SYNOPSIS

THE PERFORMANCE PREDICTION OF MACHINING

SYSTEMS USING QUEUE THEORETIC APPROACH

Submitted for the Degree of

DOCTOR OF PHILOSOPHY

IN

MATHEMATICS

BY

KUMAR PAL

Under the Supervision of

Dr. S. S. YADAV

M. Phil, Ph. D

DEPARTMENT OF MATHEMATICS,

NARAIN P.G. COLLEGE, SHIKOHABAD

Dr. B. R. AMBEDKAR UNIVERSITY, AGRA-282002.

2017

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PURPOSE OF STUDY

The objective of present study is based on the philosophy that interaction should

evolve around industrial problems which mathematicians may be able to solve in „real

time‟. In this way both industry and mathematics will be benefited; industry by increase of

mathematical knowledge and ideas brought to bear upon their concerns and mathematics

through the infusion of exciting new problems. There is a substantial number of industrial

problems to which tools of Operations Research are applicable. New techniques based on

operational research have become popular as new technology comes on line.

With a remarkable advancement in science and technology, machines have added

to almost every field in different frame works. The machines have become an integral part

of human civilization, as the life has become comfortable due to industrial development.

Our total dependence on machines can be attributed to the fact that the machining systems

today have entered into every sphere of human activities. Recent times have witnessed the

emergence of high tech machining systems required in today‟s world of competitions in

the field of manufacturing, production, telecom and IT sectors, etc.. Queueing analysis of

machining systems helps in evaluating the measures of effectiveness in such systems,

which are concerned with services or manufacturing.

The machines are unreliable and thus always prone to failures. These failed machines

are queued up for repair under the supervision of caretaker which also experience the

waiting phenomenon. Thus, the problem falls in the category of queueing theory. The

machine failures obviously affect the organization in terms of production, money, goodwill

in the market etc. For the smooth functioning of the machining system and to compensate

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the loss due to machine breakdowns, the organizations have to keep spares along with the

arrangement of repair facility. Now the role of the organization‟s system designer comes

into the picture whose task is to suggest an optimal combination of spares and repairmen,

which the organization should employ so that the expenses of keeping these facilities

should not affect the system‟s profit too much.

For this purpose, the mathematical modeling (based on queueing theory) of

machine repair problem plays an important role which helps to analyze and predict the

performance of the machining system quantitatively. This gives an opportunity to system

designers to reduce the machine interference due to failure of units of machining system to

a reasonable extent. Therefore, foreseeing a wide scope to work in this important and

interesting field, it motivates us to innovate and work in this particular field.

Our motivation to study the machining systems comes from the serious problems

being faced by the industries due to machine failures and the resulting slow progress in

technological development throughout the world. Industries are making efforts to cope up

with the increasing complexity in the machining systems, fast changing technology and

varying market patterns. Technological development and requirement of modern society

are competing with each other. Therefore, with the development of modern technology and

its role in industrial development, it is quite important to keep a pace with recent

researches and developments in order to quantify the performance and effectiveness of

various systems such as computer and communication system, manufacturing/production

system, etc. Now-a-days, it has been widely applied in various industrial service systems

including the maintenance systems, repairing of the machines, batch processing, inspection

stations, computer facilities etc. It has also been applicable in social service systems such

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as various health care systems, public transportation systems and reticulation of mass

distribution systems.

The aim of present study is to analyze the performance of machining systems

involved in some manufacturing/production or service systems. We shall attempt to

examine and provide solutions to the problems being faced due to machine failures.

Classical queueing techniques as well as numerical techniques will be used to analyze the

queueing models used for various problems of machining systems. The performance

measures in each case will be established, which may be helpful to the system designers

and decision makers to plan a strategy for the smooth functioning of the system. The

optimal control policy for some machine interference problems will be suggested so as to

ensure smooth running of the system at optimum cost under techno-economic constraints.

We hope that our study would not only help in predicting the performance and

improving the performability of the existing systems, but also support the system engineers

in creating optimum designs of the concerned architecture of machining system. Our work

will be at par with other Universities/Institutions in the country and elsewhere. The models

analyzed under proposed study will also be of applied nature and will attract the attention

of mathematicians and operations research scientists. The study will be useful as decision-

making tools to achieve maximum output subject to resource constraints.

(Dr. S.S. Yadav) (Kumar Pal)

Supervisor Research Scholar

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[A] Title: PERFORMANCE ANALYSIS OF MACHINING SYSTEMS USING

QUEUE THEORETIC APPROACH

[B] PRESENT STATE OF KNOWLEDGE

With the fast growing technology, the industries are struggling to muddle through

the mounting complexity in the machining systems due to computerization, automated

manufacturing flexibility in demand, fast change in technology, market patterns, etc..The

technological advancement and call for modern society are contesting against each other.

Therefore, with the growth of modern society and its role in industrial development, it is

indispensable to be in touch with recent research and development, which can provide a

right direction to design highly sophisticated machining systems.

Queueing theory has emerged as a powerful mathematical tool to analyze the

machining systems in different frameworks because of its ability to simulate most real time

systems. Machine repair problems are applicable in all branches of engineering,

manufacturing system/production processes, information system, transportation system,

computer and communication networks, etc.

The machining systems considered in the present study provide some tactical ideas

concerning the concept of maintenance and replacement policy via repair facility and

standby support. Here we present a survey concerning an appraisal of machine repair

problems (MRPs) highlighting the historical advancement of queueing models. The surrey

crops up historically, beginning with the developments in 1985 when the first published

review on machine interference models by Stecke and Aronson (1985) came into

existence. Machine interference problems have been studied by many queue theorists for a

variety of congestion situation. They have used different assumptions and approaches for

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this purpose. The early notable contribution in this area were due to Palm (1947), Benson

and Cox (1951), Naor (1956), Ferdinand (1971), Albright (1980), Karmeshu (1990) and

many more.

It is appropriate to review the important research works done by researchers in the

area of performance analysis of machine repair problems using queueing theory. Wang

(1995) proposed an approach to cost analysis of the machine repair problem with two types

of spares and service rates. Gupta (1997) examined machine interference problem with

warm spares, server vacations and exhaustive service. Jain and Dhyani (1999) studied

transient analysis of M/M/C machine repair problem with spares. Arulmozhi (2002)

performed the reliability of an M-out-of-N warm standby system with R repair facilities.

Jain and Sharma (2004) considered the M/M/C interdependent machining system with

mixed spares and controlled rates of failure and repair. Wang et al. (2006) suggested the

cost benefit analysis of series system with cold standby components and a repair service

station. Jain and Upadhyaya (2009) investigated threshold N- policy for degraded

machining system with multiple types of spares and multiple vacations. A stochastic model

to determine the performance of an unreliable flexible manufacturing cell (FMC) with

common cause failure under variable operational conditions was developed by Maheshwari

et al. (2010). Maheshwari et al (2010) examined diffusion approximation for M-out-of-m

machining system with group failure and repair and k type of warm spares. Cost analysis

of a machine repair problem with standbys, working vacation and server breakdown is

given by Jain and Preeti (2014). Jain et al. (2016) analyzed a time-shared machine repair

problem with mixed spares under N-policy.

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Long waiting lines at the repair counter may irritate the caretaker of the machines,

which may force him to balk or renege from the system. This issue has been considerably

incorporated in various machine repair problems. Many mathematicians have shown a

keen interest in this area also. Ancker and Gafarian (1963) considered some queueing

problems with balking and reneging. Wang and Ke (1999) did the cost analysis of the

M/M/R machine repair problem with balking, reneging and server breakdowns. Shawky

(2000) developed the M/M/C/K/N machine interference model with balking, reneging and

spares. Wang and Ke (2003) provided probability analysis of a repairable system with

warm standby plus balking and reneging. Wang et al. (2007) examined profit analysis of

the M/M/R machine repair problem with balking, reneging and standby switching failures.

Jain and Sharma (2008) considered finite capacity queueing system with queue dependent

servers and discouragement. Maheshwari et al. (2010) studied machine repair problem

with k type warm spares, multiple vacations for repairmen and reneging. Machine repair

problem with mixed spares, balking and reneging was investigated by Maheshwari and

Shazia (2013). Sharma (2015) established machine repair problem with spares, balking,

reneging and N-policy for vacation.

The appearance of queues or waiting lines is a universal phenomenon, thus making

the queueing theory applicable in almost every walk of life; the area of repairable

machining system is particularly suited to such an application. The machines along with

the spares only cannot fully serve the purpose of increasing waiting lines. To reduce the

exceeding workload, the incorporation of additional repairmen along with the permanent

repair facility is suggested. The research work including this idea has also been carried out

by many researchers. Knessl (1991) observed the transient behaviour of repairman

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problem. Wang and Hsieh (1995) discussed optimal control of a removable and non-

reliable server in Markovian queueing system with finite capacity. Jain (1996) did the

reliability analysis of M/M/C repairable system with spares and additional repairmen. Jain

(1998) examined M/M/R repair problem with spares and additional repairmen. Jain et al.

(2000) investigated M/M/C/K/N machine repair problem with balking, reneging, spares

and additional repairmen. Glazebrook et al. (2005) proposed index policies for the

maintenance of a collection of machines by a set of repairmen. Sharma (2012) explored

machine repair system with warm spares and N-policy vacation of a repairman.

Machine repair model with standbys, discouragement with vacationing unreliable

repairmen was developed by Singh and Maheshwari (2013).

The general problem of reliability of machine repair problem holds a very wide

range of problems intended to ensure and maintain a high reliability both of individual

elements and the entire system as a whole. The problem has constantly attended the

engineering and technical progress and has always found reasonable solution at the level of

technical possibilities. Various mathematicians have worked in the field of reliability

analysis of machining systems. Availability and reliability of an n-unit warm standby

redundant system was examined by Kalpakam and Hameed (1984). Availability and mean

life time prediction of multistage degraded system with partial repairs was measured by

Pham et al. (1997). Jain et al. (2004) studied reliability analysis of redundant repairable

system with degraded failure. An artificial neural network for modeling reliability,

availability and maintainability of a repairable system was proposed by Rajpal et

al. (2006). Optimal design of a maintainable cold standby system was specified by

Haiyang et al. (2007). Y and Meng (2011) investigated reliability analysis of a warm

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standby repairable system with priority in use. Reliability analysis of machining systems

by considering system cost is investigated by Zhang and Liling (2015). Zhou et al. (2017)

developed a service facility maintenance model with energy-delay tradeoff. The

maintenance model integrates individual control chart with queueing approach.

[C] BROAD OUTLINES OF THE WORK

Our investigation is particularly focused on the analysis of machine repair problems

in different frameworks. For the modeling of the problems and performance prediction, the

techniques to be incorporated will be either classical queueing techniques or numerical

techniques. The chapter wise organization of the thesis as follows:

(i) General Introduction

(ii) Review of Literature

(iii) Machine repair problem with spares and state dependent rates

(iv) Machine repair problem with spares and N-Policy vacations

(v) Machine repair problem with spares and additional repairmen

(vi) Machine repair system by generating function technique

(vii) Multi-repairmen machine repair problem with M operating units and k type of

warm spares

(viii) Machine repair problem in production system with spares and server vacations

The relevant references will be given at the end of thesis.

The nomenclature of aforesaid chapters is tentative. It may be altered in accordance

our investigation.

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Mathematical Formulation:

A basic queuing system is formed from three general elements.

(i) The arrival process of users in the system;

(ii) The order in which users obtain access to the service facility, one they join the

queue.

(iii)The service process and departure from the system.

** Arrival refers to the average number of customers who require service within a

specific period of time.

** Customers can be people, work -in-process inventory, raw materials, incoming digital

messages, or any other entities that can be modeled who are to wait for some

process to take place it may be infinite or finite also is said size of queue.

** A server can be human worker, a machine, or any other entity that can be Processor

as executing some process for waiting customers.

While analyzing a queuing system we can identify some basic components of it. Namely,

Queue Discipline refers to the priority system by which the next customer to receive

service is selected from a set of waiting customers. One common queue discipline is first-

in-first-out, or FIFO. Scheduling adopt by server refer, how to work server. Server follows

the following discipline. The server in accepting customers for service. In this context, the

rules as “first-come-first-served” (FCFS), “last-come-last-served” (LCLS), and “random

selection for service” (RS) are self explanatory. Others such as “round robin” and “shortest

processing time” may need some elaboration. In many situations customers in service

classes get priority in service over others. There are many other queue disciplines which

have been introduced for sufficient operation of computers and communication systems.

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Also, there are other factors of customer behavior such as balking, reneging, and jockeying

that require consideration as well.

Service rate (or service capacity) refers to the overall average number of customers a

system can handle in a given time period.

Utilization refers to the proportion of time that a server (or system of servers) is busy

handling customers.

Stochastic Processes are systems of events in which the times between events are random

variable. In queuing models, the patterns of customer arrivals and service are modeled as

stochastic process based on probability distribution.

Kendal (1953) classified the queues and according to this description following symbols

are used in queuing problems.

* M: Exponential

* D: Deterministic (constant)

* kH : Hyper exponential with parameter k (mixture of k exponentials)

* kE : Erlang with parameter k

* G: General (all)

A conference on standardization of notation in queuing theory agreed in 1971 to extend

Kendall‟s notation. It is as A/B/m/K/N/Z.

where,

* A indicates the distribution of interarrival times,

* B denotes the distribution of the service times,

* m is the number of servers,

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* K is the capacity of the system, that is the maximum number of customers staying at the

facility (sometimes in the queue),

* N denotes the number of sources,

* Z refers the service discipline

The some well known basic formulas of queuing theory is as follows:

Erlang’s Formulas: Erlang‟s task can be formulated as, what fraction of incoming calls is

lost because of the busy line at the telephone exchange office. Of course, the answer is not

so simple, since we first should know the inter-arrival and service time distributions. After

collecting data Erlang verified that the Poisson-process arrival and exponentially

distributed service appropriate mathematical assumptions. He considered the M/M/n/n and

M/M/n cases, that the system where the arriving calls lost because all the servers are busy,

and where the calls have to wait for service, respectively. Assuming that the arrival

intensity is he derived the formulas for loss and delay systems, called Erlang B and C

ones, respectively.

Denoting , the steady state probability that an arriving calls is lost can be obtained

in the following way

0

! ( , )

!

n

n nn

k

nP B n

k

where ( , )B n is well-known Erlang B-formula, or loss formula. It can be easily seen that

the following recurrence relation is valid

( 1, )

( , ) , 2,3,.............( 1, )

B nB n n

n B m

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(1, )1

B

.

Similarly, by using the b-formula the steady state probability that an arriving customer has

to wait can be written as

( , )

( , ) ,(1 ( , )

nB nC n

n B n

which is called Erlang C-formula, or Erlang‟s delay formula? It should be mentioned that

b-formula is insensitive to the service time distribution, in other words it remains valid for

any service time distribution with mean1 .

Little’s Law: It is also known as Little‟s result, or Little‟s theorem, is perhaps the most

widely used formula in queuing theory was published by j. little in 1961. It is simple to

state and intuitive, widely applicable, and depends only on weak assumptions about the

properties of the queuing system.

A conservation law that applies to the following general setting:

Input System Output

Input: Continuous flow or discrete units.

System: Boundary is all that is required.

Output: Same as input, call throughput.

Two possible scenarios:

(iv) System during a “cycle” (empty empty, finite horizon);

(v) System in steady state/in the long run (for example, over many cycles).

Quantities that are related via Little‟s law (long-run averages, or time-averages):

(vi) = rate at which units arrive (=long-run average rate at which units depart) =

throughput-rate, whose units are , #quantity time unit or time unit ;

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(vii) L = inventory/quantity/number in the system

(e.g., WIP: Work-In-Process, Customers)

(viii) W = time a unit spends in the system = throughput time

Thus, the Little‟s Law says that, under steady state conditions, the average number of items

in a queuing system ( L ) equals the average rate ( ) at which items arrive multiplied by

the average time (W ) that an item spends in the system, and algebraically expressed as:

L W

M/M/1 model

In this model the arrival times and service rates follow Markovian distribution or

exponential distribution which are probabilistic distributions, so this is an example of

stochastic process. In this model there is only one server. The important results of this

model are:

1. Average number of customers in the system 1

L

2. Average number of customers in the system 2

1qL

3. Expected waiting time in the system 1

(1 )L

W

.

4. Expected waiting time in the queue 2

(1 )( ) ( )

q

q

LW

.

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The PASTA Theorem

In this section we investigate the following question: when we look at the number

of customers in a queuing system at “random points” in time, do then have all “random

points” the same properties or do there exist points where the results differ fundamentally?

The answer is: yes, it makes a difference how you choose the points. A simple

example illustrating this is the D/D/1 queue (with fixed interarrival times of seconds and

fixed service times of seconds, < . If we now choose “randomly” the arrival times

of new customers as our random points it is clear that we will see zero customers in the

system every time (since customers are served faster than they arrive). If we choose the

random times with uniformly distributed time differences there is always a probability of

> 0 to find a customer in the system. Thus, the final result of this section is that

arrival times are not random times. One important exception from this rule is when the

arrival times come from a Poisson process (with exponentially distributed interarrival

times). In this case the arrival times are “random enough” and a customer arriving to a

queue in the steady state sees exactly the same statistics of the number of customers in the

system as for “real random times”. In a more condensed form this is expressed as Poisson

Arrivals See Time Averages, abbreviated with PASTA. This property comes also from the

memoryless property of the exponential distribution.

[D] PRELIMINARY WORK DONE ON THE LINE

I have gone through the literature and consulted some reference books and

textbooks related to my research topic. Some worth mentioning books consulted are as

follows:

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Operations Research: Taha, H.M.

Introduction of Queueing Theory: Cooper, R.B.

Fundamentals of Queueing Theory: Gross, D. and Harris, C.M.

Queueing Systems, Vol. 1 & 2: Klienrock, L.

Stochastic Processes: Medhi, J.

Probability & Statistics with Reliability, Queueing : Trivedi, K.S.

& Computer Science Applications

Markovian Queues: Sharma, O.P.

Reliability of Engineering System - Principles and : Ryabinin, I.

Analysis

Performance Analysis of Manufacturing Systems: Altiok, T

Springer Series

Queueing Networks with Blocking- Exact and: Perros, H.G.

Approximate Solutions

I am in constant touch with the following research journals:

Opsearch

Operations Research Letters

European Journal of Operations Research

Applied Mathematical Modeling

International Journal of Management & Systems

International Journal of Pure and Applied Mathematics

Journal of Operations Research Society

Journal of Indian Statistical Association

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Queueing Systems

Computers and Operations Research

These days the internet facility is quite beneficial for the researchers to keep a pace

with latest updates regarding the developments in their respective research fields. It has

now become an easy task to search research articles, their abstracts, references and other

necessary information. The most common websites consulted are www.sciencedirect.com,

www.inderscience.com and www.springerlink.com.

After surveying the literature, I am working on the following research problems.

Queueing system with second optional service.

Machine repair problem with mixed spares, discouragement and additional

repairmen.

The transient analysis of Markovian machine repair model with balking and

reneging by using birth-death process.

[E] Proposed research design, tools, methodology, hypothesis and tentative

conclusions:

The hypothesis of the study will be based on optimization. The

determination of highest or lowest value over some given range is called

optimization. A queuing model can be maximized to profit or minimized for loss.

The word optimization is used as general term in either case. Cost and optimization

can be classified into three categories:

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(i) Cost and optimization engineering: It deals with the topics of engineering

economy.

(ii) Operations research: It is mathematical oriented and is concerned with advance

techniques for solving complicated optimization problems using mathematical

models.

(iii) Management science: It is relatively new category originating from operations

research and is aimed at overall policy and decision-making.

(iv) The queuing models have been classified into two general types

Though queuing theory provides us a scientific method of understanding the queues

and solving such problems, the theory has certain limitations which must be understood

while using the technique and in conjunction with the other decision analysis methods like

simulation and regression. Most of the limitations are the basic assumptions for application

of queuing models. Some of the limitations of the queuing models are enumerated below:

(i) Based on assumption that service time is known.

(ii) Assumes steady state.

(iii) Service times are independent from one another.

(iv) Service rate is known.

(v) Service rate is greater than arrival rate.

(vi) Service time is described by negative exponential probability distribution.

(vii) The waiting space for the customer is usually limited.

(viii) The arrival rate may be state dependent.

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(ix) The arrival process may not be stationary.

(x) Services may not be rendered continuously.

(xi) The queuing system may not have reached the steady state. It may be, instead, in

transient state.

(xii) Mathematical distributions, which we assume while solving queuing theory

problems, are only a close approximation of the behavior of customers, time

between their arrival and service time required by each customer.

(xii) Most of the real life queuing problems are complex situation and very difficult to

use queuing theory technique, even then uncertainty will remain.

(xiii) Many situations in industry and service are multi-channel queuing problems. When

a customer has been attended to and the service provided, it may still have to get

some other service from another service and may have to fall in queue once again.

Here the departure of one channel queue becomes the arrival of the other channel

queue. In such situations the problem becomes still more difficult to analyze.

Queuing model may not be the ideal method to solve certain very difficult and

complex problems and one may have to resort to other techniques like Monte-Carlo

simulation method.

Some approaches used for the solution of mathematical models under consideration

are as follows:

Analytical techniques:

Markovian birth death process

Recursive technique

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Product type method

Supplementary variable technique

Generating Function Method

Numerical techniques:

Runge-Kutta technique

Successive Over Relaxation method (SOR)

Matrix Method, Matrix geometric method

Neuro-Fuzzy approach

To plan a fruitful strategy for the smooth functioning of the system, some

performance measures for the system under study will be obtained by using suitable

techniques which may be helpful to the system designer and decision makers. Worth

mentioning are the following:

Reliability/Availability

State probabilities.

Queue size distribution.

Expected number of spares

Expected number of repairmen

Expected number of failed machines in the queue/system

Cost function

Numerical illustration to validate the analytical results.

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To draw the conclusion, the method of plotting the graphs among different

parameters as well as interpretation for obtained results, may be of great use.

Recent computational techniques would use to solve the basic equation of the

problems. Proper programme will be prepared and existing programming will also be

applicable in some important case.

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