Chapter 1
Introduction to Queue Modelling
and Machine Repair Model
1.1
Introduction
Every moment we spent waiting for some type of service, we are part of a
queue. We wait in line in our cars in traffic jams or tool booths, we wait in line at
supermarkets to check out, we wait in line at barber shop or beauty parlor, we wait
in line at post offices etc. We, as customers, do not generally like these waits and
the managers of the establishments at which we wait also do not like us to wait,
since it may cost then business. Whey then is there waiting? The answer is, there
is facility for service available.
In the present chapter, we introduce queueing models and related some
definitions. We define machine interference problem which is finite source
queueing model.
Following are some queueing examples:
System
Customers
Server
Reception Desk
People
Receptionist
Hospital
Patients
Nurses
Airport
Aeroplanes
Runway
Road Network
Cars etc.
Traffic light
Grocery
Shoppers
Checkout
Station Computer
Jobs
CPU, Disk, CD
Table 1.1: Queueing examples.
1.2
Preliminaries of the Queueing System
Modern age is the information technology and e-governance age, require
innovations, that are based on modelling, analysing, designing, and finally
implementing new systems. The whole developing process as`sumes a well
organised team work of experts including engineers, computer scientists,
mathematicians, physicists etc. Modern info -communication networks are one
of the most complex systems where the reliability and efficiency of the
components play a very important role. For the better understanding of the
dynamic behaviour of the involved processes one has to deal with constructions
of mathematical models which describe the stochastic service of random
arriving requests. Queueing theory is one of the most commonly used
mathematical tool for the performance evaluation of such systems.
"Delay is the enemy of efficiency and waiting is the enemy of
utilization".
A flow of customers from infinite / finite population towards the service
facility forms a queue or waiting lines on accounts lack of capacity to serve
them all at a time. “The queue is the group of customer’s (people, tasks or
objects) waiting to be served”.
Though we are most aware of our own waiting time, queueing is for
most a problem for industry and government. The success of any organization
depends on maximizing the utilization of its resources. Every minute that an
employee spends waiting for another department and even minute that a job
spends waiting to be processed is money wasted. Thus, success of any
organization also depends on attracting and keeping customers. Every minute
that a customary spends waiting to be served translates into lost business and
lost revenue.
The basic components of queueing system are: queues, customers and
servers. The goal of the analysis of a queueing system is to finding analytical
expression for such performance measures as queue length, throughput and
2
utilization. Queueing is an aspect of modern life that we may encounter any
place and any time in our daily life. In the banks, people stand in a line in front
of the counters and wait for the service. Also, in the supermarkets, people wait
in the queue to pay for the goods. Although there are no people stand any line
in the barber shop, the customers will be served in the sequence they arrive.
The study of queueing theory or waiting lines is able to provide both a
theoretical background to the kind of system and the way in which the system
itself may be designed to provide some specified grade of service to its
customers. Generally, in queueing theory, a queueing model is used to imitate a
real queue situation. Also, the queueing behaviours can be analysed
mathematically by the performance measurement calculation. These measures
are very important since they are often relevant with the work efficiency or
economic losses of a real queueing system. In the queueing system, analysis is
more accurate and deterministic results are expected.
The first text book on the subject: Queues, Inventories and Maintenance
was written in [1958] by Morse,P. Satty, T.L., wrote his famous book:
Elements of Queueing Theory with Applications in [1961]. Kleinrock
completed his: Queueing Systems in [1975]. Queueing systems can be
modelled as network of queues, like waiting in line at a doctors clinic, railway
reservation centre, in a bank, a bus stop, etc. A queueing system can be
described as customers arriving for service, waiting for service, if the servers
are busy and leaving the system after own service being finished. Gross and
Harris [1985] studies Fundamentals of Queueing Theory. Little J.D.C. [1961]
and W.S. Jewell [1967] developed the basic understanding of a queueing
system and proof of the Little’s formula. Little's theorem states as: “The
average number of customers N can be determined by N = λt.
Where, λ = average customer arrival rate
And, t = average service time for a customer.
3
We explain some of the major characteristics of queueing systems which are
given below:
(i)
Arrival pattern of customers.
(ii)
Service pattern of servers.
(iii)
Customers and customer behaviour.
1.2.1 Arrival Pattern of Customers
The arrival pattern or input to a queueing system is often measured in terms
of the average number of arrivals per some unit of time (mean arrival rate) or by
the average time between successive arrivals (mean inter-arrival time).
1.2.2 Service Pattern of Servers
Service pattern can be described by a rate (number of customers served
per unit of time).
Note:
1. If the system is empty, the service facility is idle. Service may also be
deterministic or probabilistic.
2. The service rate may depend on the number of customers waiting for
service. A server may work fasten if he/she sees that the queue is building
up or conversely, he / she may get flustered and become less efficient. The
situation in which service depends on the number of customers waiting is
known as state-dependent service.
1.2.3 Customers and Customer Behaviour
(a) Customers: By customers, we mean the arriving units that require
some service to be performed. The customer may be of persons,
vehicles, machines, etc., queues stands for a number of customers
waiting to be served.
4
(b) Customer Behaviour
Some of the customer behaviours are: reneging, balking and jockeying.
Reneging is the act of leaving a queue before being served. Balking is
the act of not joining a queue upon arrival. Conceptually, the two concepts are
the same. The only difference is the timing of when the customer leaves (either
immediately in the case of balking or later in the case of reneging). Jockeying
is the act of switching from one queue to another.
Figure 1.1: Standard graphical notation for queues.
In above figure 1.1, we have shown a standard graphical notation for
queues. Until now, queueing systems have been well studied. For the shorthand
for describing queueing processes, Kendall’s [1953] notation was used. This
notation describes a single process as a series of symbols. We have presented
these notations in table 1.2.
1.3
Performance Measures in Queue
Let C be the number of identical servers and arriving unit requires
service from any of the server.
Define Pn (t) = Pr ob. {exactly n units in the system at time t}. Then
(i)
Ls (t) = Expected number of units in the system at time t is given as
follows:
∑
∞
nPn (t)
(1.1)
n =1
(ii)
L q (t) = Expected Number of units in a queue at time t given by:
5
∑
∞
n = c +1
(iii)
(n − c)Pn (t)
(1.2)
C(t) = Expected number of busy servers at time t is shown by:
c ∑ Pn (t ) + ∑ nPn (t )
(iv)
∞
c −1
n =c
n =1
(1.3)
C (t ) = Expected number of servers not busy at time
c − c(t ) = ∑ (c − n) Pn (t ).
∞
(1.4)
n =0
∑
c
(v)
Pn (t ) = Prob. {no arrival is waiting for service at time t i.e. no
n =0
queue}
Ls = Lq (t ) + c(t ).
(vi)
(1.5)
For steady state
Pn (t ) = lim Pn (t ).
t →∞
(1.6)
Ws = Expected waiting time in the system.
Wq = Expected waiting time in the queue.
1.3.1 Little’s Formula
(i)
Ls = λWs ;
Lq = LWq
(1.7)
Where, λ is an arrival rate of the system.
(ii)
Ws = Wq +
1
µ
(1.8)
Where, µ is service rate of the system.
(iii)
C=
λ
.
µ
(1.9)
6
1.4
Queueing System
Figure 1.2: Queueing system.
Steady-state (behaviour is independent of time t)
(1.10)
Pn (t ) = lim Pn (t ) = Pn
n →∞
Figure 1.3: Transient phase.
Generally, in many situations only first three symbols A / B / X are used,
because these are the three most important characteristics, the symbol Y is
omitted if there is no restriction and the queueing system is FIFO. With a single
queue, the most common performance measures are obtained as follows:
ρ :
Is the server utilization which describes the fraction of time that the
server is busy or the mean fraction of active serves, in the case of
multiple servers.
λ :
Is the throughput which describes the number of jobs where processing
is completed in a single unit of time.
Q :
Is the queue length which is the number of jobs waiting in the queue at a
given time.
7
w :
Is waiting time which is the time that the jobs spend in the queue
waiting to be served.
k :
Is the number of jobs in the system at a given time.
Pn :
Is the probability of a given number of jobs n in the system.
In the theory of queues almost every queueing system can be
represented by Kendall’s [1953] notation and analyzed as the above
performance measures.
1.4.1 Notations of Queueing Models
Due to wide range of applications, Statistical distributions
parameters and disciplines, Kendall, D.G [1953] represented Stochastic
Processes Occurring in the Theory of Queues and their Analysis by the
method of Imbedded Markov Chain and gave a short hand notation for
queueing systems. According to that notation, a queueing system is
described by the string A/B/X/Y/Z where:
A: Arrival distribution,
B: Service pattern,
X: Number of servers,
Y: System capacity, and
Z: Queueing discipline.
Standard symbols commonly used in queueing systems are presented
in Table 1.2.For example: M / D/ 5/ 50 / PRI: describes a queueing system
with exponential inter-arrival times, 5 servers each deterministic service
time, a system capacity of 50 places and a priority service discipline.
8
Note: 1. Exponential inter-arrival times directly relate to Poisson
Arrivals.
2. A system capacity of 50 places in a system with 5 servers
spacify a maximum queue size of 45.
3. All symbols are mandatory, as symbols Y and Z may be omitted
thus resulting in an abbreviated string A / B / X. In this case the
system capacity is unlimited and the queue discipline is FCFS. A
queuing system directed by M / M / 1 / ∞ / FCFS is generally
abbreviated by M / M / 1.
Characteristics
Symbol
Description
A – Inter-arrival distribution:
D
Deterministic
CK
Cox (K- phases)
EK
Enlarge (K -phases)
G
General
GI
General independent
GEO
Geometric (discrete)
HK
Hyper-exponential
M
Exponential (Markov)
ME
Matrix exponential
MAP
Markov arrival process
PH
Phase type
D
Deterministic
CK
Cox (K- phases)
EK
Enlarge (K -phases)
G
General
GI
General independent
B- Service time distribution:
9
KGEO
Geometric (discrete)
HK
Hyper-exponential
M
Exponential (Markov)
ME
Matrix exponential
MAP
Markov arrival process
PH
Phase- type
SM
Semi-Markov
X - Number of parallel servers:
1, 2, ..., ∞
Y - System capacity:
1, 2, .., ∞
Z –Queue- discipline:
FCFS
First come first server
LCFS
Last come first server
RSS
Random selection for service
PRI
Priority service
RR
Random robin
PS
Processor Sharing
GD
General
Table 1.2: Notations of queueing models.
1.5
Queue Discipline
Queue discipline is a rule, according to which customers are selected for
service when a queue has been formed. Most common queue discipline is the
FCFS or “First Come First Served” or FIFO “First in First Out” rule
under which the customers are serviced in the strict order of their arrivals.
Other queue disciplines are LIFO “Last in First Out, SIRO “Selection for
service In Random Order” and a variety of priority schemes according to
which a customer’s service is done in preference over some other customer’s
service.
Under priority discipline, the service is of two types:
(i)
preemptive
(ii)
non-preemptive
10
In preemptive discipline, the customers of high priority, are given
service over the low priority customers, while in non-preemptive discipline, a
customer of low priority is serviced before a customer of high priority. In case
of parallel channels “Fastest Server Rule” (FSR) is adopted, by parallel
channels, we mean a number of channels providing identical service facilities.
1.5.1 Cumulative Arrival and Departure Diagrams
The word cumulative means “resulting from accumulation”. A
cumulative diagram indicates how many customers have accumulated up-to
some point in time from an initial starting time. A cumulative arrival diagram
indicates that how many customers have arrived up-to a point in time and a
cumulative departure diagram indicates how many customers have departed
up-to a point in time.
Figure 1.4: Performance measures indicate whether system objectives and
goals are achieved.
Cumulative diagrams are important because they provide many of the
measures of performance in one simple picture. Now, in the next section 1.6,
we explain some major terms and definitions which are used in the theory of
queues or waiting lines which are given below:
11
1.6
Some Definitions
1.6.1 Poisson Distribution
The Poisson distribution is a probability distribution that arises from the
Poisson process. It is a discrete distribution i.e. its random variable is limited to
a set of distinct values. The Poisson distribution gets its name because it is the
probability distribution for the number of arrivals within any time period of a
Poisson process. This distribution expresses the probability of a given number
of events in a fixed interval of time and/or space, if these events occur with a
known average rate and independently of time, since the last event. Frank A.
Height [1967].
1.6.2 Characteristics of Poisson Distribution
(i)
The Poisson distribution is discrete and defined over the set of nonnegative integers.
(ii)
E[ N (t )] = λ t
(iii)
V [ N (t )] = λ t
(iv)
P[ N (t ) = 0] = e − λt
Where, E [ ] represents the expectation of the enclosed random variable, and
V[
] represents the variance of the enclosed random variable. The Poisson
distribution has the distinguishing characteristic i.e. variance is equal to its
mean. The Poisson distribution can be used for the number of events in other
specified intervals such as distance, area or volume.
1.6.3 Exponential Distribution
The exponential distribution is contineous and defined over the set of
non-negative real numbers. For exponential distribution, the probability density
function f ( x ) is given as follows:
12
f ( x) = λ e − λ x ; x ≥ 0
(1.11)
Possesses only one parameters λ > 0 describing the average rate and the
probability distribution function
F ( x ) = ∫ λ e − λt dt = 1 − e − λt ;
x ≥0.
x
0
Mean E[ x] =
1
λ
Variance V [ X ] =
(1.12)
(1.13)
1
= E 2[ X ]
2
λ
(1.14)
Figure 1.5: Explanation of exponential distribution for service facilities.
For service facilities, very often the average service time s = 1/ µ is
specified with µ the average service rate. A similar description is available for
the arrivals and departure processes. In assuming an exponential distribution
for the arrival processes one addresses the distribution of the times between
subsequent arrivals t, called inter-arrival time t.
This is graphically illustrated in the figure 1.6.
13
Figure 1.6: Inter-arrival times in an arrival process.
Consider t j as the time between two arrivals T j and T j −1
t j = T j − T j −1
(1.15)
Assuming t j for all j being exponentially distributed with parameter λ, i.e.
Pr{t j ≥ t} = e − λt
(1.16)
Then no. of arrivals Nt with in [0, t] follows a Poisson distribution.
Pr{N t = J } = f ( j , λ ) =
(λt ) j − λt
e .
j!
(1.17)
Theorem: The exponential distribution is memory-less.
Proof: The proof is based on the definition of conditions probability. A random
variable T is said to be memory-less
Pr{T > t + t0 | T > t0 } = Pr{T > t} .
(1.18)
Now a random variable t is assumed to be exponentially distributed with
parameter λ, i.e.,
Pr{T ≤ t} = 1 − e − λt .
(1.19)
Hence,
Pr{T > t + t0 | T > t0 } =
Pr{T > t + t0 }
Pr{T > t0 }
e − λ (t +t0 )
= − λt0 = e − λt = Pr{T > t}
e
14
(1.20)
Hence, the theorem.
1.6.4 Memory-less Property
A random variable (r.v.) is said to be memory-less if the time until the
next event does not depand on how much time has already elapsed since the
last event. A random variable is memory-less if
P{ X > s + t | X > t} = { X > s}
∀ s, t ≥ 0.
(1.21)
In the context of Poisson process, the memory-less property applies to
the inter-arrival time. From the law of conditional probability,
P{ X > s + t | X > t} =
=
P{ X > s + t and X > t}
P{ X > t}
P{ X > s + t}
P{ X > t}
(1.22)
Substituting the exponential distribution function in above equation
We obtain
P{ X > s + t | X > t} =
e − λ ( s +t )
= e − λs = P{ X > s }.
− λt
e
(2.23)
1.6.5 Gamma Probability Distribution Function
FYn (t ) = 1 − ∑
n −1
j =0
e − λt ( λ t ) j
; n ≥1
j
(1.24)
Gamma Probability Density function is given by
fYn (t ) =
λ n n −1 − λt
t e ; t≥0
n
Where, n is the gamma function.
[Note: The gamma function is not a probability distribution function].
When n is an integer then the gamma function is defined as
15
(1.25)
n = n − 1;
n ≥ 1 and integer.
The gamma function is continuous and defined over the domain t ≥ 0 . The
mean and variance of the gamma distribution are:
E[T ] =
n
λ
V [T ] =
n
λ2
1.6.6 Bernoulli Distribution
In probability theory and statistics, Bernoulli distribution, named after
Swiss scientist Jacob Bernoulli, is the probability distribution of a random
variable which takes value 1 with success probability p and value 0 with failure
probability q = 1 − p.
1.6.7 Bernoulli Random Variable
A random variable X is said to be a Bernoulli random variable if its
probability mass function (p.m.f.) is given as following:
P ( X = 0) = 1 − p
P ( X = 1) = p
We assume X = 1 when the experimental outcome is successful and
X = 0 when it is failed.
E[ X ] = 1. p ( X = 1) + 0. p ( X = 0) = p
(1.26)
var[ X ] = E[ X 2 ] − E[ X ]2 = p (1 − p)
(1.27)
1.6.8 Binomial Distribution
Suppose we repeat a Bernoulli p experiment n times and count the
number X of successes, the distribution of X is called the Binomial B(n, p)
random variable
16
n
pmf ; p( X = k ) =   p k q n −k
k 
Where, q = 1 − p;
(1.28)
k = 0,1, 2,..., n
E[ X ] = np
var[ X ] = np (1 − p ) .
Note:
1.
Binomial distribution is a sum of independent and evenly distributed
Bernoulli trials.
2.
Bernoulli trial is a random experiment with only two possible
outcomes.
3.
Binomial experiment is a sequence of Bernoulli trials performed
independently.
1.6.9 Renewal Process
A renewal process is defined as a discrete time independent process
{ X n | n = 1, 2,....} where X1, X2, .... are independent, identically distributed,
non-negative random variables.
Ex. Consider a system in which the repair (or replacement) after a failure is
performed, requiring negligible time. The times between successive
failure might well be independent, identically distributed random
variables {Xn | n = 1, 2, ....} of a renewal process.
1.6.10
Z-transform: An application in Queuing Systems. Let X is a
discrete random variable
P ( X = i ) = Pi , i = 0,1,...
P0 , P1 , P2 ...
17
g ( z ) = ∑ Pi z i
∞
i =0
1.6.11 Properties of Z-transform:
g (1) = 1,
P0 = g ( 0 ) , P1 = g ′ ( 0 )
1
g ′′ ( 0 )
2
P2 =
E(X ) =
( )
d
g ( z ) |z =1
dz
E X2 =
d2
d
g ( z ) |z =1 + g ( z ) |z =1
2
dz
dz
1.6.12 Matrix Geometric Method
In probability theory, the matrix geometric method is a method for the
analysis of a quasi-birth-death process. Continuous time Markov chain whose
transition rate matrices with a repetitive block structure. Harrison Peter G, Patel
Naresh, M.[1992].
1.6.13
Method Description
This method required a transition rate matrix with tri-diagonal block
structure which is given as follows:
 B00
B
 10
Q=



B01
A1
A0


A2


A1 A2

A0 A1 A2 
⋱ ⋱ ⋱ 
(1.29)
Where, B00, B01, A0, A1, A2 are matrices. To compute the stationary distribution
π, we have π Q = 0 and the balance equations are considered for sub-vectors πi
18
π 0 B00 + π 1B10 = 0
π 0 B01 + π 1 A1 + π 2 A0 = 0
⋮
⋮
⋮
π i −1 A2 + π i A1 + π i +1 A0 = 0
We observe that
π i = π 1 R i −1
(1.31)
Above equation holds true where, R is the Neutes’ rate matrix. Amussen, S.R.
[2003] and Neutes, M [1981]. Numerically, we have
[π 0
B
π 1 ]  00
 B10

= [0 0]
A1 + RA0 
B01
(1.32)
By using this relation, we can find π 0 , π 1 , and calculate iteratively all
the πi.
1.7
Simulation
Simulation is the technique that creates the data. According to Wesbters
dictionary, a simulation is the act of imitation. It is a word used in many
contexts. NASA uses chambers to simulate zero gravity for astronauts. United
airline uses computer programs to simulate Jet flight. The key characteristics of
Simulation are dynamic and time varying behaviour.
In the domain of Operations Research, Simulation is a powerful
technique for evaluating the behaviour. Simulation is a very powerful and
widely used management science technique for the analysis and study of
complex system. A simulation model usually takes the form of a set of
assumptions about the operation of the system, expressed as mathematical or
logical relations between the objects of interest in the system.hus the process of
designing a mathematical or logical model of a real-system and then
19
conducting computer based experiments with the model to describe, explain
and predict the behaviour of the real system.
Simulation fits in the following:
Figure 1.7: Simulation.
A system is a collection of entities that act and interact towards the
accomplishment of some logical end. Systems generally tend to be dynamic,
their status changes over time. To describe this status, we use the concept of the
state of a system.
Examples (Simulation Model)
1.
Emergency room (beds, doctors, nurses)
2.
Power integration-Semiconductor capacity, and random machine
downtime,
1.7.1 Value of Simulation
1.
Empirical Method versus mathematical Model.
2.
Allow we to calculate the extreme values not just the expected value.
i.
Simulation is the actual running of the Model System to gain
insight into its performance.
ii.
Simulation is used to better understand the expected performance
of the real system and to test the effectiveness of the system
design.
iii.
Why use Simulation: Experimental system, new concepts, costly
experimentation, unsafe experimentation. Determines limits of
stress.
20
1.7.2
Simulation vs. Analytical Modelling
Advantage
i.
Various performance measures
ii.
Greater realism.
iii.
Easier to understand.
iv.
Model the steady state as well as the transit behaviour.
Disadvantage
i.
May not provide, with the optimal solution.
ii.
Time to construct model will be longer.
1.7.3 Simulation vs. Physical
Advantage
High speed, not descriptive, replication easy, control variations,
generally less costly.
Disadvantage
Realism
Validity
Representing System:
A collection of mutually interacting objects designed to accomplish a goal
(machine repair system).
1.7.4 Entities
Denotes an element / object with in boundary of the system such as
(machines, operators and repairmen) etc.
21
Entity: Work being performed an object.
Resource: Performing the work. Characteristics or property or an entity
(machine ID, Type of breakdown, time that machine went down).
Activity: Transform the state of an object usually over some time (repairman
service time, machine run time).
1.7.5
Queue: It is a set, used to model waiting
1.7.6 Types of Simulation Models
There are two types of simulation models which are given below:
1. Static and
2. Dynamic
1. A static simulation model is a representation of a system at a particular
point in time. Example: Monte Carlo Simulation.
2. Dynamic simulation is a representation of a system as it evolves over time.
With these two classifications, a simulation may be deterministic or
stochastic.
i.
A deterministic simulation model is one that contains no random
variables.
ii.
A stochastic simulation model contains one or more random
variables.
iii.
Discrete event: State of system change only at discrete points in
time. Example: machine repair problem.
The simulation model provides a very flexible mechanism for evaluating
alternative policies. Simulation runs usually result in variation due to its
property and the random number used. Therefore, instead of depending on an
average output the confidence interval of output is usually used to illustrate the
numerical results. As mentioned by Banks [2004] “a confidence interval for the
22
performance measures being estimated by the simulation model is a basic
component of output analysis.
Simulation is an alternative approach to solve the machine interference
problem. Simulation is often used to solve complex problems that are too
complicated to be derived by mathematical modeling or through the analytical
approach. Simulation might not provide the exact solution, but it provided a
good estimate for existing problems. Guild and Harnett [1982] has worked on
the simulation study of the machine interference problem that involved N
machines and R operators with two type of stoppage. In continuation to the
application of the machine interference problem, Sztrik and Moller [2001]
discussed characteristics of stochastic simulation of the Markov modulated
finite source queueing systems also in [2002] Sztrik and Moller has studied on
simulation of machine interference on randomly changing environments.
1.8 Queueing Problems
Queueing problems can be classified into three types:
(i)
Perpetual queue.
(ii)
The predictable queue
(iii)
The Stochastic queue
1.8.1 The Perpetual Queue
It is a worst type queue. A perpetual queue is a queue for which all
customers have to wait for service. All customers have to wait, because the
servers have insufficient capacity to handle the demand for the service.
1.8.2 The Predictable Queue
This type queue occurs when the arrival rate is known to exceed the
service capacity over finite intervals of time.
23
1.8.3 The Stochastic Queue
The Stochastic queues are not predictable. They occur by chance when
customers happen to arrive at a faster rather then they are served. Stochastic
queues can occur whether or not the service capacity exceeds the average
arrival rate.
1.9
Markov Chains
The stochastic process { X n , n = 0,1, 2,...} is called a Markov chain, if
for j, k, j1 ,...., jn−1 (for any subset of I).
Pr { X n = k | X n−1 = j , X n−2 = j1 ,..., X 0 = jn−1}
= Pr { X n = k | X n−1 = j} = Pjk
(1.32)
The outcomes are called the states of the Markov chain. The transition
probabilities p jk are basic to the study of the structure of the Markov-chain.
1.9.1 Transition Matrix : The transition probabilities p jk satisfy
∑
p jk ≥ 0;
p jk = 1,
∀j ,
where,
(1.33)
k
 p11
P =  p21

 ⋯
p12
p22
⋯
p13 ⋯
p23 ⋯

⋯ ⋯
1.9.2 Classification of States and Chains
A property defined for all states of a chain is a class property if its
possession by one state in a class implies its possession by all states of the
same class. One such property is the periodicity of a state.
1.9.2.1 Periodicity : State i is a return state if,
pii( n ) > 0 , for some n ≥ 1.
24
(1.34)
The period di of a return to state i is defined as the g.c.d., of all m such that
pii( m ) > 0
i.e.,
(1.35)
di = gcd{m : pii( m ) > 0}
1.9.2.2 Aperiodic: State i is said to be aperiodic if, di = 1.
1.9.2.3 Periodic: State i is said to be periodic if, di > 1.
1.9.2.4 Absorbing state: The state j is said to be absorbing if,
p jj = 1, p jk = 0,
k≠ j
1.9.3 Reducible and non-reducible Markov Chains
In an irreducible Markov chain, every state can be reached from every
other state and chains which are not irreducible are called reducible.
In the next section 1,10, we shall describe M / M / 1: (The Classical
Queueing System).
1.10 M / M / 1: The Classical Queueing System
M/M/1 queue is the simplest non-trivial interesting queueing system and
described by selecting the birth-death coefficients as follows:
λn = λ;
n = 0,1, 2,....
µn = µ ;
n = 1, 2,3,....
Figure 1.8: State-transition rate diagram for M/M/1 queue.
25
We assume that customers are served in FCFS (first comes-first served)
basis. Balance equations are given as follows:
0 = −λ0 p0 + µ1 p1;
k =0
(1.36)
0 = −(λn + µn ) pn + λn−1 pn−1 + µ n+1 pn+1;
Applying pn = p0 ∏
n −1
i =0
λi
;
µi +1
n = 0,1, 2,...
n −1
i =0
Or,
And,
p0 =
1+ ∑
n =1
1
λ
µ
 
λ
.
µ
k ≥0
n
(1.37)
(1.38)
Using equation (1.38), we obtain, pn = p0 ∏
λ
pn = p0   ;
µ
k ≥1
(1.39)
(1.40)
n
The necessary and sufficient condition for ergodicity in the M / M / 1
queue is λ < µ.In equation (1.40) sum converges, since λ < µ therefore
p0 =
ρ=
1
λ
= 1−
λ/µ
µ
1+
1− λ / µ
(1.42)
λ
and 0 ≤ ρ < 1
µ
Finally, we have from equation (1.39),
p n = (1 − ρ)ρn ;
n = 0,1, 2,....
1.11 Machine Interference in Queueing Theory
With the advancement of latest technology, machining system has
pervaded in every nook and corner of our lives. Thus, ensuring our almost
26
dependence on machining system. As the time progresses a machine may fail
due to wear and tear or due to some unpredicted fault and thus requires a
corrective measure by providing the repair facility, after which it can be again
start functioning properly. If at any time, failed machines need the repairman's
attention, a queue of failed machines may develop due to unavailability of idle
repairmen. This phenomenon needs attention of mathematicians as such
machine repair models have captivated the interest of many renowned
researchers working in the area of queueing theory.
Thus, the loss of production due to the period of machines having to
wait for repair or service is called machine interference. Machine interference
occurs when a single repairman repairs two or more machine or a group of R
servers services a lot of N machines in case R < N. The time that is spent
between the call for service and the start of service is known as interference
time. Some notable work was done by Elsayed, E.A. [1981], Jain, M. [1998],
Jain, Singh and Beghel [2000] and Showky, A.I. [2000].
In many fast growing industries the operations may be interrupted due to
failure of machines involved in the system. The service facility should be so
adjusted that the machines are repaired immediately, so as to work
continuously. The factors, which effect the solution of machine interference
problems are running time of machine before break down, or while the
repairman is engaged, the second machine will remain in waiting until the
service of the first machine is completed. In case of several repairmen, if the
machines are failed these are repaired by repairman, the excess number of
machines will have to wait until repairmen are available.
1.12 Machine Repairmen Model: (M/M/1/m/m)
This is an interesting special case of birth-death process occurs when the
birth rate λn is of the form (M – n)λ; n = 0, 1, …, M and the death rate is
µn = µ.
27
This type of situation occurs in the modelling of a server where an
individual client issues a request at the rate λ whenever it is in the "thinking
state". If n out of total M clients are currently waiting for a response to a
pending request, then the effective request rate is (M – n)λ. Where µ is the
request completion rate.
( M − n ) λ ; 0 ≤ n ≤ M
λn = 
0
; n≥M

(1.43)
µ
µn = 
0
(1.44)
; 1≤ n ≤ M
; n>M
.
A similar situation arises when M machines share a repair facility. The
failure rate of each machine is λ and repair rate is µ.The state diagram of such
finite population system is given in the following figure 1.9.
Figure 1.9: The state diagram of a finite population queueing system.
The expression for steady state probabilities, which are obtained using
above equations, we have
pn = p0 ∏
λi
; n = 1,2,...
µ i +1
(1.45)
pn = p0 ∏
λ (M − i)
; 0≤n≤M
µ
(1.46)
n −1
i =0
n −1
i =0
or
λ
pn = p0  
µ
= p0 ρn
n
M!
( M − n )!
(1.47)
M!
( M − n )!
28
p0 =
Hence,
∑
M
n =0
1
ρn
M!
( M − n )!
M
M! 
P0 =  ∑ ρn

( M − n )!
 n =0
−1
(1.48)
1.13 M/M/2 Queuing System in Steady-State
λP0 = µP1
( λ + µ ) P1 = λP0 + 2µP2
( λ + 2µ ) Pn = λPn−1 + 2µPn+1 ; ( n = 2,3,...)
Therefore
Pn = 2ρn P0
Where,
ρ=
L=
(1.49)
λ
and
2µ
P0 =
Also,
( n = 1,2,...) .
1− ρ
1+ ρ
2ρ
1 − ρ2
Lq =
2ρ3
1 − ρ2
(1.50)
2ρ 2
And probability of having to wait =
.
1+ ρ
(1.51)
Now, we consider finite population queueing model.
1.14 Finite Population Model (M/M/C/m/m):
This is a simple matter to extend the limited population model to
cover systems with multiple servers. The parameters of this model are given
below:
29
( m − n ) λ
λn = 
0

; 0≤n≤m
; n≥m
(1.52)
 nµ ; 1 ≤ n ≤ c

µ n = cµ ; c < n ≤ m
0 ; n > m

(1.53)
Substituting these values in the following equations:
∏
n −1
pn =
i =0
n
∏
λi
µi
(1.54)
p0
i =1

∞

p0 = 1 + ∑
 n=1

And
∏
n −1
i =0
n
∏
i =1

λi 

µi 

−1
(1.55)
We obtain,
 c −1
p0 =  ∑
 j =0
m
λ
m!
+
( m − j )! j !  µ  ∑
j =c
i
λ
m!
j −c 
( m − j )!c !c  µ 
i



−1
(1.56)
And,
n

λ
m!

  p0 ; 0 < n ≤ c
 ( m − n )!n!  µ 
pn = 
n
λ

m!
 m − n !c!c n−c  µ  p0 ; c < n ≤ m
)
 
(
(1.57)
The expected number in the system and the expected number in the
queue and calculated from the following definitions.
Lq = ∑
m
n =c
( n − c ) pn
(1.58)
30
L = ∑ npn
m
(1.59)
n =0
= m − ∑ r pm −r
m
(1.60)
r =0
The effective arrival rate is the product of the failure rate of
individual machines and the expected number of machines running.
λe = λ E ( r )
= λ∑ r pm−r
m
(1.61)
r =0
Waiting times are
Wq =
Lq
λe
∑ (n − c) p
m
=
n
n=c
(1.62)
λ ∑ r pm − r
m
r =0
And,
W=
=
L
λe
m
λ ∑ r pm − r
m
−
1
λ
(1.63)
r =0
In case of single channel, the arriving customer distribution is equal
to (m – 1) customer system.
Then
qn ( m ) = pn ( m − 1)
(1.64)
31
The corresponding waiting times for an arriving customer are
Wq′ = ∑
m −1
n =c
n − c +1
qn ( m )
cµ
W ′ = Wq′ +
And,
(1.65)
1
µ
(1.66)
Example
In a networking conference, each speaker has 15 min to give his talk,
otherwise, he is rudely removed from podium, Given that time to give a
presentation is exponential, With mean 10 minutes. What is the probability,
that a speaker will not finish his talk.
Solution: Given that
E( X ) =
1
1
= 10 minutes ⇒ λ =
λ
10
Let t be the time required to give a presentation that a speaker will
not manage to finish his presentation if t exceeds 15 minutes.
P ( t > 15 ) = e −1.5 = 0.22
=
22
= 22%
100
This means speaker will not finish his presentation with in time is 22%
while he/she will finish his/her presentation with in time, that probability is
78%.
In the next section 1.15, we shall explain some applications of queueing
theory in industry, organization and in our daily life.
32
1.15 Some Applications of Queueing Theory
“Each of us had spent a great deal of time waiting in lines”
Where the time goes:
In a life time, the average person will spend (Approx.)
Six months
:
Waiting at stoplights.
Eight months :
Opening junk mail.
One year
:
Looking for misplaced objects.
Two years
:
Reading - e-mail.
Four years
:
Doing house work.
Five years
:
Waiting in line.
Six years
:
Eating, etc.
Queueing theory or waiting lines is useful crowed safety
management, entry and exist systems, concession planning and crows flow
assessment, venue ticket sales, transport loading (to and from avenue),
traffic control and planning, finding the sequence of computer operations,
telecommunication like waiting calls when server is busy, air port traffic,
layout of manufacturing systems, capacity planning for buses and trains,
health services (e.g. control of hospital bed assignments). Also queueing
theory is useful in game e.g. when to remove a goalie in a hockey game. We
have application of queueing theory in machine repairing system.
1.16 Structure of the Rest of the Thesis
This thesis is organised, according to the specific models that we
study. More concretely, in every chapter we first motivate the model under
consideration and then describe the previous contributions reported in the
literature. In chapter 2, we present an overview of the general approaches
33
proposed in the literature for queueing models and machine repairing
system. In this chapter, firstly, we give some general approaches undertaken
by Operational Researchers in dealing with queueing theory and machine
repairing systems and related areas. Secondly, some texts past and present
work have been incorporated.
In chapter 3, we discuss detailed explanations of machine repairing
system for queuing models. In this chapter we have explained some
performance measures, distributions and illustrated with numerical
examples.
In chapter 4, we present mathematical formulation and analysis of
different queueing models. In this chapter we performed birth and death
model and related results. Then we explain M / M / C (C > 1) model,
M / M / 1 / N (single-server) model. Also we have find results on M / G / 1
queue, M / Ek / 1 queue and M / D / 1 queue.
.
In Chapter 5, we focus on comparative study of different queueing
models for machine repairing system. In this chapter firstly, we have
developed M/M/R machine repair problem with spares secondly we gave
machine repair problem with spares and N-policy vacations. Finally, we
analyzed queuing model for machine repairing system with Bernoulli
vacation schedule.
In chapter 6, we develop queueing networks modelling software and
computing language(s) and we present software like MATLAB which has
been used for our calculation of numerical results.
Finally, in chapter 7, we summaries the conclusion of this research
and suggests issues for further research. Bibliography (alphabetical order)
and list of publications and presentations are placed at the end of the thesis.
34
1.17 Summary
In this chapter, we undertook a theoretical investigation of queueing
model and machine repair model. We have explained basics of queueing
systems and performance measures of queues, general notations, classical
queueing system and machine repairmen model. We have considered some
practical and general life examples and applications of queueing theory.
35