Discrete time queue

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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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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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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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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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*, ,

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


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


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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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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*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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`^

( )*( 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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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:


12/25


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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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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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).


17/25


*(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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,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


20/25


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


22/25


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


23/25


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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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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Discrete time queue

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