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Int. J. Mathematics in Operational Research, Vol. 2, No. 2, 2010 233
Copyright 2010 Inderscience Enterprises Ltd.
Discrete-time batch service GI/Geo/1/Nqueue with
accessible and non-accessible batches
Veena Goswami*
School of Computer Application,
KIIT University,
Bhubaneswar 751024, India
E-mail: [email protected]
*Corresponding author
K. Sikdar
Department of Mathematics,
BMS Institute of Technology,
PO Box 6443,
Doddaballapura Main Road,
Yelahanka, Bangalore 560064, India
E-mail: [email protected]
Abstract: Discrete-time queues are extensively used in modelling theasynchronous transfer mode environment at cell level. In this paper, weconsider a discrete-time single-server finite-buffer queue with general inter-arrival and geometric service times where the services are performed inaccessible or non-accessible batches of maximum size b with a minimumthreshold value a. We provide a recursive method, using the supplementaryvariable technique and treating the remaining inter-arrival time as thesupplementary variable, to develop the steady-state queue/system lengthdistributions at pre-arrival and arbitrary epochs under the early arrival system.
The method is depicted analytically for geometrical and deterministic inter-arrival time distributions, respectively. Various performance measures andoutside observers observation epochs are also discussed. Finally, somecomputational results have been presented.
Keywords: AB; accessible batch; discrete-time queue; finite buffer; NAB;non-accessible batch; supplementary variable.
Reference to this paper should be made as follows: Goswami, V. andSikdar, K. (2010) Discrete-time batch service GI/Geo/1/N queue withaccessible and non-accessible batches, Int. J. Mathematics in Operational
Research, Vol. 2, No. 2, pp.233257.
Biographical notes: Veena Goswami is currently a Professor in the School ofComputer Application, KIIT University, Bhubaneswar, India. She received herPhD from Sambalpur University, India, in the year 1994 and then worked as a
Research Associate at Indian Institute of Technology, Kharagpur for two years.Her research interests include continuous- and discrete-time queues. She has
published research articles in INFORMS Journal on Computing, Computersand Operations Research, RAIRO Operations Research, Computers and
Mathematics with Applications, Computers and Industrial Engineering,Applied Mathematical Modelling, Applied Mathematics and Computation, etc.
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234 V. Goswami and K. Sikdar
K. Sikdar is currently a Lecturer in the Department of Mathematics at BMSInstitute of Technology, Bangalore. She received her PhD from Indian Instituteof Technology, Kharagpur in the year 2004. From August 2005 to July 2008,she worked at Indian Institute of Science, Bangalore as a Post Doctoral Fellow.
Her main research interests include continuous-time queueing theory and itsapplications. She has published research papers in various journals such as
Performance Evaluation, Computers and Operations Research, AppliedMathematics and Computation, and Journal of Applied Mathematics andStochastic Analysis.
1 Introduction
Discrete-time queueing systems have been receiving a notable interest in the last few
decades due to their applications to a variety of slotted digital communication systems
and other related areas. Their importance has grown due to the advent of the broadband
integrated services digital network (B-ISDN), which can support transfer of video, voice
and data communication with varying characterisations and different quality of service(QoS) requirements through high-speed local area networks (LANs), on-demand video
distribution and video telephony communications. The asynchronous transfer mode
(ATM) is conceived as the basic transfer mode for implementing B-ISDN. In these
systems, the time axis is slotted and fixed length packets, called cells are used to
transfer information. Readers are referred to Bruneel and Kim (1993), Takagi (1993),
Woodward (1994) and references therein. In discrete-time queueing systems, the arrivals
and departures can occur simultaneously at a boundary epoch of a slot. In the case of
simultaneity, their order may be taken care of by either arrival-first (AF) or departure-
first (DF) management policies, which are also known as late arrival system with delayed
access (LAS-DA) and early arrival system (EAS), respectively, and both have potentials
for applications. For more details on this topic, see Hunter (1983) and Gravey and
Hbuterne (1992).
Batch-service queueing models are often encountered in applications. Queueing
systems with batch service are common in transportation processes involving trains,
buses, ships, airplanes, elevators, cable cars and intra-campus shuttles that run on
schedule and have considerable economic implications. In semiconductor manufacturing
processes, in service mechanisms of a web server and computer operating systems, jobs
are frequently processed in batches whose sizes usually vary depending on the total
number of jobs accumulated.
Queues with finite buffer space are more realistic in real-life situations than queues
with infinite-buffer space as the former is used to store arrived customers if the server is
busy. However, if the buffer space is full, the arrived customer is considered to be lost. In
such situations, one of the main concerns of a system designer is the estimation of the
blocking probability (PBL) of the customers which, in general, is kept small to avoid loss
of customers. Also, it is widely recognised that the results of the infinite-buffer queuescan be obtained from those of the corresponding finite-buffer counterparts by taking the
finite-buffer parameter sufficiently large.
In this paper, we focus on a more general discrete-time single-server finite-buffer
batch-service queue with accessible and non-accessible batches (NABs) wherein
inter-arrival time and service time of batches are, respectively, arbitrarily and
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Discrete-time batch service 235
geometrically distributed. We provide a recursive method, using the supplementary
variable technique and treating the remaining inter-arrival time as the supplementary
variable, to develop the steady-state probability distributions of the number of customers
in the queue/system. The method is illustrated analytically for geometric anddeterministic inter-arrival time distributions, and the results for geometric distribution
match with Goswami et al. (2006). The distributions of the number of customers in the
queue/system at pre-arrival epochs and at arbitrary epochs, as well as the outside
observers distributions are established.
The rest of this paper is organised as follows. In Section 2, we review the related
work. Section 3 presents the description of the queueing model and provides a recursive
method using the supplementary variables technique, to obtain the steady-state
probability distributions of the number of customers in the queue. We illustrate a
recursive method by presenting simple examples for geometric and deterministic inter-
arrival time distributions. Section 4 presents the outside observers distribution. Section 5
discusses various performance measures and some special cases of our system. Section 6
contains computational results to demonstrate the effectiveness of the model parameters.
Section 7 concludes our paper.
2 Review of related work
Batch-service queues have numerous potential applications in the areas of production
systems, transportation systems, loading and unloading of cargoes at a seaport, traffic
signal systems, computer networks and telecommunication systems where the processor
processes packets in batch. In such batch-service systems, jobs arriving one at a time
must wait in the queue until a sufficient number of jobs get accumulated. There are many
instances where the services are carried out in batches to increase the service rate.
Batch-service queues have been discussed extensively over the last several years.
Note that many papers on batch-service queues have mainly concentrated on the
continuous-time models. The first study on batch-service queues was due to Bailey
(1954) in which the solution to the fixed-size batch-service queue with Poisson arrivals
has been discussed. Neuts (1967) proposed the general bulk service rule in which service
initiates only when a certain number of customers in the queue is available. More
extensive studies on batch-service queues are available in Chaudhry and Templeton
(1983) and Medhi (1984, 1991).
Computational aspects of single-server finite-capacity queue with general bulk
service rule where customers arrive according to a Poisson process and service times of
the batches are arbitrarily distributed have been discussed in Chaudhry and Gupta (1999).
The finite buffer continuous-time queues with general arrivals and batch service have
been studied by Laxmi and Gupta (1999). The queue was analysed using both the
supplementary variable and imbedded Markov chain techniques. Chang and Choi (2006)
have analysed a single-server batch arrival batch-service queue with setup times where
customers arrive according to a Poisson process and service times of the batches are
arbitrarily distributed. Batch service with deterministic service times has been studied by
Chaudhry and Templeton (1983). Chang (2006) obtained an explicit expression for the
mean steady-state waiting time and an estimate of the optimal batch size minimising the
mean steady-state waiting time ofM/DN/1 queue. Increasing convex ordering of queue
length in batch queues has been discussed in Cai and Zhang (2008).
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236 V. Goswami and K. Sikdar
Further, batch service may be with accessible batches (ABs). If a batch being served
does not utilise its full capacity for service, it may remain accessible for customers
arriving during the service time of the batch until its full capacity is attained. The service
time is not changed by inclusion of such joining customers in course of ongoing service.This has been considered by Gross and Harris (1998), Kleinrock (1975) and Medhi
(1984, 1991). The infinite-buffer queue with accessible and non-accessible batch-service
rule has been studied by Sivasamy (1990), where the arrivals and service times are
exponentially distributed.
Discrete-time single-server finite and infinite queueing models with generally
distributed service times have been studied in Gravey and Hbuterne (1992). Gupta and
Goswami (2002) discussed analytic and computational aspects of Geo/Ga,b/1/N queue.
Computational study of a discrete-time batch-service queue with variable capacity and
finite waiting space has been discussed in Chaudhry and Chang (2004) and Yi et al.
(2007). Yi et al. (2007) discussed the model of a single-server having variable capacity
which serves the customers only when the number of customers in a system is at least a
certain threshold value. Some analytic computational results for discrete-time batch-
service queues have been reported in Janssen and Leeuwaarden (2005). Algorithmicanalysis of the discrete-time single-server batch arrival batch-service queue has been
studied by Alfa and He (2008), where the arrivals and service times are generally
distributed. Claeys et al. (2008) computed the probability generating function of the delay
in a discrete-time batch-service queueing model with batch arrivals and single-slot
service time.
The finite and infinite-buffers queues with accessible and non-accessible batch-
service rules in discrete-time systems have been studied by Goswami et al. (2006), where
the arrivals and service times are geometrically distributed. A general uncorrelated arrival
process appears to be more appropriate and reasonable than geometrical distribution, as
the memoryless property of the arrival process does not always meet the need of
applications and also it can include the special cases of geometrical, deterministic, etc.
Therefore, the main purpose of this paper is to do both analytic and computational
analysis of the discrete-time GI/Geo(a,d,b)/1/Nqueue.Discrete-time queueing systems are better suited than their continuous-time
counterparts to evaluate system performance measures in computer and digital
telecommunication networks, because of the clock-driven operation of those systems.
Furthermore, the modelling of discrete-time queues is more involved and quite different
from the analysis used for the corresponding continuous-time queueing models. The
advantage of analysing a discrete-time queue is that one can obtain the continuous-time
results from it as a limiting case but the converse is not true. However from an applied
and a theoretical point of view, both the discrete-and continuous-time queueing models
have importance.
3 The model description
Let us consider a finite buffer GI/Geo(a,d,b)/1/N queue where customers (packets) are
served (transmitted) by a single-server in batches of maximum size b with a minimum
threshold value a. However, if the number of customers in the queue is less than the
minimum threshold value a, the server remains idle until the number of customers in the
queue reaches a. Ifb or more customers are present in the queue at service initiate epoch
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Discrete-time batch service 237
then only b of them are taken into service. It is further assumed to allow the late entries to
join a batch in course of ongoing service as long as the number of customers in that batch
is less than d < b (called maximum accessible limit). At every departure epoch, that is,
before initiating service of the next batch, the server may find the system in any one ofthe following three cases:
1 0 n a1
2 a n d1
3 n d.
In Case 1, the server cannot initiate service, it remains idle. In Case 2, the server takes the
entire queue for batch service and admits the subsequent arrivals in the batch while the
service is on, till the accessible limit dis reached, and such a batch is called an accessible
batch. In Case 3, i t takes min(n; b) customers for the service and does not allow further
arrivals into the batch being served even if the current batch size is not b, that is, when
the batch size is greater than or equal to d, the batch becomes non-accessible for late
arriving customers. The system has finite buffer (queue) capacity of size N(> b), that is,maximum number of customers allowed in the system at any time is (N+ b). The inter-
arrival times {An, n 1} of customers (packets) are independent and identically
distributed (i.i.d.) random variable (r.vs.) with common probability mass function (p.m.f.)
ai =P(An = i), i 1, probability generating function (p.g.f.)1
( ) ,iiiA z a z
f
and mean
inter-arrival time a =A(1)(1) whereA(1)(1) is the first derivative ofA(z) with respect to z
atz= 1. The service times {Sn, n 1} are independent and geometrically distributed with
common p.m.f. 1( ) , 0 1, 1,inP S i nP P P t where 1P P and mean service
time 1/sT P . The traffic intensity is given by 1/ ( )abU PT . In Section 3.1, we discuss
this queue under EAS.
3.1 The GI/Geo(a,d,b)
/1/N queue with EAS
Let us assume that the time axis is slotted into intervals of equal length with the length of
a slot being unity. Further, let the time axis be marked by 0, 1, 2, , t and assume that
the potential arrivals and departures occur in the time interval (t, t+) and (t , t),
respectively. For the sake of understanding, various time epochs at which events
(arrival/departure) occur are depicted in Figure 1.
Figure 1 Various time epochs in EAS
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238 V. Goswami and K. Sikdar
The state of the system prior to a potential arrival (at t), that is before the beginning of the
slot, is described by the following random variables, namely,
x Ns(t) = number of customers present in the system including those in service.
x Nq(t) = number of customers present in the queue not counting those in service.
x U(t) = remaining inter-arrival time for the next arrival.
x0, if the server is idle or busy with an accessible batch,
( )1, if the server is busy with a non-accessible batch.
t[-
Let us define joint probabilities by
,0 ( , ) ( ) , ( ) , ( ) 0 , 0 1, 0,n sQ u t P N t n U t u t n d u[ d d t
,1( , ) ( ) , ( ) , ( ) 1 , 0 , 0.n qQ u t P N t n U t u t n N u[ d d t
In the steady-state, let us define
, ,( ) lim ( , ), 0,1.n j n jt
Q u Q u t jof
To obtain the queue length distribution at arbitrary epochs and performance measures of
the system, we develop the difference equations using the remaining inter-arrival time as
the supplementary variable. Observing the state of the system at two consecutive time
epochs t and (t+ 1), using definitions and probabilities defined above, we have in the
steady-state the following difference equations foru 1
1 1
0,0 0,0 ,0 0,1 ,0
1
( 1) ( ) ( ) ( ) (0),
d d
k u k
k a k a
Q u Q u Q u Q u a QP P P
(1)
,0 ,0 ,1 1,0 1,1( 1) ( ) ( ) (0) (0), 1 1,n n n u n u nQ u Q u Q u a Q a Q n aP P d d (2)
,0 ,0 ,1 1,0 1,1( 1) ( ) ( ) (0) (0), 1,n n n u n u nQ u Q u Q u a Q a Q a n d P P P P d d (3)
1
0,1 0,1 ,1 1,0 ,1
1
( 1) ( ) ( ) (0) (0),
b b
k u d u k
k d k d
Q u Q u Q u a Q a QP P P P
(4)
,1 ,1 ,1 1,1 1,1( 1) ( ) ( ) (0) (0),
1 1,
n n n b u n u n bQ u Q u Q u a Q a Q
n N b
P P P P
d d (5)
,1 ,1 ,1 1,1
1,1 ,1
( 1) ( ) ( ) (0)
(0) (0) ,
N b N b N u N b u
N N
Q u Q u Q u a Q a
Q Q
P P P P
(6)
,1 ,1 1,1( 1) ( ) (0), 1 1,n n u nQ u Q u a Q N b n N P P d d (7)
,1 ,1 1,1 ,1( 1) ( ) (0) (0) .N N u N NQ u Q u a Q QP P (8)
Let us define the p.g.f. ofQn,j (u) by
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Discrete-time batch service 239
*, ,
0
( ) ( ) , | | 1.u
n j n j
u
Q z Q u z z
f
d (9)
Let Qn,0 denotes the probability ofn customers in the system when the server is idle orbusy with an AB and let Qn,1 denotes the probability ofn customers in the queue when the
server is busy with a NAB at an arbitrary epoch. They are given by
* *,0 ,0 ,0 ,1 ,1 ,1
0 0
( ) (1), 0 1, ( ) (1), 0 .n n n n n nu u
Q Q u Q n d Q Q u Q n N
f f
d d d d
Multiplying (1) to (8) byzuand summing overu from 1 to , and using (9), we obtain
1 1* * *0,0 ,0 0,1 ,0 0,0
1
1
,0 0,1
( 1) ( ) ( ) ( ) (0) ( ) (0)
(0) (0),
d d
k k
k a k a
d
k
k a
z Q z Q z Q z Q A z Q
Q Q
P P P
P P
(10)
* *,0 ,1 1,0 1,1 ,0
,1
( 1) ( ) ( ) (0) ( ) (0) ( ) (0)
(0), 1 1,
n n n n n
n
z Q z Q z Q A z Q A z Q
Q n a
P P
P
d d (11)
* *,0 ,1 1,0 1,1 ,0
,1
( ) ( ) ( ) (0) ( ) (0)(0)
(0), 1,
n n n n n
n
z Q z Q z Q A z Q A z Q
Q a n d
P P P P P
P
d d (12)
1
* *0,1 ,1 1,0 ,1
1
,1 0,1
( ) ( ) (0) ( ) (0) ( )
(0) (0),
b b
k d k
k d k d
b
kk d
z Q z Q z Q A z Q A z
Q Q
P P P P
P P
(13)
* *,1 ,1 1,1 1,1 ,1
,1
( ) ( ) (0) ( ) (0) ( ) (0)
(0), 1 1,
n n b n n b n b
n
z Q z Q z Q A z Q A z Q
Q n N b
P P P P P
P
d d (14)
* *,1 ,1 1,1 1,1 ,1,1 ,1
( ) ( ) (0) ( ) (0) (0) ( )
(0) (0),
N b N N b N N
N b N
z Q z Q z Q A z Q Q A z
Q Q
P P P P
P P
(15)
*,1 1,1 ,1( )= (0) ( ) (0), 1 1,n n nz Q z Q A z Q N b n NP P P d d (16)
* ,1 1,1 ,1 ,1( )= (0) (0) ( ) (0).N N N Nz Q z Q Q A z QP P P (17)
One important result which is used frequently can be obtained using (10)(17). This is
given below:
Theorem 3.1: The mean number of entrances into the system per unit time equals the
mean arrival rate that is
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240 V. Goswami and K. Sikdar
1
,0 ,1
0 0
1(0) (0) (say).
d N
n nan n
Q Q OT
(18)
Proof: Adding (10)(17), we get
1 1* *,0 ,1 ,0 ,1
0 0 0 0
( ) 1( ) ( ) (0) (0) .
1
d N d N
n n n n
n n n n
A zQ z Q z Q Q
z
-
Taking the limit asz 1 and using the normalisation condition:
1
,0 ,1
0 0
1,
d N
n n
n n
Q Q
after simplification we get the desired result.
3.1.1 Steady-state distribution at pre-arrival epochs
Let ,0 ,1( )n nQ Q represents the probability that there are n customers present in the system
(queue) prior to an arrival epoch of a customer when the server is idle or busy with an
accessible (a non-accessible) batch. To obtain the steady-state distribution of the number
of customers in the queue/system at pre-arrival epochs, we first connect pre-arrival epoch
probabilities ,0 (0 1)nQ n d d d and ,1(0 )nQ n N
d d with the rates Qn,0(0) (0 n d 1)
and Qn,1(0) (0 n N). These are given by
,
, ,1
,0 ,1
0 0
(0) 1(0), 0,1,
(0) (0)
n j
n j n jd N
n n
n n
QQ Q j
Q QO
(19)
where O is given by (18) and ,0 ,1( )n nQ Q represents the probability that there are n
customers present in the system (queue) prior to an arrival epoch of a customer when the
server is idle or busy with an accessible (a non-accessible) batch.
Now, we first evaluate Qn,0(0) (0 n d1) and Qn,1(0) (0 n N) from (11)(17) inthe following manner. Settingz= 1 in (17) and z P in (17) and (16), we finally obtain
,1 ,1,(0) ,n n NQ Q N b n N I d d (20)
where
^ `1
, ,1
, 1, 2, , 1.n
N n
An N
A
n N N N bA
P P
P PI
P
P P
-
!
(21)
Setting z P in (12)(15), we get
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Discrete-time batch service 241
*1,1 1 ,1 2 ,1(0) ,N b N b N N NQ Q QI I P (22)
1,1 *
,1 2 1,1 1,1 1 ,1
(0)(0) (0) (0),
2, 3,...,1,0,
n
n N n b n b N n b
QQ Q Q Q
A
n N b N b
I P IP
(23)
10,1 *
1,0 2 ,1 ,1 1 ,1
1
(0)(0) (0) (0),
b b
d N k k N k
k d k d
QQ Q Q Q
AI P I
P
(24)
1,0 *, 2 1,1 1,1 1 ,1
(0)(0) (0) (0),
2, 3,..., 1,
n
n o N n n N n
QQ Q Q Q
A
n d d a
I P IP
(25)
where *,1( ), , 1,..., ,iQ i N N aP appearing in (22)(25) can be obtained from (14) to
(17) in the following manner.We obtain *,1( ), , 1,...,iQ i N N aP by differentiating (14)(17) with respect to z
and setting z P . Differentiating (17) with respect to z, j (=N 1) times and
simplifying, we have
( )*( 1) *( )
,1.,1 ,1
( )( ) ( )
1
jj j
NN N
A zjQ z z Q z Q
A
PP
P
(26)
Differentiating (16) with respect toz,j (= n 1) times, we obtain
`^
( )*( 1) *( )
,1,,1 ,1
( )( ) ( ) 1, 2,..., 1.
jj j
Nn n N n
A zjQ z z Q z Q n N N N b
A
PP
P
(27)
Differentiating (15) with respect toz,j (=N b 1) times, we get
2*( 1) *( ) *( ) ( )
1,1 ,1,1,1 ,1( ) ( ) ( ) (0) ( ).1
j j j jN b NNN b N bjQ z z Q z Q z Q Q A z
A
PP P P
P P
-
(28)
Differentiating (14) with respect toz,j (= n 1) times, we obtain
^ `*( 1) *( ) *( ) ( )1,1 1,1,1 ,1 ,1( ) ( ) ( ) (0) (0) ( ),1, 2,..., 2,1,
j j j jn n bn n n bjQ z z Q z Q z Q Q A z
n N b N b
P P P P
(29)
where *(0) *,1,1 ( ) ( ).iiQ z Q z Now substituting z P in (26)(29), we obtain
*,1( ), , 1,....,iQ i N N aP and given by
( )*( 1)
,1,,11
jj
NN
AQ Q
j A
P PP
P
(30)
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242 V. Goswami and K. Sikdar
`^
( )*( 1)
,1,,1 1, 2, , 1,
jj
Nn N n
AQ Q n N N N b
j A
P PP
P
! (31)
*( ) ( )2,1*( 1)
1,1 ,1,1 (0) ,1
j jNj
N b NN b
Q AQ Q Q
j jA
P P PPP P
P P
-
(32)
^ `
*( ) ( ),1*( 1)
1,1 1,1,1 (0) (0) ,
1, 2,..., 2,1.
j jn bj
n n bn
Q AQ Q Q
j j
n N b N b
P P PP P P
(33)
Now, substitutingz= 1 in (11), we obtain
,0 1,0 1,1 ,1 1,1(0) (0) (0) (0) , 2, 3,...,1,0.n n n n nQ Q Q Q Q n a aP (34)
Settingz= 1 in (14)(16), respectively, then Qn+1,1 n = 0, 1, , a2, occurred in (34) are
given by
^ `,1 ,1,
11 1,n NN n
AQ Q N b n N
A
P
P
d d (35)
,1
,1 1,1 ,1(0) (0) ,N
N b N b N b
QQ Q Q
P
P P (36)
,1 ,1 1,1 ,1 1,1 ,1(0) (0) (0) (0) ,
1, 2,..., 2,1.
n n b n b n b n nQ Q Q Q Q Q
n N b N b
PP
P
(37)
It may be noted that to evaluate ,n jQ from (19) we do not require the value ofQN,1, since
it cancels out in the numerator and denominator of RHS of (19).
3.1.2 Steady-state distribution at arbitrary epochs
To obtain the queue/system length distribution at arbitrary epochs, we develop relations
between distributions of number of customers in the queue/system at pre-arrival and
arbitrary epochs. This is discussed in the following theorem.
Theorem 3.2: The arbitrary epoch probabilities are given by
,1 1,1,N NQ b QUP (38)
,1 1,1 ,1 , 1 1,n n nQ b Q Q N b n N UP d d (39)
,1 ,1 1,1 ,1 1,1,N b N N b N b NQ Q b Q Q QUP O (40)
,1 ,1 1,1 ,1 1,1 ,1 ,1, 2,...,1,
n n b n n n b n bQ Q b Q Q Q Q
n N b N b
UP O
(41)
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Discrete-time batch service 243
0,1 ,1 1,0 0,1 1,1 ,1 ,b
k d d b
k d
Q Q b Q Q Q QUP O
(42)
,0 ,1 1,0 ,0 1,1 ,1 , 1.n n n n n nQ Q b Q Q Q Q a n d UP O d d (43)
Proof: Settingz= 1 in (12)(17), dividing both sides by O and using (19), we obtain theresult of the theorem.
One may note here that from this Theorem 3.2, we cannot get 1,0 0{ }a
nQ . However,
these can be obtained using the following theorem.
Theorem 3.3: The arbitrary epoch probabilities 1,0 0{ }a
nQ are given by
1 1*(1) *(1)
0,0 ,00,1,0
1
(1) (1) ,
d d
kk
k a k a
Q Q Q QP P P
(44)
*(1),0 1,0 1,1,1 (1) , 1 1,n n nnQ Q Q Q n aP P d d (45)
where *(1),0 (1) ( 1)kQ a k d d d and*(1),1 (1) (0 1)nQ n ad d can be obtained from
*(1) 1,1 ,1 ,1,11
(1) ,N N NNQ Q Q QPP
(46)
*(1) 1,1 ,1,11
(1) , 1 1,n nnQ Q Q N b n N PP
d d (47)
,1*(1) *(1) 1,1 ,1 1,1,1,11
(1) (1) ,NN b N b NNN bQ Q Q Q Q QPP
(48)
*(1) *(1) 1,1 ,1 1,1,1 ,11
(1) (1) , 1 1,n n n bn n bQ Q Q Q Q n N bPP
d d (49)
1
*(1) *(1)1,0 0,1 ,10,1 ,1
1
1(1) (1) ,
b b
d kk
k d k d
Q Q Q Q QPP
(50)
*(1) *(1) 1,0 ,0 1,1,0 ,11
(1) (1) , 1.n n nn nQ Q Q Q Q a n d PP
d d (51)
Proof: Differentiating (10)(17) with respect to z and setting z= 1, we get the desired
results.
3.2 Algorithm for computing state probabilities
To demonstrate the working schemes of the recursive method, we describe the solution
algorithm for calculating the steady-state probabilities. Given the values of P, a, d, b,N
and the p.g.f. expression of the inter-arrival time distribution, namely,A(z), the steps of
the solution algorithm are stated as follows:
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Step 1 Set QN,1 = 1.
Step 2 Compute In forN b 1 n Nusing (21).
Step 3 Compute Qn,1(0) forn =N,N 1, ,N b and QNb,1(0) from (20).
Step 4 For n =N, N 1, , a, and j (= n 1) times, compute *( 1),1 ( )j
nQ P from
(30)(33).
Step 5 Compute Qn,1 (0) forn =N b 1, , 1, 0 from (22) and (23), respectively.
Step 6 Compute Qn,0 (0) forn = d 1, , 0 from (24), (25) and (34), respectively.
Step 7 Compute ,n jQ from (19).
Step 8 Compute Qn,1 for 0 n Nand Qn,0 fora n d 1 using (38)(43).
Step 9 Compute *(1),1 (1)nQ for 0 j a 1 and*(1),0 (1)nQ fora n d 1, from (46)(51).
Step 10 Compute Qn,0 for 0 n d 1 from (44) and (45).
3.3 Simple examples
We use the solution algorithm to illustrate a recursive method. We discuss two simple
examples for two different inter-arrival time distributions such as geometric and
deterministic, respectively.
Example 1: For Geo/Geo(a,d,b)
/1/N queueing system, we set the mean inter-arrival time
a = 1/O, where O is the inter-arrival time. Assume that a = 2, d = 4, b = 5 and N = 7. In
this case, we have
( ) .1
zA z
z
O
O
Step 1 Set Q7,1 = 1.
Step 2 For 1 n 7, compute In using (21).
Using (21), we have6
7
1, , 1 6.
n
n nP OP
I O IP OP
d d
Step 3 For 2 n 7, compute Qn,1 (0) using (20).
From (20), we obtain
,1 7,1,
(0) for 2 7.n n
Q Q nI d d
Step 4 Forn = 7, 6, , 2, compute *( ),1 ( )j
nQ P using (30)(33).
From (30)(33), we finally get
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*( )7,17,1 1
7*( )7,1,1 2
5
*( )7,12,1 2
,1
1, 3 6,
1
11 .
1 1
jj
j
njjn j
jj
j
Q Q
Q Q n
Q j j Qj
OOP
OP
OPO OP P
OPOP
OPO OP P O
OPOP
d d
-
Step 5 Forn = 1, 0, compute Qn,1(0) using (22) and (23), respectively.
From (22) and (23), it follows that
5
*1,1 1 7,1 5 7,1 7,1
1,1 *0,1 5 6,1 6,1 6 5,1
6
7,12 2
1 1(0) ,
(0)(0) (0) (0)
1 1.
Q Q Q Q
QQ Q Q Q
A
Q
P OPI I P
P OP P
I P IP
P OP OP P
P OP OP OP
Step 6 Forn = 3, 2, 1, 0, compute Qn,0 (0) using (24), (25) and (34).
Using (24), (25) and (34), it follows that
6
3,0 3 7,1 3 2 2
22
2,0 2 7,1 2 3 3
42 2
1,0 1 7,1 1 2 3 3
1 1 2 1, where 1 ,(0)
1 1(0) , where ,
1 1(0) , where
Q Q
Q Q
Q Q
P OP OP OP P P OP Z Z
P OP OP OP OP OP
OP P OP Z Z Z
OP OP OP
OP P OP P Z Z Z
OP OP OP P
-
5 6
0,0 0 7,1 0 1 2 2
,
1 1 1 1(0) , where 1 .Q Q
P OP OP OP P OP Z Z Z
P OP P OP OP OP OP
-
Step 7 From (19), compute , ,n jQ
,0 ,0 ,1 ,1(0) / , 0 3,and (0) / , 0 7.n n n nQ Q n Q Q nO O d d d d
Step 8 Compute Qn,1 for 0 n 7 using (38)(42) and Qn,0 forn = 2, 3 using (43).
Using (38)(42), we get
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246 V. Goswami and K. Sikdar
6
,1 7,1
1, 3 6,
n
nQ Q nP OP
OP OP
d d
4
2,1 7,1
1,Q Q
P OPO
OP OP
5
1,1 7,1
1,Q Q
P OP P
OP OP P
5 2
30,1 7,12
1 1 1 1.Q Q
Z POP OP P OP P OP P OP P
OP OP OP OP P OP OP P OP
- Using (43), we get
3
3,0 2 3 7,121 ,Q QOP P P Z Z
OP POP
4
2,0 1 2 7,12 2
1 1.Q Q
OP P P P Z Z
OP POP P
Step 9 Compute *(1),1 (1)nQ forn = 1, 0, and*(1),0 (1)nQ and forn = 3, 2 from (46)(51).
It yields from (46)(51) that
*(1)5,1 0,1 6,1 1,11,1
*(1)4,1 3,1 3,0 5,1 4,1 0,10,1
*(1)2,1 2,0 3,1 3,03,0
*(1)7,1 6,1 1,1 1,0 7,1 2,1 2,02,0
1(1) ,
1(1) ,
1(1) ,
1(1) .
Q Q Q Q Q
Q Q Q Q Q Q Q
Q Q Q Q Q
Q Q Q Q Q Q Q Q
P
P
P
P
P
P
P
P
Step 10 Compute Q1,0 and Q0,0 using (44) and (45).
Using (44) and (45), it follows that
*(1)1,0 0,1 0,01,1
52 0
1,0 7,12
4 3 5 3
0,0 7,1 6,1 ,1 ,0 7,1 ,1 0,1 ,0
1 1 2 2
(1) ,
1 111 ,
.n n n nn n n n
Q Q Q Q
Q Q
Q Q Q Q Q Q Q Q Q
P
OP P OP ZP OP P
OP OP OP O OP
- -
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The above results are in accordance with the expressions given in Goswami et al. (2006).
Example 2: For D/Geoa,d,b
/1/N queueing system, we set the mean inter-arrival time
a = k. Assume that a = 2, d = 4, b = 5 and N = 7. In this case, we have
( ) , ,1 1k k kA z z A AP P P P
Step 1 Set Q7,1 = 1.
Step 2 For 1 n 7, compute In using (21).
Using (21) we have
7 6
, , 1 6.1
k
n nk kn
PP PI I
P P P P
d d
Step 3 For 2 n 7, compute Qn,1 (0) using (20).
From (20) we have
,1 7,1(0) , for 2 7.n nQ Q nI d d
Step 4 Forn = 2, 3, , 7, compute *( ),1 , 1 7j
nQ nP d d using (30)(33).
1*( )
7,17,1
1*( )
7,1,1 7
2 2 17,1*( )
2,1 4
!,
( 1) 1 ( 1)!
!, 3 6,
( 1) ( 1)!
( 1)! 1.1 11 ( 2)!
k jj
k
k jj
n nk
k j kj
k k kk
kQ Q
j k j
kQ Q n
j k j
Q k kQ j k j
PPP
P
PPP
P
P P P PP P P P PP
d d
-
Step 5 Forn = 1, 0, compute Qn,1 (0) using (22) and (23), respectively.
From (22) and (23), it follows that
1
1,1 7,141
1 2
0,1 7,14 22 1
1(0) ,
1
1(0) .
1
k
kk k
k
k kk k
kQ Q
k kQ Q
P PP
PP P
P PP P
P PP P
-
Step 6 Forn = 3, 2, 1, 0, compute Qn,0 (0) using (24), (25) and (34).
Using (24), (25) and (34), it follows that
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248 V. Goswami and K. Sikdar
1 2 2
3,0 3 7,1, 3 3 1 4 2 2 3 2
2
2,0 2 7,1, 2 3 3 2
33 2
1,0 1 7,1, 1 2 4 2
0,0
1 2(0) where ,
1
1(0) where ,
( 1)1(0) where ,
2 1
(0)
k
k k k k k
k k
k
k kk
k k kQ Q
kQ Q
k k kQ Q
Q
P PP P PZ Z
P P P P P
PZ Z Z
P P
P P PZ Z Z
P PP
-
210 7,1, 0 1 2 1 4 2
1 11where .
1
k kk
k k kk
kkQ
P P P PP P P PZ Z Z
P P PP
-
Step 7 From (19), compute , ,n jQ
,0 , 0 ,1 ,1(0) / , 0 3, and (0) / , 0 7.n n n nQ Q n Q Q nO O d d d d
Step 8 Compute Qn,1 for 0 n 7 using (38)(42) and Qn,0, forn = 2, 3 using (43).
Using (38)(42), we get
25 1
,1 7,1 2,1 7,17 5
1
1,1 7,11 2 4
1 21
0,1 4 3 4
1 11, 3 6, ,
1
1 1 1,
1
1 1 1 1
1
k k kk
n n k kk
k k k
k k k k
k kk k k
k k k k
kQ Q n Q Q
k kQ Q
k k k
Q
P P P P P
P PP
P P P PP
P P P P
PP P P PP PP P
P P P P
d d
-
7,13 12
.k
k
Q
PP
P
Using (43) we get
3,0 2 3 7,13 1
2 1 3 5 1 6 1
2,0 1 2 7,16
1 1,
1 1 2 1.
1
k
k k
k k k k k
k k
Q Q
k kQ Q
P PZ Z
PP P
P P P P P P P PZ Z
PP P
Step 9 Forn = 1, 0, compute *(1),1 (1)nQ and forn = 3, 2, compute*(1),0 (1)nQ from (46)(51).
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*(1)5,1 0,1 6,1 1,11,1
*(1)
4,1 3,1 3,0 5,1 4,1 0,10,1
*(1)2,1 2,0 3,1 3,03,0
*(1)7,1 6,1 1,1 1,0 7,1 2,1 2,02,0
1(1) ,
1
(1) ,
1(1) ,
1(1) .
Q Q Q Q Q
Q Q Q Q Q Q Q
Q Q Q Q Q
Q Q Q Q Q Q Q Q
PP
PP
PP
PP
Step 10 Compute Q1,0 and Q0,0 using (44) and (45).
Using (44) and (45), it follows that
11 2 11,0 0 7,12 4 2 1
4 3 5 3
0,0 7,1 6,1 ,1 ,0 7,1 ,1 0,1 ,0
1 1 2 2
1(1 ) (1 ) 1,
1
.
kk k
k k k k
n n n n
n n n n
kk k k k Q k Q
Q Q Q Q Q Q Q Q Q
P PPP PP P PP Z
P P P P
4 Outside observers distribution
Since an outside observers distribution plays an important role in evaluating
performance measures, its discussion seems important too. For example, in order to use
Littles rule to get the average waiting time in the queue/system, the average number of
customers in the queue/system at the outside observers observation epoch is needed.
Since an outside observers observation epoch falls in a time interval after a potential
arrival and before a potential batch departure, the probability 0 0,0 ,1( )n nQ Q that the outside
observer sees n customers in the system (queue) with an accessible (a non-accessible)
batch when the server is idle (busy) can be obtained by using the relation
1
0,0 0,0 ,0 0,1
,0 ,0 ,1
,0 ,0 ,1
0,1 0,1 ,1
,1 ,1 1
,1 ,1
,
, 1 1,
, 1,
,
, 1 ,
, 1 .
d
k
k a
n n n
n n n
b
k
k d
n n n b
n n
Q Q Q Q
Q Q Q n a
Q Q Q a n d
Q Q Q
Q Q Q n N b
Q Q N b n N
R R R
R R
R R
R R
R R
R
P P
P
P P
P P
P P
P
d d
d d
d d
d d
The above relations have been obtained by considering arbitrary and outside observers
observation epochs in Figure 1. Now, solving for ,0nQR and ,1nQ
R , we obtain
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250 V. Goswami and K. Sikdar
,1
,1 , 1 ,n
n
QQ N b n N R
P d d (52)
,1 ,1 ,11 , , 1, ,1,n n n bQ Q Q n N b N bR RP
P ! (53)
0,1 0,1 ,1
1,
b
k
k d
Q Q QR RPP
(54)
,0 ,0 ,1
1, 1,n n nQ Q Q a n d
R RP
P d d (55)
,0 ,0 ,1, 1 1,n n nQ Q Q n aR R
P d d (56)
1
0,0 0,0 ,0 0,1 .
d
k
k a
Q Q Q QR R R
P
(57)
Now, ,0nQR and ,1nQ
R can be computed by recursion using Qn,0 and Qn,1.
5 Performance measures and special cases
5.1 Performance measures
Performance measures are important features of queueing systems as they reflect the
efficiency of the queueing system under consideration. Once the state probabilities at pre-
arrival, arbitrary and outside observers observation epochs are known, we can evaluate
various performance measures such as the average number of customers in the queue at
an arbitrary epoch (Lq), average waiting time in the queue (Wq) and the PBL are given by
1
,0 ,1 ,1
1 1
, PBL .
a N
q n n N
n n
L nQ nQ Q
It may be noted that to obtain average waiting time in the queue (Wq) of a customer, we
use Littles rule. However, to use this rule, we need to evaluate average queue length at
outside observers observation epoch which is denoted by qLR and is given by
10,0 ,1
1 1
.
a N
q n n
n n
L nQ nQR R
Therefore, '/q qW LR O where ,1(1 )NQO O c being the effective arrival rate.
5.2 Special cases
In this section, some special cases are deduced from our model by taking specific values
for the parameters a, dand b.
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Case 1: a = d= b = 1, that is, the batch size is one. The model reduces to GI/Geo/1/N
queue formulated by Chaudhry and Gupta (1999).
Let us define Q0 = Q0,0 and Qn = Qn1,1, 1 n N+ 1, where Qn = Pr{number in system
is n}.Then from (38)(43), we obtain
1 ,
1 , 1, , 1.
UP
UP UP
!
N N
n n n
Q Q
Q Q Q n N N
Using normalisation condition, we get
0 11 1 .o NQ Q QUP U
It is to be noted that (39) and (43) will not occur in this case. Then from equations (52)
(57), we obtain outside observers observation epochs as
1 1
1
0 0 1
1 ,
1, , 1, , 2,1,
.
N N
n n n
Q Q
Q Q Q n N N
Q Q Q
R
R R
R R
P
PP
P
!
It is to be noted that (55) and (56) will not occur in this case. Finally, the average waiting
time in the queue which is equivalent to Chaudhry and Gupta (1999) is given by
/ ,q qW LR
Oc where 1 1 1and (1 )N
q n n N L nQ QR R
O O
c being the effective arrival rate.
Case 2: a = d, that is, the server is non-accessible. The model reduces to standard
GI/Geo(a,b)/1 =N queue. Substituting a = d in Equations (20), (22)(24) and (34),
(38)(42) and (44)(45), (52)(54) and (56)(57), the steady-state distributions at pre-arrival, arbitrary and outside observers observation epochs can be deduced. It is to be
noted that (25), (43) and (55) will not occur in this case. The result of this model is not
available in the literature.
6 Computational results
In this section, we present some computational results in the form of tables and graphs, to
demonstrate the effectiveness of the model parameters. The results for the geometric
inter-arrival time distributions at pre-arrival, arbitrary and outside observers observation
epochs were obtained and are presented in Table 1, taking the following input parameters
O= 0.4, P= 0.2, a = 2, d= 4, b = 6, N= 10 and O= 0.5, P= 0.1, a = 5, d= 7, b = 10,
N= 15. The results for the deterministic and the arbitrary inter-arrival time distributionsat pre-arrival, arbitrary and outside observers observation epochs are presented in
Table 2, taking O= 0.5, P= 0.25, a = 2, d= 5, b = 7,N= 15 and O= 0.049261, P= 0.018,
a = 5, d= 7, b = 10, N= 15, respectively. Various performance measures such as the
blocking probabilities, the average queue-length at outside observers observation epoch
and the average waiting times in the queue using Littles rule are given at the bottom of
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252 V. Goswami and K. Sikdar
Tables 1 and 2. It can be seen from Table 1 that, for the geometrical inter-arrival times,
the queue length distributions at pre-arrival and arbitrary epoch probabilities are the same
due to the memoryless property of Bernoulli arrivals. We have matched our results with
Chaudhry and Gupta (1996) by taking a = d= b = 1. Further, the results obtained havebeen matched with Goswami et al. (2006) by taking geometric inter-arrival time
distribution.
Table 1 Distributions of system length at various epochs forGeo/Geo(a,d,b)/1/Nqueue
O= 0.4, P= 0.2, a = 2
d = 4, b = 6, N=10
O= 0.5, P= 0.1, a = 5
d = 7, b = 10, N=15
(n, j) ,n jQ
,n jQ ,n jQR
,n jQ
,n jQ ,n jQR
0,0 0.238805 0.238805 0.086795 0.031018 0.031018 0.025049
1,0 0.276542 0.276542 0.256775 0.051614 0.051614 0.038306
2,0 0.170385 0.170385 0.250286 0.069069 0.069069 0.057792
3,0 0.104978 0.104978 0.154863 0.083844 0.083844 0.0742994,0 0.096336 0.096336 0.088267
5,0 0.086047 0.086047 0.091192
6,0 0.076844 0.076844 0.081445
0,1 0.076741 0.076741 0.088036 0.078870 0.078870 0.077857
1,1 0.048670 0.048670 0.059898 0.066892 0.066892 0.072881
2,1 0.030839 0.030839 0.037971 0.056662 0.056662 0.061777
3,1 0.019525 0.019525 0.024051 0.047941 0.047941 0.052301
4,1 0.012890 0.012890 0.015544 0.040518 0.040518 0.044229
5,1 0.007933 0.007933 0.009916 0.038972 0.038972 0.039745
6,1 0.004882 0.004882 0.006102 0.031886 0.031886 0.035429
7,1 0.003004 0.003004 0.003755 0.026089 0.026089 0.0289888,1 0.001849 0.001849 0.002311 0.021345 0.021345 0.023717
9,1 0.001138 0.001138 0.001422 0.017464 0.017464 0.019405
10,1 0.001820 0.001820 0.002275 0.014289 0.014289 0.015877
11,1 0.011691 0.011691 0.012990
12,1 0.009565 0.009565 0.010628
13,1 0.007826 0.007826 0.008696
14,1 0.006403 0.006403 0.007115
15,1 0.028815 0.028815 0.032016
sum 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000
Lq = 0.693459, Wq = 1.736809,
PBL = 0.00182
Lq = 3.360797, Wq = 6.921021,
PBL = 0.028815
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Table 2 Distributions of system length at various epochs forD/Geo(a, d ,b)/1/Nand GI/Geo(a, d, b)
/1/Nqueue with a10 = 0.5, a15 = 0.1, a24 = 0.2, a45 = 0.2
D/Geo(a,d,b)/1/N GI/Geo(a,d,b)/1/N
O= 0.5, P= 0.25, a = 2
d = 5, b = 7, N=15
O= 0.049261, P= 0.018, a = 5
d = 7, b = 10, N=15
(n, j) ,n jQ
,n jQ ,n jQR
,n jQ
,n jQ ,n jQR
0,0 0.292197 0.197521 0.049769 0.094311 0.073792 0.069049
1,0 0.309089 0.302859 0.296067 0.112371 0.107618 0.106766
2,0 0.173884 0.218289 0.285892 0.125352 0.121936 0.121323
3,0 0.097822 0.122916 0.160948 0.134681 0.132226 0.131786
4,0 0.055032 0.069213 0.090608 0.141386 0.139622 0.139305
5,0 0.104182 0.112842 0.114675
6,0 0.076701 0.083104 0.084457
0,1 0.030963 0.041528 0.053563 0.059372 0.063500 0.0643531,1 0.017644 0.020509 0.027168 0.042680 0.046537 0.047359
2,1 0.010056 0.011688 0.015482 0.030677 0.033450 0.034042
3,1 0.005730 0.006660 0.008823 0.022047 0.024041 0.024466
4,1 0.003264 0.003795 0.005027 0.015843 0.017276 0.017582
5,1 0.001860 0.002162 0.002864 0.011685 0.012654 0.012859
6,1 0.001059 0.001231 0.001631 0.008305 0.009083 0.009250
7,1 0.000603 0.000701 0.000929 0.005903 0.006456 0.006574
8,1 0.00349 0.000404 0.000531 0.004195 0.004589 0.004673
9,1 0.000196 0.000229 0.000305 0.002982 0.003261 0.003321
10,1 0.000110 0.000129 0.000172 0.002119 0.002318 0.002360
11,1 0.000062 0.000072 0.000097 0.001506 0.001647 0.001678
12,1 0.000035 0.000041 0.000054 0.001071 0.001171 0.001192
13,1 0.000020 0.000023 0.000031 0.000761 0.000832 0.000848
14,1 0.000011 0.000013 0.000017 0.000541 0.000592 0.000602
15,1 0.000014 0.000017 0.000022 0.001329 0.001454 0.001480
Sum 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000
Lq = 0.442784, Wq = 0.885581,
PBL = 0.000014
Lq = 1.892262, Wq = 38.464045
PBL = 0.001329
Figure 2 compares the effect of buffer size (N) on the probability of blocking (PBL) for
various inter-arrival time distributions with the following parameters: O= 0.05,
P = 0.018, a = 5, d= 7 and b = 10. As an effect of buffer size on the probability ofblocking, we observed that probability of blocking decreases as buffer size increases.
However, we further noted that the improving effect of buffer size on probability of
blocking is magnified as inter-arrival time variability increases. This is because the effect
of assigning buffer capacity is greater in longer lines in which the frequency of coupling
events is relatively higher. With the same reasoning, the extra buffer capacity yields a
reduction in probability of blocking in the high inter-arrival time variability case as
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254 V. Goswami and K. Sikdar
compared to the low inter-arrival time variability case. It can be seen that the geometrical
distribution gives the highest value for the PBL and the deterministic distribution gives
the smallest value. This is intuitively clear since the PBL of the inter-arrival times is
higher for the geometric distribution. This observation can be understood from the factthat the higher the buffer contents, more customers get lost for a given amount of buffer
space. We further observe that for all distributions considered here, the PBL decreases as
buffer sizeNincreases and finally reaches to its minimum value zero as it should be. This
is due to the fact that the model becomes an infinite-buffer queue. Hence, we can setup an
admissible buffer size in the system in order to have lower PBL.
Figure 3 depicts the effect ofa on the average waiting time (Wq) when inter-arrival
time is geometric with O= 0.05, P= 0.018, b = 40 andN= 50. We varied the AB size d.
It can be observed that, for fixed AB size d, Wq monotonically increases with the increase
of minimum threshold value a. Further, with fixed minimum threshold value a, the
average waiting time decreases when the AB size d increases. For large minimum
threshold value a the increase is almost linear. The effect of minimum threshold value a
on average waiting time in the queue does not terminate as minimum threshold value a
increases. The above findings suggest that minimum threshold value a should not beincreased beyond a certain limit. The minimum threshold value a and the AB size din the
system can be carefully setup to minimise average waiting time in the queue.
Figure 2 Effect ofNon PBL
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Discrete-time batch service 255
Figure 3 Effect ofa on Wq for different values ofd
7 Conclusion
In this paper, we have carried out an analysis of a discrete-time single-server finite-buffer
renewal input batch-service queue with accessible or NABs that have potential
applications in modelling computer and telecommunication systems, computer networks,
etc. We have developed a recursive method, using the supplementary variable technique
and treating the remaining inter-arrival time as the supplementary variable, to find the
steady-state queue/system length distributions at pre-arrival, arbitrary and outside
observers observation epochs under the EAS. The recursive method is powerful and easy
to implement. We have illustrated a recursive method by presenting two simple examples
for geometrical and deterministic inter-arrival time distributions. Various performance
measures such as the PBL, average queue-length at outside observers observation epoch
and analysis of average waiting time in the queue have been carried out. The results for
the LAS-DA model can also be obtained in a similar manner. The techniques used in this
paper can be applied to analyse more complex models such as DMAP/Geo(a,d,b)/1/Nand
GI/Geo(a,d,b)
/1/Nqueues which are left for future investigations.
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256 V. Goswami and K. Sikdar
Acknowledgements
The authors are thankful to the referees for their valuable comments and suggestions
which have helped in improving the quality of the presentation of this paper.
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