEGGOpt OptMsa

)]([log)]([ '0'' BTENrBTE SRPTMOptM

)]([log)]([ 0'' BTENrBTE SRPTMOptM

)]([log2)( 0, BTENrGGOpt SRPTMsa


Summary The paper proves that for any GI/GI/1 queuing

system, the performance of the best blind policy for that system achieves average sojourn close to the best possible by any other (non-blind) policy.

The result, however, does not give a construct way to produce the optimum (deterministic) blind scheduling policy for an arbitrary GI/GI/1 system.

The dependence on the sixth coefficient of variations of inter-arrival times and sizes is the artifact of the analysis. The finiteness of the sixth moments allows to show that the probability a busy period has more than n jobs decays at least as 1/n3.


References[1] On the average sojourn time under M/M/1/SRPT, Nikhil Bansal, to

appear in Operations Research Letters, Volume 33, 2, March 2005, 195-200.

[2] Achievable sojourn times by non-size based policies in a GI/GI/1 queue, Nikhil Bansal, submitted for publication.

[3] D.R. Smith. A new proof of the optimality of the shortest remaining processing time discipline. Operations Research, 26:197–199, 1976.

[4] L.E. Schrage. A proof of the optimality of the shortest processing remaining time discipline. Operations Research, 16:678–690, 1968.

[5] R. W. Conway, W. L. Maxwell, and L. W. Miller. Theory of Scheduling. Addison-Wesley Publishing Company, 1967.

[6] R. Motwani, S. Phillips, and E. Torng. Nonclairvoyant scheduling. Theoretical Computer Science,130(1):17–47, 1994.

[7] B. Kalyanasundaram and K. Pruhs. Minimizing flow time nonclairvoyantly. In IEEE Symposium on Foundations of Computer Science, pages 345–352, 1997.

[8] A. C-C. Yao. Probabilistic computations: Toward a unified measure of complexity (extended abstract). In IEEE Symposium on Foundations of Computer Science (FOCS), 1977.


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CS 3150 Presentation 1 Ihsan Ayyub Qazi “Achievable sojourn times in M/M/1 and GI/GI/1 systems: A...

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