synopsis -...
TRANSCRIPT
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
µ*(Opimal srevice rate)
E(Nq): Length of queue
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 −
−⎟⎟⎠
⎞⎜⎜⎝
⎛α−+μα+μα+μμ
λ++
α+μα+μμλ
+μ
λ+
μλ
+μλ
+=N
bbbbbbk Nn
( )( ) ( ) .)2(2......2222
)........()( 2
132⎟⎟⎠
⎞⎜⎜⎝
⎛α−+μα+μα+μμ
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
µ*(Opimal srevice rate)
E(Nq): Length of queue
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
µ*(Opimal srevice rate)
E(Nq): Length of queue
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.
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. 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
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.
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.
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 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
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.
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.