synopsis -...

Synopsis-1

SYNOPSIS

INTRODUCTION: Queueing theory is the mathematical study of waiting lines or queue. The theory enables

mathematical analysis of several related processes, including arriving at the queue,

waiting in the queue and being served by the server(s) at the front of the queue. The

theory permits the derivation and calculation of several performance measures including

the average waiting time in the queue or the system, the expected number waiting or

receiving service and the probability of encountering the system in certain states, such as

empty, full, having an available server or having to wait a certain time to be served.

Queueing theory is generally considered a branch of operations research because the

results are often used when making business decisions about the resources needed to

provide service. There are many valuable applications of the theory, most of which

have been well documented in the literature of probability, operations research,

management science, and industrial engineering. Some examples are traffic flow

(vehicles, aircraft, people, communications), scheduling (patients in hospitals, jobs on

machines, programs on a computer), and facility design (banks, post offices,

amusement parks, fast-food restaurants).

CHARACTERISTICS OF QUEUEING PROCESSES: In most cases, six basic characteristics of queueing processes provide an adequate

description of a queueing system: (1) Arrival pattern of customers,

(2) Service pattern of servers,

(3) Queue discipline,

(4) System capacity,

(5) Number of service channels,

(6) Number of service stages.

Synopsis-2

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

NOTATIONS OF QUEUEING MODELS: As a shorthand for describing queueing processes, a notation has evolved, due for

the most part to Kendall (1953), which is now rather standard throughout the

queueing literature. A queueing process is described by a series of symbols and

slashes such as A/B/X/Y/Z, where A indicates in some way the interarrival- time

distribution, B the service pattern as described by the probability distribution for

service time, X the number of parallel service channels, Y the restriction on system

capacity, and Z the queue discipline. Some standard symbols for these characteristics

are presented in Table.

Table of Queueing Notation A/B/X/Y/Z

Characteristic Symbol Explanation Interarrival-time distribution (A) Service-time distribution (B) Number of parallel servers (X) Restriction on system capacity (Y) Queue discipline (Z)

M D Ek Hk PH G 1,2,…,∞ 1,2,…,∞ FCFS LCFS RSS PR GD

Exponential Deterministic Erlang type k (k=1,2,..∞) Mixture of k exponentials Phase type General First come, first served Last come, first served Random selection for service Priority General discipline

PERFORMANCE OF QUEUEING MODELS: Up to now the concentration has been on the physical description of the queueing

processes. Generally there are three types of system responses of interest. These

are: (1) some measure of the waiting time that a typical customer might be forced

to endure; (2) an indication of the manner in which customers may accumulate; (3)

a measure of the idle time of the servers. Since most queueing systems have

Synopsis-3

stochastic elements, these measures are often random variables and their probability

distributions, or at the very least their expected values, are desired. There are two

types of customer waiting times, the time a customer spends in the queue and the

total time a customer spends in the system (queue plus service). Depending on the

system being studied, one may be of more interest than the other. For example, if

we are studying an amusement park, it is the time waiting in the queue that

makes the customer unhappy. On the other hand, if we are dealing with machines

that require repair, then it is the total down time (queue wait plus repair time)

that we wish to keep as small as possible. Correspondingly, there are two customer

accumulation measures as well: the number of customers in the queue and the

total number of customers in the system. The former would be of interest if we

desire to determine a design for waiting space (say the number of seats to have

for customers waiting in a hair- styling salon), while the latter may be of interest

for knowing how many of our machines may be unavailable for use. Idle- service

measures can include the percentage of time any particular server may be idle, or the

time the entire system is devoid of customers. The task of the queueing analyst is

generally one of the two things. He or she is either to determine the values of

appropriate measures of effectiveness for a given process, or to design an

“optimal” (according to some criterion) system. To do the former, one must relate

waiting delays, queue lengths, and such to the given properties of the input stream

and the service procedures. On the other hand, for the design of a system the

analyst might want to balance customer waiting time against the idle time of

servers according to some inherent cost structure. If the costs of waiting and idle

service can be obtained directly, they can be used to determine the optimum

number of channels to maintain and the service rates at which to operate these

channels. Also, to design the waiting facility it is necessary to have information

regarding the possible size of the queue to plan for waiting room. There may also

be a space cost which should be considered along with customer- waiting and

idle- server costs to obtain the optimal system design. In any case, the analyst will

strive to solve this problem by analytical means; however, if these fail, he or she

must resort to simulation. Ultimately, the issue generally comes down to a trade-

Synopsis-4

off of better customer service versus the expense of providing more service

capability, that is, determining the increase in investment of service for a

corresponding decrease in customer delay.

CHAPTER-1

PERFORMANCE ANALYSIS OF AN M/M/1/N QUEUE WITH BALKING AND RENEGING

1.1. INTRODUCTION: In real life, many queueing situations arise in which there may be a tendency for

customers to be discouraged by a long queue. As a result, the customers either decide not

to join the queue (i.e. balk) or depart after joining the queue without getting service due

to impatience (i.e. renege). Balking and reneging are not only common phenomena in

queues arising in daily activities, but also in various machine repair models. Many

practical queueing systems especially those with balking and reneging have been widely

applied to many real-life problems, such as the situations involving impatient telephone

switchboard customers, the hospital emergency rooms handling critical patients, and the

inventory systems with storage of perishable goods [1]. In this chapter, we consider an

M/M/1/N queueing system with balking and reneging.

Queue system M/M/1/N means: arrival of queue is Poisson, departure of queue is

Exponential, number of server is one and capacity of system is N. Queueing systems with

balking, reneging or both have been studied by many researchers. Haight [2] first

considered an M/M/1 queue with balking. An M/M/1 queue with customers reneging was

also proposed by Haight [3]. The combined effects of balking and reneging in an

M/M/1/N queue have been investigated by Ancker and Gafarian [4], [5]. Abou-EI-Ata

and Hariri [6] considered the multiple servers queueing system M/M/c/N with balking and

reneging.

In the next part of this chapter, we give a description of the queueing model. In part 1.3

of this chapter, we derive the steady-state equations by the Markov process (It is a

random process which has the property that the probability of transition from a given

Synopsis-5

state to any state depend only on the present state and not on the manner in which it was

reached.) method. By writing the transition rate matrix as block matrix, we get the matrix

form solution of the steady-state probabilities and present a procedure for calculating the

steady-state probabilities. In part 1.4 of this chapter, we give some performance measures

of the system. Based on the performance analysis, we formulate a cost model to

determine the optimal service rate. Some numerical examples are presented to

demonstrate how the various parameters of the model influence the behavior of the

system. Conclusion is given in last of this chapter.

1.2. SYSTEM MODEL In this chapter, we consider an M/M/1/N queueing system with balking and reneging. The

assumptions of the system model are as follows: 1. Customers arrive at the system one by one according to a Poisson process with rate λ.

On arrival a customer either decides to join the queue with probability bn or balk with

probability 1- bn when n customers are ahead of him (n=0,1,…N-1), where N is the

maximum numbers of the customers in the system, and

0 ≤ bn-1 ≤ bn <1, 1 ≤ n ≤ N-1,

b0 =1, and bn = 0, n ≥ N.

2. After joining the queue each customer will wait a certain length of time T for service to

begin. If it has not begun by then, he will get impatient and leave the queue without

getting service. This time T is a random variable whose density function is given by

d(t) = αe-αt, t ≥ 0, α >0,

where α is the rate of time T. Since the arrival and departure of the impatient customers

without service are independent, the average reneging rate of the customer can be given

by (n - i)α. Hence, the function of customer’s average reneging rate is given by

r(n) = (n-i)α, i ≤ n ≤ N, i= 0,1

r(n) = 0, n>N.

3. The customers are served on a first-come, first served (FCFS) discipline. Once service

commences it always proceeds to completion. The service times are assumed to be

distributed according to an exponential distribution with density function as follows:

s(t) = µe-µt , t ≥ 0, µ > 0, where µ is the service rate.

Synopsis-6

( )

( ) ( ) ( ) ( )[ ]

( ) ( )[ ] .),(11

.1,,2,1),(111

0,0)1(

1

1

NnNPnNPb

NnnPnbnPnnPb

nPP

N

nn

=−+=−

−=−++=+++−

==

−

−

αμλ

αμλαμλ

λμ

LLL

( )

( ) ( ) ( ) [ ]

( ) ( ) ( ) [ ]

( ) ( ).

.3),3(22

2),2(321

1),1(20

0,0)1(

2

21

1

=α+μ=λ

=α+μ+λ=α+μ+λ

=μ+λ=α+μ+λ

=λ=μ

nPPb

nPbPPb

nPbPP

nPP

( )

( )( ) .2)3(

)2(

)1(

)0(

213

12

α+μα+μμλ

=

α+μμλ

=

μλ

=

=

kbbP

kbP

kP

kP

,1)3()2()1()0( =+++ PPPP

( ) ( )( ) .2

11

213

12 −

⎟⎟⎠

⎞⎜⎜⎝

⎛α+μα+μμ

λ+

α+μμλ

+μλ

+=bbbk

1.3. STEADY STATE PROBABILITY In this part, we derive the steady-state probabilities by the Markov process method. Let

P(n) be the probability that there are n customers in the system when the server is on

available. Applying the Markov process theory, we obtain the following set of steady-

state equations.

For N = 3 steady state difference equations are:

On solving these we get,

Using

we get,

Synopsis-7

( ) ( ) ( )

( ) ( )

( )

( ) ( ) )4.1(1..

)3.1()1(..

)2.1(

)1.1(1

1

1

1

1

∑

∑

∑

∑

=

=

=

=

−=

−=

=

−=

N

n

N

nn

N

n

N

nq

nPnRR

nPbRB

nnPNE

nPnNE

α

λ

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

)( 121

α−+μα+μα+μμλ

= −

nkbbb

nP nn

( ) ( ) ( ) ( ) ,1.................210 =+++ NPPPP

( ) ( )( ) ( )( ) ( )( ) .1.....2

..............

21

1

121213

12 −

−⎟⎟⎠

⎞⎜⎜⎝

⎛α−+μα+μα+μμ

λ++

α+μα+μμλ

+α+μμ

λ+

μλ

+=N

bbbbbbk NN

In general for N = n

and using we get

1.4. PERFORMANCE MEASURES AND COST MODEL

In this part, we give some performance measures of the system. Based on these

performance measures, we develop a cost model to determine the optimal service rate.

1.4.1. PERFORMANCE MEASURES

Using the steady-state probability presented in part 1.3 of this chapter, we can obtain

some performance measures of the system, such as the busy probability of the server PB,

the expected number of the waiting customers E(Nq) and the expected number of the

customers in the system E(N) as follows:

L.R. = B.R. + R.R. (1.5)

Synopsis-8

1.4.2. COST MODEL

In this subpart, we develop an expected cost model, in which service rate µ is the control

variable. Our objective is to control the service rate to minimize the system’s total

average cost per unit. Let

C1 cost per unit time when the server is busy,

C2 cost per unit time when a customer joins in the queue and waits for service,

C3 cost per unit time when a customer balks or reneges.

Using the definitions of each cost element listed above, the total expected cost function

per unit time is given by

F(µ) = C1 PB + C2 E(Nq) + C3 L.R.,

where E(Nq), E(N), B.R., R.R. and L.R. are given in Eqs. (1.1) − (1.5) .The first item is the

cost incurred by the server. The second item C2 E(Nq) is the cost incurred by the

customer’s waiting. The last item C3 L.R. is the cost incurred by the customer loss.

1.4.3. NUMERICAL RESULTS

In this subpart of chapter, we present some numerical examples to demonstrate how the

various parameters of the model influence the optimal service rate µ*, the optimal

expected cost of the system F(µ*) and other performance measures of the system. We fix

the maximum number of customers in the system N = 3, the probability bn = 1/(n + 1) and

the cost elements C1 = 15, C2 = 12, C3 = 18. Table 1.1: Performance measures for α = 0.1.

λ 0.4 0.5 0.6 0.7 0.8 0.9

µ* 0.9468 0.9089 0.8754 0.8444 0.8155 0.7882

F(µ*) 17.2026 18.4446 19.9224 21.6150 23.4906 25.5186

E(Nq) 0.0658 0.1039 0.1495 0.2016 0.2592 0.3211

E(N) 0.4047 0.5180 0.6327 0.7481 0.8627 0.9757

L.R. 0.0785 0.1221 0.1738 0.2331 0.2989 0.3703

≡≡

≡

Synopsis-9

Table1.2: Performance measures for λ = 0.5.

α 0.05 0.1 0.2 0.3 0.4 0.5

µ* 0.9652 0.9089 0.8010 0.6993 0.6041 0.5160

F (µ*) 18.2430 18.4572 18.9180 19.4208 19.9734 20.5152

E(Nq) 0.1006 0.1039 0.1105 0.1173 0.1239 0.1302

E(N) 0.5002 0.5180 0.5550 0.5956 0.6830 0.6825

L.R. 0.1131 0.1228 0.1440 0.1674 0.1937 0.2196

First, we select the rate of the waiting time α = 0.1 and change values of arrival rate of

customers λ. The numerical results are summarized in Table 1.1. This shows that: (i) the

optimal service rate µ decreases with the increasing λ and its minimum expected cost

F(µ*) increases with the increasing λ; (ii) the expected number of the waiting customers

E(Nq), the expected number of customers in the system E(N) and the average rate of

customer loss L.R. all increases with the increase of λ. This is because the number of the

customers in the system increases with the increase of λ. Thus, E(Nq) and L.R. all

increases which result in the increase of the optimal cost.

Finally, we select λ = 0.5, and change values of α. The numerical results are summarized

in Table 1.2. This shows that: (i) the optimal service rate µ* decreases with the increasing

α, and its minimum expected cost F(µ*) increases with the increasing α; (ii) E(Nq), E(N)

and L.R. all increases with the increasing α. Thus E(Nq) and L.R. all increases which

result in the increase of the optimal cost. The following graph shows the performance of

the model at α = 0.1 and λ = 0.5. The below Figure 1.1 and 1.2 are for α = 0.1 and shows

performance of the model. Similarly, Figure 1.3 and 1.4 are for λ = 0.5 and shows

performance of the model.

Synopsis-10

0 0.2 0.4 0.6 0.8 1

λ

F(µ

*)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

λ

µ*(Opimal srevice rate)

E(Nq): Length of queue

E(N): Length of queueingsystemL.R.(Average rate ofcustomer loss)

Figure 1.1: Graph between arrival rate and µ*, E(Nq), E(N), L.R.

Figure 1.2: Graph between arrival rate and expected optimal cost function

Synopsis-11

18

18.5

19

19.5

20

20.5

21

0 0.1 0.2 0.3 0.4 0.5 0.6

α

F(µ

*)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6

α



E(N): Length of queueingsystemL.R.(Average rate ofcustomer loss)

Figure 1.3: Graph between α and µ*, E(Nq), E(N), L.R.

Figure 1.4: Graph between α and F(µ*)

Synopsis-12

1.5. CONCLUSION In this chapter, we considered an M/M/1/N queueing system with balking and reneging.

We developed the equations of the steady state probabilities and derived the matrix form

solution of the steady-state probabilities. We also gave some performance measures of

the system, and formulated a cost model to determine the optimal service rate. Although

the function of the cost is too complicated to derive the explicit expression of the optimal

service rate, the performance measures and the optimal service rate can be numerically

evaluated by the formula in Section 1.4.2. Some numerical examples were presented to

demonstrate how the various parameters of the model influence the behavior of the

system.

CHAPTER-2

PERFORMANCE ANALYSIS OF AN M/M/c/N QUEUE WITH BALKING AND RENEGING:

2.1. INTRODUCTION: We have found performance of M/M/1/N queueing system with balking and reneging.

In M/M/1/N model no. of server is one. In this chapter, we consider an M/M/c/N queueing

system with balking and reneging. In M/M/c/N model no. of server is c and each server

has an independently and identically distributed exponential service time distribution. Queueing systems with balking, reneging or both have been studied by many researchers.

Haight [2] first considered an M/M/1 queue with balking. An M/M/1 queue with

customers reneging was also proposed by Haight [3]. The combined effects of balking

and reneging in an M/M/1/N queue have been investigated by Ancker and Gafarian [4],

[5]. Abou-EI-Ata and Hariri [6] considered the multiple servers queueing system

M/M/c/N with balking and reneging. Queue system M/M/c/N means: arrival of queue is

Poisson, departure of queue is Exponential, number of server is c and capacity of system

is N.

Synopsis-13

In the next part of this chapter, we give a description of the queueing model. In part 2.3

of this chapter, we derive the steady-state equations by the Markov process method. By

writing the transition rate matrix as block matrix, we get the matrix form solution of the

steady-state probabilities and present a procedure for calculating the steady-state

probabilities. In part 2.4 of this chapter, we give some performance measures of the

system. Based on the performance analysis, we formulate a cost model to determine the

optimal service rate. Some numerical examples are presented to demonstrate how the

various parameters of the model influence the behavior of the system. Conclusion is

given in part 2.5 of this chapter.

2.2. SYSTEM MODEL In this chapter, we consider an M/M/c/N queueing system with balking and reneging. The

assumptions of the system model are as follows:

1. Customers arrive at the system one by one according to a Poisson process with rate λ.

On arrival a customer either Rdecides to join the queue with probability bn or balk with

probability 1- bn when n customers are ahead of him (n = 0,1,…N-1), where N is the

maximum numbers of the customers in the system, and

0 ≤ bn-1 ≤ bn <1, 1 ≤ n ≤ N-1,

b0 =1, and bn = 0, n ≥ N.

2. After joining the queue each customer will wait a certain length of time T for service to

begin. If it has not begun by then, he will get impatient and leave the queue without

getting service. This time T is a random variable whose density function is given by

d(t) = αe-αt, t ≥ 0, α >0,

where α is the rate of time T. Let i denote the number of severs being busy and n

represent the number of customers in the system. If n is less than or equal to i, the

customers will get service instantly upon arrival to the server, and the phenomenon of

reneging will not occur. If n is greater than i, then there are n − i customers who have to

wait in the queue. Since the arrival and the departure of the impatient customers without

service are independent, the average reneging rate in this state is given by (n − i) α .

Synopsis-14

( )( )

( )( )

.4)4()22()3(3)3(2)4()22()2(2)2(2)3()2()1(1)1()2(2)0(00)1(

3

32

2

=α+μ=λ=α+μ+λ=α+μ+λ=μ+λ=α+μ+λ=μ+λ=μ+λ=λ=μ

nPPbnPbPPbnPbPPnPPPnPP

( )

( )

( )

( )

. ),())(()3(

),()()1())1(()1(

),2()1()()1(

11 ),()1()1()1(

0 ,0)1(

1

1

NnNPcNcPb

NncnPcncbnPcncnPb

cnPcbcPccP

cnnPnnPnnP

nPP

N

nn

c

=α−+μ=λ

<<α−+μ+λ=+α−++μ+−λ

=μ+λ=+α+μ+−λ

−≤≤μ+λ=++μ+−λ

=λ=μ

−

−

3. The customers are served on a first-come, first served (FCFS) discipline. Once service

commences it always proceeds to completion. The service times are assumed to be

distributed according to an exponential distribution with density function as follows:

s(t) = µe-µt, t ≥ 0, µ > 0, where µ is the service rate.

2.3. STEADY-STATE PROBABILITY In this part of chapter, we derive the steady-state probabilities by the Markov process

method. Let P(n) be the probability that there are n customers in the system. Applying the

Markov process theory, we obtain the following set of steady-state equations.

For N = 4 steady state difference equations are

Synopsis-15

( )( ) .222222

11

232

4

22

3

2

2 −

⎟⎟⎠

⎞⎜⎜⎝

⎛α+μα+μμ

λ+

μλ

+μλ

+μλ

+=bbbk

,1)4()3()2()1()0( =++++ PPPPP

( )( )

( )( )

( )( )

( ) ( )( )

( )( ) ( )( )

1

232

4

22

3

2

2

232

4

1

232

4

22

3

2

2

22

3

1

232

4

22

3

2

2

2

2

1

232

4

22

3

2

2

1

232

4

22

3

2

2

2222221

2222)4(

2222221

22)3(

2222221

2)2(

2222221)1(

2222221)0(

−

−

−

−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛++

++++++

=

⎟⎟⎠

⎞⎜⎜⎝

⎛++

+++++

=

⎟⎟⎠

⎞⎜⎜⎝

⎛++

++++=

⎟⎟⎠

⎞⎜⎜⎝

⎛++

++++=

⎟⎟⎠

⎞⎜⎜⎝

⎛++

++++=

αμαμμλ

μλ

μλ

μλ

αμαμμλ

αμαμμλ

μλ

μλ

μλ

αμμλ

αμαμμλ

μλ

μλ

μλ

μλ

αμαμμλ

μλ

μλ

μλ

μλ

αμαμμλ

μλ

μλ

μλ

bbbbbP

bbbbP

bbbP

bbbP

bbbP

( )

( )( ) .2222)4(

22)3(

2)2(

)1(

)0(

232

4

22

3

2

2

α+μα+μμλ

=

α+μμλ

=

μλ

=

μλ

=

=

bbkP

bkP

kP

kP

kP

On solving these we get,

Using

we get,

Therefore,

Synopsis-16

( ) ( )

( )

( )

( )

( ).5.2.....

4.2)()1(.

3.2)()1(..

2.2)()(

1.2)()1(

1

1

1

1

RRRBRL

nPnRR

nPbRB

nnPNE

nPnNE

N

n

N

nn

N

n

N

nq

+=

−=

−=

=

−=

∑

∑

∑

∑

=

=

=

=

α

λ

,1)(.........)4()3()2()1()0( =++++++ NPPPPPP

( )( ) ( )( ) ( ) .)2(2......2222

..................

2222221

1

2132

232

4

22

3

2

2 −

−⎟⎟⎠

⎞⎜⎜⎝


λ++

α+μα+μμλ

+μ

λ+

μλ

+μλ

+=N

bbbbbbk Nn

( )( ) ( ) .)2(2......2222

)........()( 2

132⎟⎟⎠

⎞⎜⎜⎝


λ= −

nkbbb

nP nn

For N = n Using we get,

2.4. PERFORMANCE MEASURES AND COST MODEL In this part, we give some performance measures of the system. Based on these

performance measures, we develop a cost model to determine the optimal service rate.

2.4.1. PERFORMANCE MEASURES Using the steady-state probability presented in the part 2.3 of this chapter, we can obtain

some performance measures of the system, such as the busy probability of the server PB,

the expected number of the waiting customers E(Nq) and the expected number of the

customers in the system E(N) as follows

Synopsis-17

2.4.2. COST MODEL In this subpart of this chapter, we develop an expected cost model, in which service rate

µ is the control variable. Our objective is to control the service rate to minimize the

system’s total average cost per unit. Let

C1 cost per unit time when the server is busy,

C2 cost per unit time when a customer joins in the queue and waits for service,

C3 cost per unit time when a customer balks or reneges.

Using the definitions of each cost element listed above, the total expected cost function

per unit time is given by

F(µ) = C1 PB + C2 E(Nq) + C3 L.R.

where E(Nq), E(N), B.R., R.R. and L.R. are given in Eqs. (2.1) − (2.5). The first item is the

cost incurred by the server. The second item C2 E(Nq) is the cost incurred by the

customer’s waiting. The last item C3 L.R. is the cost incurred by the customer loss.

2.4.3. NUMERICAL RESULTS In this subpart of this chapter, we present some numerical examples to demonstrate how

the various parameters of the model influence the optimal service rate µ*, the optimal

expected cost of the system F(µ*) and other performance measures of the system. We fix

the maximum number of customers in the system N = 3, the probability bn = 1/(n + 1) and

the cost elements C1 = 15, C2 = 12, C3 = 18.

Table 2.1: Performance measures for α = 0.1.

λ 0.4 0.5 0.6 0.7 0.8 0.9

µ* 0.1301 0.1549 0.1929 0.2479 0.3060 0.3647

F(µ*) 21.3084 22.6902 23.466 23.6616 23.8824 24.1074

E(Nq) 0.2296 0.2615 0.2630 0.2403 0.2215 0.2074

E(N) 0.6193 0.6985 0.7000 0.6406 0.5913 0.5546

L.R. 0.1974 0.2529 0.2950 0.3210 0.3458 0.3712

≡≡

≡

Synopsis-18

Table 2.2: Performance measures λ = 0.4

α 0.05 0.1 0.2 0.3 0.4 0.5

µ* 0.1559 0.1301 0.0901 0.0581 0.0352 0.0212

F(µ*) 24.198 21.3084 22.3338 23.1978 23.8278 24.1056

E(Nq) 0.2154 0.2296 0.2402 0.2414 0.2312 0.2122

E(N) 0.5759 0.6193 0.6591 0.6715 0.6505 0.6023

L.R. 0.3674 0.1974 0.2473 0.2945 0.3363 0.3644

First, we select the rate of the waiting time α = 0.1 and change values of arrival rate of

customer λ. The numerical results are summarized in Table 2.1. This shows that: (i) the

optimal service rate µ increases with the increasing λ and its minimum expected cost

F(µ*) increases with the increasing λ; (ii) the expected number of the waiting customers

E(Nq), the expected number of customers in the system E(N) first increases with the

increase of λ and then decreases with the increase of λ; (iii) average rate of customer loss

L.R. increases with the increasing λ this is because the number of the customers in the

system increases with the increasing λ. Thus, F(µ*) increases with the increasing λ.

Finally, we select λ = 0.4, and change values of α. The numerical results are summarized

in Table 2.2. This shows that: (i) the optimal service rate µ* decreases with the

increasing α, and its minimum expected cost F(µ*) increases with the increasing α ; (ii)

E(Nq), the expected number of customers in the system E(N) first increases with the

increase of α and then decreases with the increase of α. Thus F(µ*) first decreases with

the increase of α and then increases with the increase of α. The following graph shows

the performance of the model at α = 0.1 and λ = 0.4. The below Figure 2.1 and 2.2 are for

α = 0.1 and shows the performance of the model. Similarly, Figure 2.3 and 2.4 are for

λ = 0.4.

Synopsis-19

00.10.20.30.40.50.60.70.8

0 0.2 0.4 0.6 0.8 1

λ



E(N): Length of queueingsystemL.R.(Average rate ofcustomer loss

21

21.5

22

22.5

23

23.5

24

24.5

0 0.2 0.4 0.6 0.8 1

λ

F(µ

*)

Figure 2.1: Graph between arrival rate and µ*, E(Nq), E(N), L.R.

Figure 2.2: Graph between arrival rate and expected optimal cost function

Synopsis-20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6

α



E(N): Length of queueingsystemE(N): Length of queueingsystem

21

21.5

22

22.5

23

23.5

24

24.5

0 0.1 0.2 0.3 0.4 0.5 0.6

α

F(µ

*)

Figure 2.3: Graph between α and µ*, E(Nq), E(N), L.R.

Figure 2.4: Graph between α and F(µ*),

Synopsis-21

2.5. CONCLUSION In this chapter, we considered an M/M/c/N queueing system with balking and reneging.



the system, and formulated a cost model to determine the optimal service rate. Some

numerical examples were presented to demonstrate how the various parameters of the

model influence the behavior of the system. Although the function of the cost is too

complicated to derive the explicit expression of the optimal service rate, the performance

measures and the optimal service rate can be numerically evaluated by the formula in

Section 2.4.2. Some numerical examples were presented to demonstrate how the various

parameters of the model influence the behavior of the system.

CHAPTER-3

PERFORMANCE ANALYSIS OF AN M/M/1/k QUEUE

WITH NON-PREEMPTIVE PRIORITY:

3.1. INTRODUCTION In chapter 1 and 2 we discussed the case of balking and reneging on the model M/M/1/N

and M/M/c/N. Now we take criterion of priority, obviously a considerable portion of real-

life queueing situations contains priority considerations. In chapter 3 and 4 we will

discuss the case of different priorities. The priority discipline followed may be either non-

preemptive or preemptive in nature. When the priority discipline is non-preemptive in

nature, a job in service is allowed to complete its service normally even if a job of higher

priority enters the queue while its service is going on. In the preemptive case, the service

to the ongoing job will be preempted by the new arrival of higher priority. If the priority

discipline is preemptive resume, then service to the interrupted job, when it restarts,

continues from the point at which the service was interrupted. For the preemptive non

Synopsis-22

resume case, service already provided to the interrupted job is forgotten and its service is

started again from the beginning. Note that there may be loss of work in the preemptive

non-resume priority case. Such loss of work will not happen in the case of the other two

priorities. Since the service times are assumed to be exponentially distributed, they will

satisfy the memory-less property and that, therefore, the results will be the same both for

the preemptive resume and preemptive non-resume cases. We consider here queues with

preemptive priority disciplines. The approach suggested may be applied to both single

server and multi-server queues. Moreover, queues with both finite and infinite buffers

may be analysed using the suggested approach.

The solution of queueing system is given by L. Kleinrock [7] in 1975.The idea about the

specially push out priority queue discipline given by D. P. Heyman [8] in 1985.After that

the performance evaluation of queueing system shown by H. Takagi [9] in 1991 and

mixing time and loss priorities in a single server queue is presented by A. Gravey and G.

Hebuterne [10] in1991.

Many papers of queueing model on priority is given by many authors time to time. The

idea of finite buffer on two class queue with different schemes is shown by X. Cheng and

Akyildiz [11] in 1992. In year 2006 Takagi [12] described explicit delay distribution in

first come first served M/M/m/k and M/M/m/k/n queue and a mixed loss delay system.

We have also worked on performance analysis of queueing model [13], [14].

In this chapter we define the way in which such a queue may actually be analysed by

solving its balance equations. As usual with any M/M/1/k queue, the basic approach to its

solution is to define the system-states appropriately so that a state transition diagram of

the queue may be drawn. Using the exponential (memory-less) distributions of the inter-

arrival and service times, the probability flows in the state transition diagram are

identified. The balance equations are then obtained by choosing proper closed boundaries

and equating the flow across each such boundary. These balance equations are then

solved along with the normalisation condition (i.e. sum of the probabilities of all states

equal to unity) to obtain the state probabilities. Once these have been obtained, the

performance parameters of interest may be suitably calculated. In the following, we

consider the case of non-preemptive priority. For each case, we first give the analytical

approach for the M/M/1 case and then show how it may be used to handle the M/M/1/k

Synopsis-23

queue and the M/M/n/K queue. Queue system M/M/1/k means: arrival of queue is

Poisson, departure of queue is Exponential, number of server is one and capacity of

system is k.

3.2. ASSUMPTIONS In this chapter, we consider an M/M/1/k queueing system with non-preemptive priority.

The assumptions of the system model are as follows:

1. The system at any instant of time is one of the two modes of operation.

2. All distribution are negative exponential, ie, all transition rates are constants.

3. The probability of two or more random events occurring simultaneously is negligible.

3.3. NOTATIONS i mode of system operation, i = 1, 2 N maximum number of customers in the system

n current number of customers in the system

λi arrival rate in mode i

µi service rate in mode i

L mean number in the system

W mean waiting time in the system

Lq mean number in the queue

Wq mean waiting time in the queue

3.4. SYSTEM MODEL We first consider the 2- priority M/M/1 queue with infinite buffers operating with a non-

preemptive priority discipline. Since all the flows are known, the equilibrium

probabilities of each of the states may be found by writing and solving the corresponding

balance equations with the normalisation condition. As an example, consider the

2- priority M/M/1/2 queue with non-preemptive priority.

Synopsis-24

11,0,22,1,11,1,11,0,12,2,02,1,00 =++++++ ppppppp

( ) ( ) ( ) ( ) .)(

11

22112

22

1

2212

12

22

21

2212

1

212

1

2212

1

22

1

1

2

1

1

−

⎥⎦

⎤⎢⎣

⎡

μ+λ+λλμλμ

+μ+λ+λλμ

λμ+

μ+λ+λλμ+λμ

+μ+λ+λλ

λμ+

λλ

+λμ

+=k

11,0,111,0,2

12,1,022,1,1

22,1,022,2,0

21,0,111,1,1

22,1,111,0,2101211,0,1

22,2,0202212,1,0

11,0,122,1,0210

)()(

)(

λμλμλμλμ

μμλμλλμλμλλ

μμλλ

pppppp

pppppp

pppppp

=

=

=

=

++=++

+=++

+=+

1

21,1,1

1

11,0,1

1,0,2

λλλμ

kp

kp

kp

=

=

=

( )

( )

( )

( ).

)(

22112

22

12,1,1

2212

12

22

21

2,2,0

2212

1

212

10

2212

1

22

12,1,0

μ+λ+λλμλμ

=

μ+λ+λλμλμ

=

μ+λ+λλμ+λμ

=

μ+λ+λλλμ

=

kp

kp

kp

kp

3.5. STEADY-STATE PROBABILITY In this part of chapter, we derive the steady-state probabilities by the Markov process



On solving these equation we get;

Using :

We get:

Synopsis-25

00.5

11.5

22.5

33.5

44.5

0 0.2 0.4 0.6 0.8

λ2/µ2

L(Length of thesystem)

Lq(Length of thequeue)

W(Waiting time ofsystem)

Wq(Waiting time ofqueue)

3.5.1. NUMERICAL RESULTS

In this subpart of this chapter, we present some numerical examples to demonstrate how

the various parameters of the model influence the performance measures of the system.

We fix the maximum number of customers in the system N = 2.

Table 3.1.The case for λ1/µ1 = 0.8

λ2/µ2 0.1 0.2 0.3 0.4 0.5 0.6 0.7

L 0.9206 0.9877 1.0469 1.1213 1.1473 1.1952 1.2369

Lq 0.2954 0.3275 0.3574 0.326 0.4107 0.4362 0.4593

W 4.1849 4.1157 4.0266 4.0050 3.8240 3.7352 3.6380

Wq 1.3430 1.3645 1.3747 1.4022 1.3689 1.3629 1.3509

First, we select the λ1/µ1 = 0.8 and change values of arrival rate of customer λ2/µ2 .The

numerical results are summarized in Table 3.1. This shows that: (i) the length of queueing

system increases with the increasing of λ2/µ2. (ii) The length of the queue increases with

the increasing of λ2/µ2. (iii) The Waiting time in queueing system decreases with

increasing of λ2/µ2. (iv) The Waiting time in queue first increases with increases of λ2/µ2

and then decreases. The following graph is showing the performance of the model at

λ1/µ1 = 0.8.

Figure 3.1: Graph between λ2/µ2 and L , Lq ,W, Wq

Synopsis-26

3.6. CONCLUSION In this chapter, we considered an M/M/1/k queueing system with non-preemptive priority.



the system. Some numerical examples were presented to demonstrate how the various

parameters of the model influence the behavior of the system as given above.

CHAPTER-4

PERFORMANCE ANALYSIS OF AN M/M/1/k QUEUE

WITH PREEMPTIVE PRIORITY:

4.1. ASSUMPTIONS In this chapter, we consider an M/M/1/K queueing system with preemptive priority.

The assumptions of the system model are as follows:

1. The system at any instant of time is one of the two modes of operation.

2. All distribution are negative exponential, i.e., all transition rates are constants.

3. The probability of two or more random events occurring simultaneously is negligible.

4.2. NOTATIONS i mode of system operation, i = 1, 2

N maximum number of customers in the system

n current number of customers in the system

λi arrival rate in mode i

µi service rate in mode i

L mean number in the system

W mean waiting time in the system

Lq mean number in the queue

Wq mean waiting time in the queue

Synopsis-27

.22

21

222

10,2

22

1

22

11,0

21

21

0

1

10,1

0,2

μλλμ

=

μλλμ

=

λμ

=

λμ

=

=

kp

kp

kp

kp

kp

,12,00,21,00,10 =++++ ppppp

.11

221

2

22

21

22

1

22

12

1

21

1

1

−

⎥⎦

⎤⎢⎣

⎡

μλλμ

+μλλμ

+λμ

+λμ

+=k

.

)()(

)(

10,110,2

21,022,0

1010,2110,1

22,020221,0

10,121,0210

λ=μ

λ=μ

λ+μ=λ+μ

μ+λ=λ+μ

μ+μ=λ+λ

pppp

ppppppppp

4.3. SYSTEM MODEL We first consider the 2- priority M/M/1 queue with infinite buffers operating with a

preemptive priority discipline. Since all the flows are known, the equilibrium

probabilities of each of the states may be found by writing and solving the corresponding

balance equations with the normalisation condition. As an example, consider the 2-

priority M/M/1/2 queue with preemptive priority.

4.4. STEADY-STATE PROBABLITY In this part of this chapter, we derive the steady-state probabilities by the Markov process



On solving these equation we get;

Using

we get,

Synopsis-28

4.4.1. NUMERICAL RESULTS In this subpart, we present some numerical examples to demonstrate how the various

parameters of the model influence the performance measures of the system. We fix the

maximum number of customers in the system N = 2.

Table 4.1: Performance measures for λ1/µ1 = 0.8 λ2/µ2 L Lq W Wq

0.1000 0.8067 0.2543 3.9125 1.5596

0.2000 0.8656 0.2537 3.6087 1.4571

0.3000 0.9044 0.3667 3.4784 1.4101

0.4000 0.9331 0.2666 3.3325 0.9521

0.5000 0.9653 0.2789 3.2176 0.9521

0.6000 0.9998 0.2940 3.1240 0.9189

07000 1.0081 0.2113 2.9647 0.9155

0.8000 1.0721 0.3298 2.9582 0.9161

First, we select the λ1/µ1 =0.8 and change values of arrival rate of customer λ2/µ2 .The

numerical results are summarized in Table 4.1. This shows that: (i) the length of queueing

system increases with the increasing λ2/µ2 .(ii) The length of the queue increases upto

λ2/µ2 from 0.1 to 0.3 , it decrease at 0.4 , again increases from 0.5 to 0.6, again decrease

at 0.7 and then increases. (iii) The Waiting time in queueing system decreases with

increasing λ2/µ2.(iv) The Waiting time in queue again decreases with the increasing λ2/µ2.

The following graph shows the performance of the model at λ1/µ1 = 0.8.

Synopsis-29

00.5

11.5

22.5

33.5

44.5

0 0.2 0.4 0.6 0.8 1

λ2/µ2

L(Length of the system)Lq(Length of the queue)W(Waiting time of system)Wq(Waiting time of queue)

Figure 4.1: Graph between λ2/µ2 and L , Lq ,W, Wq

4.5. CONCLUSION In this chapter, we considered an M/M/1/k queueing system with priority. We developed

the equations of the steady state probabilities and derived the matrix form solution of the

steady-state probabilities. We also gave some performance measures of the system. Some

numerical examples were presented to demonstrate how the various parameters of the

model influence the behavior of the system.

Synopsis-30

REFERENCES

[1] Robert E., “Reneging phenomenon of single channel queues”, Mathematics of

Operations Research 4(2), pp. 162-178, 1979.

[2] Haight F. A., “Queueing with balking”, Biometrika 44, pp. 360-369, 1957.

[3] Haight F. A., “Queueing with reneging”, Metrika 2, pp. 186-197, 1959.

[4] Ancker Jr. C. J., Gafarian A. V., “Some queueing problems with balking and

reneging”, I. Operations Research 11, pp. 88-100, 1963.

[5] Ancker Jr. C. J., Gafarian A. V., “Some queueing problems with balking and

reneging”, II. Operations Research 11, pp. 928-937, 1963.

[6] Abou El-Ata M. O., Hariri A. M. A., “The M/M/C/N queue with balking and

reneging” Computers and Operations Research 19, pp. 713-716, 1992.

[7] Kleinrock L., “Queueing Systems”, vol. 1:Theory, Wiely, New York, 1975.

[8] Heyman D. P., “The push-out priority queue discipline”, Operations Research, vol.33,

no. 2, pp. 397-403, 1985.

[9] Takagi H., “Queueing Analysis: A Foundation of Performance Evaluation”, vol.2,

Finite System, North-Holland, Amsterdam, 1991.

[10] Gravey A., Boyer P., Hebuterne G., “Tagging versus strict rate enforcement in ATM

networks”, Proceedings of IEEE GLOBECOM ’91, pp. 271-275, 1991.

Synopsis-31

[11] Cheng X., Akyildiz I. F., “A finite buffer two class queue with different scheduling

and push-out schemes”, Proceedings of IEEE INFOCOM ’92, pp. 231-241, 1992.

[12] Takagi H., “Explicit delay distribution in first-come first-served M/M/m/K and

M/M/m/K/n queue and a mixed loss-delay system”, Proceedings of Asia-Pacific

Symposium on Queueing Theory and Its Applications to Telecommunication Networks,

Korea, 2006.

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