Performance Evaluation 64 (2007) 1–19www.elsevier.com/locate/peva

Discrete-time single-server finite-buffer queues under discreteMarkovian arrival process with vacations

U.C. Guptaa,∗, S.K. Samantaa, R.K. Sharmab, M.L. Chaudhryc

a Department of Mathematics, Indian Institute of Technology, Kharagpur-721302, Indiab Department of Mathematics, Indian Institute of Technology, New Delhi-110016, India

c Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. Box 17000, STN Forces, Kingston Ont.,Canada K7K 7B4

Received 18 October 2003; received in revised form 21 December 2005Available online 10 February 2006

Abstract

This paper treats a discrete-time single-server finite-buffer exhaustive (single- and multiple-) vacation queueing system withdiscrete-time Markovian arrival process (D-MAP). The service and vacation times are generally distributed random variables andtheir durations are integral multiples of a slot duration. We obtain the queue-length distributions at departure, service completion,vacation termination, arbitrary and prearrival epochs. Several performance measures such as probability of blocking, averagequeue-length and the fraction of time the server is busy have been discussed. Finally, the analysis of actual waiting time under thefirst-come-first-served discipline is also carried out.c© 2006 Elsevier B.V. All rights reserved.

Keywords: Discrete-time Markovian arrival process (D-MAP); Finite buffer; Multiple vacations; Queue; Single vacation

1. Introduction

The discrete-time Markovian arrival process (D-MAP) can represent a variety of arrival processes which include,as special cases, the Bernoulli arrival process, discrete-time PH-renewal process, and Markov modulated Bernoulliprocess (MMBP). For more details on D-MAP and related topics, see [3,14,2]. This arrival process is a goodrepresentation of bursty and correlated traffic arising in telecommunication networks based on the AsynchronousTransfer Mode (ATM) environment. The ATM has been adopted as the transport mechanism for the implementation ofBroadband Integrated Services Digital Networks (B-ISDN) and provides high flexibility of network access, dynamicbandwidth allocation on demand, and flexible bearer capacity allocation. In such an environment all information isdigitized and segmented into small packets, called ‘cells’. The time is slotted and packets are transmitted around slotboundaries, see, e.g., [5]. Usually, there are two models in discrete time (i) late arrival system with delayed access(LAS-DA) and (ii) early arrival system (EAS), which are also known as arrival-first (AF) and departure-first (DF)

∗ Corresponding author. Tel.: +91 3222 283654; fax: +91 3222 155303.E-mail addresses: umesh@maths.iitkgp.ernet.in (U.C. Gupta), sujit@maths.iitkgp.ernet.in (S.K. Samanta), rksharma@maths.iitd.ernet.in

(R.K. Sharma), chaudhry-ml@rmc.ca (M.L. Chaudhry).

0166-5316/$ - see front matter c© 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.peva.2006.01.001

http://www.elsevier.com/locate/pevamailto:umesh@maths.iitkgp.ernet.inmailto:sujit@maths.iitkgp.ernet.inmailto:rksharma@maths.iitd.ernet.inmailto:chaudhry-ml@rmc.cahttp://dx.doi.org/10.1016/j.peva.2006.01.001

2 U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19

policies, respectively and both have potentials for applications. For details, see [12] and [16]. In recent years severalauthors have analyzed various discrete-time queueing models and a number of new results have been reported in theliterature, see [6] and references therein. However, very few of them have considered such queues with input processas D-MAP. Blondia [4] discusses a D-BMAP/G/1/N queue with AF policy and obtains distributions of buffer contentat departure and arbitrary epochs. Later Herrmann [11] carries out a complete analysis of a D-BMAP/G/1/N queuewith both AF and DF policies under the partial admission strategy. Recently, Chaudhry and Gupta [7] have analyzeda D-MAP/G/1/N queue using the supplementary variable technique and obtain the distributions of buffer contentat various epochs. Further in [8] they also carry out the analysis of a finite-buffer bulk-service D-MAP/Ga,b/1/Nqueue.

In most of the queueing models, on completion of service to the existing customers, the server stays in the emptysystem awaiting for a new arrival. But there are situations where if the server after completing the service of a customerfinds the queue empty, it goes away for a length of time called ‘vacation’. This time may be utilized by the server tocarry out some additional work. On return from a vacation if it finds one or more customers waiting, it takes themfor service on a one-by-one basis until the system empties, after which time it takes another vacation. However, if, onreturn from a vacation, it finds no customer waiting, then, in the case of single vacation, it remains dormant until atleast one customer arrives, whereas in the case of multiple vacation it immediately proceeds for another vacation andcontinues in this manner until it finds at least one waiting customer upon return from a vacation. Discrete-time queueswith server’s vacation also have wide applications in the areas mentioned above. Recently, Zhang and Tian [18] havecarried out analysis of the Geo/G/1 queue with multiple adaptive vacations. In [17] they discussed the G I/Geo/1queue with multiple vacations. The GeoX/G/1 queue subjected to multiple vacations governed by a geometricallydistributed timer is analyzed by Fiems and Bruneel [9]. Further, in [10] they have considered an exhaustive and gatedqueue with single and multiple vacations. Using a matrix-analytic method a class of discrete-time vacation modelsin which distributions of interarrival, service, vacation and operational times are of phase type, has been studied byAlfa [1].

In this paper, we consider a discrete-time single-server finite-buffer exhaustive (single- and multiple-) vacationqueueing system with input process D-MAP. The service and vacation times are independently identically distributed(i.i.d.) random variables with general distributions and their durations are integral multiples of a slot. We present aunified approach to analyze both single- and multiple-vacation models, and obtain the distributions of the number ofcustomers in the queue at departure, service completion, vacation termination, arbitrary and prearrival epochs using thesupplementary variable and the embedded Markov chain methods. We also obtain the probability generating function(p.g.f.) of actual waiting time in the queue of a customer under the first-come-first-served (FCFS) discipline. Variousperformance measures such as probability of blocking, average queue-length, probability that the server is busy andaverage waiting time in queue have been discussed. Finally, some numerical results have been presented in the formof tables and graphs for a wide range of model parameters.

The paper is structured as follows. Section 2 introduces the D-MAP as well as the assumptions and notations ofqueueing models considered in this paper. The analytic analysis of queue-length distributions at various epochs, andp.g.f. of actual waiting time in queue are carried out in Section 3. In Section 4, some numerical results are presented.Further, the effectiveness of the model parameters on key performance measures is also discussed. Finally, Section 5concludes the paper.

2. Assumptions and notations

The arrival process of the queueing system of interest is a discrete-time Markovian arrival process (D-MAP).Formally, in D-MAP the arrivals are governed by an underlying m-state Markov chain having probability ci j , 1 ≤i, j ≤ m, with a transition from state i to j without an arrival and having probability di j , 1 ≤ i, j ≤ m, with atransition from state i to j with an arrival. Let C = (ci j ), D = (di j ) be the m × m non-negative matrices both havingat least one positive entry with (C + D)e = e, where e is a column vector of ones with an appropriate dimension.The sum (C + D) is a stochastic matrix corresponding to an irreducible Markov chain underlying D-MAP. We callthe actual state of this chain the “phase” of the arrival process. Let π be the 1 × m stationary vector of the underlyingMarkov chain, i.e., π(C + D) = π , πe = 1. The fundamental arrival rate of this process is given by λ∗ = πDe.

Let us consider a discrete-time single-server finite-buffer queue of size N excluding the one in service withexhaustive (single and multiple) vacation. We present a unified approach to analyze both single- and multiple-vacation

U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19 3

Fig. 1. Various time epochs in late arrival system with delayed access.

models, and in order to do that we define an indicator function (δM ) as follows: δM = 0 yields the results for single-vacation policy, and δM = 1 gives the results for multiple-vacation policy. Assume that the time axis is slotted intointervals of equal length with the length of a slot being unity. To be more specific, let the time axis be marked by0, 1, 2, . . . , t, . . .. Here we discuss the model for the late arrival system with delayed access (LAS-DA) and therefore,a potential arrival takes place in (t−, t) and a potential departure occurs in (t, t+). More specifically, various timeepochs at which events occur are depicted in Fig. 1. The service (vacation) can start only at slot boundaries andtheir durations are integral multiples of a slot. Further, the service [vacation] times S[V ] are i.i.d. random variableshaving common probability mass function (p.m.f.) sn = P(S = n), n ≥ 1 [vn = P(V = n), n ≥ 1], p.g.f.S(z) =

∑∞

n=1 snzn

[V (z) =∑

∞

n=1 vnzn] and mean service [vacation] time E(S)[E(V )]. Let ρ′ be defined as the

carried load, i.e., the probability that the server is busy at arbitrary time, whereas the offered load ρ is defined as usualto be ρ = λ∗E(S).

The state of the system just before a potential arrival (at t−) is described by the following random variables:

Nt−: number of customers in the queue excluding the one in service,

Jt−: phase of the arrival process,

Ut−: remaining service time of the customer in service excluding the current service slot,

Vt−: remaining vacation time of the server excluding the current vacation slot,

ξt−: state of the server, i.e., ξt− = 2, 1, or 0 corresponding to whether the server is busy, on vacation, or in dormancy,respectively.

Let us define the joint probabilities, for 1 ≤ i ≤ m,

πi (n, u, t−) = P[Nt− = n, Jt− = i, Ut− = u, ξt− = 2], 0 ≤ n ≤ N , u ≥ 0,

ωi (n, u, t−) = P[Nt− = n, Jt− = i, Vt− = u, ξt− = 1], 0 ≤ n ≤ N , u ≥ 0,

νi (0, t−) = P[Nt− = 0, Jt− = i, ξt− = 0].

In steady state, we have

πi (n, u) = limt−→∞

πi (n, u, t−), ωi (n, u) = limt−→∞

ωi (n, u, t−) and νi (0) = limt−→∞

νi (0, t−).

Further, let π(n, u), ω(n, u) and ν(0) be the row vectors of order m whose i-th components are πi (n, u), ωi (n, u) andνi (0), respectively. Define the vector-generating functions (v.g.fs.) of π(n, u) and ω(n, u), respectively, by

π∗(n, z) =∞∑

u=0

π(n, u)zu and ω∗(n, z) =∞∑

u=0

ω(n, u)zu, | z |≤ 1

with π∗(n, 1) = π(n) and ω∗(n, 1) = ω(n), where π(n)[ω(n)] is the 1×m vector whose i-th component πi (n)[ωi (n)]denotes the probability of n customers in the queue and the arrival process in phase i , when the server is busy [on

4 U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19

vacation] at arbitrary epoch. The i-th component νi (0) of ν(0) represents the probability that the server is in thedormant state and the arrival process in phase i at an arbitrary epoch.

3. Basic equations and analysis

To begin with observing the state of the system at two consecutive epochs t− and (t + 1)−, considering variouspossible phase transitions and using matrices and vectors, we have, in steady state, for u ≥ 1

π(0, u − 1) = π(0, u)C + π(1, 0)Csu + π(0, 0)Dsu + ω(1, 0)Csu + ω(0, 0)Dsu+ (1 − δM )ν(0)Dsu, (1)

π(n, u − 1) = π(n, u)C + π(n, 0)Dsu + π(n + 1, 0)Csu + π(n − 1, u)D + ω(n + 1, 0)Csu+ ω(n, 0)Dsu, 1 ≤ n ≤ N − 2, (2)

π(N − 1, u − 1) = π(N − 1, u)C + π(N , 0)(C + D)su + π(N − 1, 0)Dsu + π(N − 2, u)D

+ ω(N − 1, 0)Dsu + ω(N , 0)(C + D)su, (3)

π(N , u − 1) = π(N , u)(C + D) + π(N − 1, u)D, (4)

ω(0, u − 1) = ω(0, u)C + δMω(0, 0)Cvu + π(0, 0)Cvu, (5)

ω(n, u − 1) = ω(n, u)C + ω(n − 1, u)D, 1 ≤ n ≤ N − 1, (6)

ω(N , u − 1) = ω(N , u)(C + D) + ω(N − 1, u)D, (7)

ν(0) = ν(0)C + ω(0, 0)C. (8)

It may be remarked here that Eq. (8) will not appear in the case of multiple vacations due to the absence of the dormantstate. Multiplying Eqs. (1)–(7) by zu and summing over u = 1 to ∞, we get

zπ∗(0, z) = π∗(0, z)C − π(0, 0)C + π(1, 0)CS(z) + π(0, 0)DS(z) + ω(1, 0)CS(z)

+ ω(0, 0)DS(z) + (1 − δM )ν(0)DS(z), (9)

zπ∗(n, z) = π∗(n, z)C − π(n, 0)C + π(n, 0)DS(z) + π(n + 1, 0)CS(z) + π∗(n − 1, z)D

− π(n − 1, 0)D + ω(n + 1, 0)CS(z) + ω(n, 0)DS(z), 1 ≤ n ≤ N − 2, (10)

zπ∗(N − 1, z) = π∗(N − 1, z)C − π(N − 1, 0)C + {π(N , 0) + ω(N , 0)}(C + D)S(z)

+ {π(N − 1, 0) + ω(N − 1, 0)}DS(z) + π∗(N − 2, z)D − π(N − 2, 0)D, (11)

zπ∗(N , z) = π∗(N , z)(C + D) − π(N , 0)(C + D) + π∗(N − 1, z)D − π(N − 1, 0)D, (12)

zω∗(0, z) = ω∗(0, z)C − ω(0, 0)C + δMω(0, 0)CV (z) + π(0, 0)CV (z), (13)

zω∗(n, z) = ω∗(n, z)C − ω(n, 0)C + ω∗(n − 1, z)D − ω(n − 1, 0)D, 1 ≤ n ≤ N − 1, (14)

zω∗(N , z) = ω∗(N , z)(C + D) − ω(N , 0)(C + D) + ω∗(N − 1, z)D − ω(N − 1, 0)D. (15)

The normalizing equation and condition are, respectively,N∑

n=0

{π(n) + ω(n)} + (1 − δM )ν(0) = π and

{N∑

n=0

{π(n) + ω(n)} + (1 − δM )ν(0)

}e = 1.

Now we obtain a few important results in the form of lemmas which are used in deriving other results and the relationsamong state probabilities at various epochs.

Lemma 1.

{π(0, 0) + δMω(0, 0)}Ce =N∑

n=0

ω(n, 0)e.

The left-hand side represents the entering rate to the vacation state in a random slot, while the right-hand siderepresents the departure rate from the vacation state in a random slot.

Proof. Setting z = 1 in Eqs. (13)–(15), we obtain

ω(0) = ω(0)C − ω(0, 0)C + δMω(0, 0)C + π(0, 0)C,

U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19 5

ω(n) = ω(n)C − ω(n, 0)C + ω(n − 1)D − ω(n − 1, 0)D, 1 ≤ n ≤ N − 1,

ω(N ) = ω(N )(C + D) − ω(N , 0)(C + D) + ω(N − 1)D − ω(N − 1, 0)D.

Post multiplying the above equations by e and adding them, we get, after simplification, the desired result. �

Lemma 2.

N∑n=0

π(n)e = E(S)N∑

n=0

π(n, 0)e = ρ′

N∑n=0

ω(n)e + (1 − δM )ν(0)e = E(V )N∑

n=0

ω(n, 0)e + (1 − δM )ν(0)e = 1 − ρ′.

These results have probabilistic interpretations: since∑N

n=0 π(n, 0)e denotes the mean number of customers servedper slot, multiplying this by E(S) gives ρ′, which is the probability that the server is busy. Similarly,

∑Nn=0 ω(n, 0)e

denotes the mean number of vacations terminated per slot, multiplying this by E(V ) gives the probability that theserver is on vacation. Therefore (1−ρ′) is the probability that the server is in an unavailable period. The unavailableperiod corresponds to the time taken for vacations in the case of multiple vacations, and in the case of a singlevacation it corresponds to the time taken for vacation plus dormancy.

Proof. Post multiplying Eqs. (9)–(12) and (8) by e, adding them and using Lemma 1, we get

N∑n=0

π∗(n, z)e =S(z) − 1

z − 1

N∑n=0

π(n, 0)e.

Taking the limit as z → 1, yields the first result.Similarly, from Eqs. (13)–(15) and using Lemma 1, we get

N∑n=0

ω∗(n, z)e =V (z) − 1

z − 1

N∑n=0

ω(n, 0)e.

Taking the limit as z → 1 and using the normalization condition, we obtain the second result. �

3.1. Queue-length distributions at various epochs

In this section, we obtain the queue-length distributions at various epochs, viz., departure, service completion,vacation termination, arbitrary and prearrival epochs.

3.1.1. Queue-length distribution at departure epochIn order to obtain the queue-length distribution at the departure epoch, first we derive a few basic results between

various matrices which are used later.Let A(k)n be the matrix of order m×m whose element [A

(k)n ]i j is the conditional probability given a departure, which

leaves at least one customer in the system with arrival process in phase i and there are n arrivals during a service timelasting k slots with arrival process in phase j .

Let B(k)n be the matrix of order m × m whose element [B(k)n ]i j is the conditional probability given a departure,

which leaves the system empty with arrival process in phase i and there are n arrivals during a vacation time lasting kslots with arrival process in phase j .

Let An and Bn denote the matrices of order m × m which represent that n customers arrive during a service time Sof a customer and a vacation time V , respectively. Then

An =∞∑

k=1

skA(k)n and Bn =∞∑

k=1

vkB(k)n , n ≥ 0.

6 U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19

Let A(k)(z) be the matrix-generating function of A(k)n , then

A(k)(z) =∞∑

n=0

A(k)n zn

= [A(1)(z)]k = [C + Dz]k,

where C + Dz is the matrix-generating function of the number of arrivals during a slot.If A(z) is the matrix-generating function of An with A = A(1), then

A(z) =∞∑

n=0

Anzn =∞∑

n=0

∞∑k=1

skA(k)n zn

=

∞∑k=1

sk[C + Dz]k .

Setting z = 1 in the above equation, we get A =∑

∞

k=1 sk[C + D]k . Applying similar arguments, we can obtain

B(k)(z) = [C + Dz]k , B(z) =∑

∞

k=1 vk[C + Dz]k and B =

∑∞

k=1 vk[C + D]k .

Further, let

Âi =∞∑

n=i

An = A −i−1∑n=0

An, i ≥ 1 and B̂N =∞∑

n=N

Bn = B −N−1∑n=0

Bn .

It may be noted here that in order to know An and Bn , we need to compute both A(k)n and B(k)n efficiently. Following

the arguments of Chaudhry and Gupta [8], it can be seen that A(k)n and B(k)n satisfy the following equations:

A(k)n = CA(k−1)n + DA

(k−1)n−1 , k ≥ 1, n ≥ 0, (16)

B(k)n = CB(k−1)n + DB

(k−1)n−1 , k ≥ 1, n ≥ 0, (17)

with A(0)0 = B(0)0 = I, A

(k)−1 = B

(k)−1 = 0 and A

(k)n = B

(k)n = 0, n > k ≥ 0, where I and 0 are the Identity and Zero

matrices of order m × m, respectively.In the case of multiple vacations, let 8n be the matrix of order m × m whose element [Φn]i j is the probability

that during the server unavailable period (which consists of several vacations) there are n arrivals and that at theend, the phase of the arrival process is j , given that the vacation started with the arrival process in phase i . Thus,we have

8n =

∞∑r=0

Br0Bn = (I − B0)−1Bn, 1 ≤ n ≤ N − 1,

8N =

∞∑r=0

Br0B̂N = (I − B0)−1B̂N .

In the case of a single vacation, let D(k)

be the matrix of order m × m whose element [D(k)

]i j is the conditionalprobability that given a vacation is terminated with system empty (the server is in dormant state) and the arrival processin phase i , the next arrival occurs after (k − 1) slots with the arrival process in phase j . Arguing probabilistically,we have

D(k)

= Ck−1D, k ≥ 1.

Let D denote a matrix of order m × m which represents that the dormancy period is terminated with an arrival.Then

D =∞∑

k=1

D(k)

=

∞∑k=1

Ck−1D = (I − C)−1D.

We now study the imbedded Markov process, obtained by considering the queue-length and the phase of the D-MAP atdepartures, where vacation termination epochs are also considered but only departure epochs are the embedded points.Therefore, the number of customers in the queue and the phase of the arrival process immediately after the departureepoch is a bivariate imbedded Markov chain with state space {0, 1, 2, . . . , N } × {1, 2, . . . , m} and the correspondingone-step transition probability matrix

U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19 7

P =

Q0 Q1 Q2 Q3 · · · · · · QN−3 QN−2 QN−1 QNA0 A1 A2 A3 · · · · · · AN−3 AN−2 AN−1 ÂN0 A0 A1 A2 · · · · · · AN−4 AN−3 AN−2 ÂN−10 0 A0 A1 · · · · · · AN−5 AN−4 AN−3 ÂN−2...

......

......

......

......

......

......

......

......

......

...

0 0 0 0 · · · · · · A1 A2 A3 Â40 0 0 0 · · · · · · A0 A1 A2 Â30 0 0 0 · · · · · · 0 A0 A1 Â20 0 0 0 · · · · · · 0 0 A0 Â1

,

where elements of the first row are obtained from

Q j = (1 − δM )B0DA j +j+1∑k=1

{(1 − δM )Bk + δM8k}A j+1−k, 0 ≤ j ≤ N − 2,

QN−1 = (1 − δM )B0DAN−1 +N−1∑k=1

{(1 − δM )Bk + δM8k}AN−k + {(1 − δM )B̂N + δM8N }A0,

QN = (1 − δM )B0DÂN +N−1∑k=1

{(1 − δM )Bk + δM8k}ÂN+1−k + {(1 − δM )B̂N + δM8N }Â1.

Let p+(n) be the 1 × m vector whose i-th component p+i (n) represents the probability that there are n customers inthe queue immediately after the departure epoch and arrival process in phase i . These probabilities can be obtained bysolving the system of equations p+P = p+ and p+e = 1 using the GTH (Grassmann, Taksar and Heyman) algorithmgiven in [15, p. 123], or otherwise, where p+ = [p+(0), p+(1), . . . , p+(N )] of order 1 × (N + 1)m is a stationaryprobability vector of the transition probability matrix P.

3.1.2. Queue-length distribution at service completion and vacation termination epochs

Let π+(n) [ω+(n)] be the 1 × m vector whose i-th component π+i (n) [ω+

i (n)] is the probability that there are ncustomers in the queue at the service completion [vacation termination] epoch and the arrival process is in phase i .Applying Bayes’ theorem, we have for 1 ≤ i ≤ m

π+i (n) = P{n or (n − 1) customers in the queue just prior to service completion epoch and arrival process in

phase i |≤ N customers in the queue just prior to service completion or vacation termination epoch}

=

1σ

{m∑

j=1

π j (n, 0)c j i +m∑

j=1

π j (n − 1, 0)d j i

}, 0 ≤ n ≤ N − 1,

1σ

{m∑

j=1

π j (N , 0)(c j i + d j i ) +m∑

j=1

π j (N − 1, 0)d j i

}, n = N ,

where π j (x, 0) = 0, for x < 0 and

σ = P{≤N customers in the queue just prior to service completion or vacation termination epoch}

=

N∑n=0

{π(n, 0) + ω(n, 0)}e. (18)

The above results have been obtained by observing the events at epochs t− and t+ of Fig. 1. Similarly, we can obtainan expression for ω+i (n). Finally, in matrix and vector notations, we have

8 U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19

π+(n) =1σ

{π(n, 0)C + π(n − 1, 0)D} , 0 ≤ n ≤ N − 1, (19)

π+(N ) =1σ

{π(N , 0)(C + D) + π(N − 1, 0)D} , (20)

ω+(n) =1σ

{ω(n, 0)C + ω(n − 1, 0)D} , 0 ≤ n ≤ N − 1, (21)

ω+(N ) =1σ

{ω(N , 0)(C + D) + ω(N − 1, 0)D} . (22)

It can be seen from (19) to (22) that to obtain π+(n) and ω+(n) we first need to find π(n, 0) and ω(n, 0). As π(n, 0)and ω(n, 0) are cumbersome to evaluate directly from (9) to (15), we obtain π+(n) and ω+(n) using the followinglemmas. One may note here that Eqs. (19)–(22) are used in subsequent parts to obtain other various results.

Lemma 3. The expression for σ is given as

σ =ρ′E(V ) + {1 − ρ′ − (1 − δM )ν(0)e}E(S)

E(S)E(V ).

Proof. Substituting the values of∑N

n=0 π(n, 0)e and∑N

n=0 ω(n, 0)e from Lemma 2 in (18), we obtain the aboveresult. One may note here that the unknown quantities ρ′ and ν(0), respectively, will be obtained from Lemmas 6 and7 given below. �

Lemma 4. The relation between π+(n) and p+(n) is given by

π+(n) =ρ′E(V )

ρ′E(V ) + {1 − ρ′ − (1 − δM )ν(0)e}E(S)p+(n), 0 ≤ n ≤ N .

Proof. Since the probability vector π+(n) (considered only at the service completion epochs) is proportional to theprobability vector p+(n) with

∑Nn=0 p

+(n)e = 1, they are related by the following relation

p+(n) =π+(n)

N∑n=0

π+(n)e

, 0 ≤ n ≤ N . (23)

Post multiplying Eq. (19) to (20) by e and adding them, we obtain

N∑n=0

π+(n)e =1σ

N∑n=0

π(n, 0)e =ρ′E(V )

ρ′E(V ) + {1 − ρ′ − (1 − δM )ν(0)e}E(S).

Substituting the value of∑N

n=0 π+(n)e in (23), we get the desired result. �

Lemma 5. The vacation termination epoch probabilities {ω+(n)}N0 are given by

ω+(0) = π+(0)B0(I − δM B0)−1, (24)

ω+(n) = {π+(0) + δMω+(0)}Bn, 1 ≤ n ≤ N − 1, (25)

ω+(N ) = {π+(0) + δMω+(0)}B̂N . (26)

Proof. Observing the service completion epoch with the system empty, and the epochs of vacation terminations, usingthe probabilistic argument, we obtain

ω+(n) = {π+(0) + δMω+(0)}Bn, 0 ≤ n ≤ N − 1,

ω+(N ) = {π+(0) + δMω+(0)}B̂N .

For n = 0, the first equation gives (24) after simplification. �

U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19 9

Lemma 6. The expression for ρ′ (probability that the server is busy) is given by

ρ′ =E(S)

E(S) + p+(0){(1 − δM )B0(I − C)−1 + E(V )(I − δM B0)−1}e.

Proof. Let ΘB[ΘI ] be the random variable denoting the busy [unavailable] period and E(ΘB) [E(ΘI )] thecorresponding mean. From the definition of the carried load ρ′ (the fraction of time that the server is busy), wehave

ρ′ =E(ΘB)

E(ΘB) + E(ΘI ). (27)

Using Lemma 2 and Eq. (27), we have

E(ΘI )E(ΘB)

=1 − ρ′

ρ′=

(1 − δM )ν(0)e + E(V )N∑

n=0ω(n, 0)e

E(S)N∑

n=0π(n, 0)e

.

Following the arguments given in [16, p. 291] for the Geo/G/1//N queue, it can be shown that E(ΘB) = E(S)p+(0)e , andusing this in the above yields

E(ΘI ) =(1 − δM )ν(0)e + E(V )

N∑n=0

ω(n, 0)e

p+(0)eN∑

n=0π(n, 0)e

.

Now using Eq. (8) and Lemma 1, we have

E(ΘI ) =(1 − δM )ω(0, 0)C(I − C)−1e + E(V ){π(0, 0) + δMω(0, 0)}Ce

p+(0)eN∑

n=0π(n, 0)e

.

Dividing numerator and denominator by σ , and using Eqs. (19)–(22), we obtain

E(ΘI ) =(1 − δM )ω+(0)(I − C)−1e + E(V ){π+(0) + δMω+(0)}e

p+(0)eN∑

n=0π+(n)e

.

After using (24) and (23) in the above equation for E(ΘI ), we get

E(ΘI ) =p+(0){(1 − δM )B0(I − C)−1 + E(V )(I − δM B0)−1}e

p+(0)e.

Finally, using expressions for E(ΘB) and E(ΘI ) (obtained above) in (27), after simplification we get the desiredresult. �

3.1.3. Queue-length distribution at an arbitrary epochTo obtain the queue-length distributions at arbitrary epoch we develop below relations among distributions of

number of customers in the queue at departure, service completion, vacation termination and arbitrary epochs.

Lemma 7. The expression for ν(0) (probability vector that the server is in dormancy) is given by

ν(0) =ρ′p+(0)B0(I − C)−1

E(S).

10 U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19

Proof. Using Eq. (21), for n = 0, in Eq. (8), we have

ν(0) = σω+(0)(I − C)−1.

Further, using Eqs. (24) and (23) in the above result, we obtain

ν(0) = σN∑

n=0

π+(n)e p+(0)B0(I − C)−1 =N∑

n=0

π(n, 0)e p+(0)B0(I − C)−1.

Applying Lemma 2 yields the stated result. �

Lemma 8. The arbitrary epoch probabilities, when the server is on vacation, are given by

ω(0) = σ[π+(0) − (1 − δM )ω+(0)

](I − C)−1,

ω(n) =[ω(n − 1)D − σω+(n)

](I − C)−1, 1 ≤ n ≤ N − 1.

Proof. Setting z = 1 in Eqs. (13) and (14), we get

ω(0)(I − C) = π(0, 0)C − (1 − δM )ω(0, 0)C,

ω(n)(I − C) = ω(n − 1)D − ω(n, 0)C − ω(n − 1, 0)D, 1 ≤ n ≤ N − 1.

Dividing the above equations by σ and using (19)–(22), we obtain the desired results. �

Lemma 9. The arbitrary epoch probabilities, when the server is busy, are given by

π(0) =[(1 − δM )ν(0)D + σ {π+(1) + ω+(1) − π+(0)}

](I − C)−1,

π(n) =[π(n − 1)D + σ {π+(n + 1) + ω+(n + 1) − π+(n)}

](I − C)−1, 1 ≤ n ≤ N − 1.

Proof. Setting z = 1 in Eqs. (9)–(11), we get

π(0)(I − C) = −π(0, 0)C + π(1, 0)C + π(0, 0)D + ω(1, 0)C + ω(0, 0)D + (1 − δM )ν(0)D,π(n)(I − C) = −π(n, 0)C + π(n, 0)D + π(n + 1, 0)C + π(n − 1)D − π(n − 1, 0)D

+ ω(n + 1, 0)C + ω(n, 0)D, 1 ≤ n ≤ N − 2,

π(N − 1)(I − C) = −π(N − 1, 0)C + π(N , 0)(C + D) + π(N − 1, 0)D + π(N − 2)D− π(N − 2, 0)D + ω(N − 1, 0)D + ω(N , 0)(C + D).

Dividing the above equations by σ and using (19)–(22), we obtain the desired results. �

Lemma 10.

π(N )e = ρ′ −N−1∑n=0

π(n)e and ω(N )e = 1 − ρ′ − (1 − δM )ν(0)e −N−1∑n=0

ω(n)e.

It may be noted here that due to the singularity of (I − (C + D)), we cannot obtain π(N ) and ω(N ) from (12) and(15), respectively, after setting z = 1. But if we find the row sums π(N )e and ω(N )e, then these are enough for thecalculations of key performance measures. The above results are obtained using Lemma 2.

Lemma 11. Let p(n) be the 1 × m vector whose i-th component pi (n) is the probability that there are n customers inthe queue at an arbitrary epoch. Then, in vector notation, we have

p(0) = π(0) + ω(0) + (1 − δM )ν(0),

p(n) = π(n) + ω(n), 1 ≤ n ≤ N − 1,

p(N ) = π −N−1∑n=0

p(n).

U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19 11

3.1.4. Queue-length distribution at prearrival epochLet π−(n) {ω−(n)} [ν−(0)] be the 1 × m vector whose i-th component π−i (n) {ω

−

i (n)} [ν−

i (0)] is the probabilitythat an arrival finds n customers in the queue, when the server is busy {on vacation} [in dormancy] and the arrivalprocess is in phase i .

Lemma 12. The prearrival probability vectors π−(n), ω−(n) and ν−(0) are given by

π−(n) =π(n)D

λ∗, ω−(n) =

ω(n)Dλ∗

, 0 ≤ n ≤ N − 1,

ν−(0) =ν(0)D

λ∗and π−(N ) + ω−(N ) =

{π(N ) + ω(N )}Dλ∗

.

Proof. We just prove the derivation of π−(n), since it is easy to prove the remaining parts. Applying Bayes’ theorem,we have for 1 ≤ i ≤ m

π−i (n) = P {n customers in the queue at arbitrary epoch and arrival process in phase i

when server is busy | arrival is about to occur}

=

m∑j=1

π j (n)d j i{N∑

n=0{π(n) + ω(n)} + (1 − δM )ν(0)

}De

, 0 ≤ n ≤ N .

In matrix and vector notations, we have

π−(n) =π(n)D

λ∗, 0 ≤ n ≤ N .

Finally, one may note here that we cannot evaluate π−(N ) and ω−(N ) separately as π(N ) and ω(N ) are not availableindividually. Since π(N ) + ω(N ) is known, we have

π−(N ) + ω−(N ) ={π(N ) + ω(N )}D

λ∗. �

Lemma 13. Let p−(n) be the 1 × m vector whose i-th component p−i (n) is the probability that there are n customersin the queue at prearrival epoch. Then, in vector notation, we have

p−(0) = π−(0) + ω−(0) + (1 − δM )ν−(0),

p−(n) = π−(n) + ω−(n), 1 ≤ n ≤ N .

One may note here that the probability of blocking (PBL) in single as well as multiple vacations is given byPBL = {π−(N ) + ω−(N )}e.

3.2. Waiting time analysis

In this section, we obtain the vector-generating function (v.g.f.) of actual waiting time of a customer who is acceptedin the queue under the FCFS discipline. For this, the stationary queue-length distribution just prior to an arrival epochis needed. Let π̂−(n, u, t−)[ω̂−(n, u, t−)] be the 1 × m vector whose i-th component π̂−i (n, u, t−)[ω̂

−

i (n, u, t−)] isthe probability that there are n (0 ≤ n ≤ N − 1) customers in the queue with remaining service [vacation] time uslots when the server is busy [on vacation] and arrival process is in phase i (1 ≤ i ≤ m) just prior to an arrival of acustomer. Further, let ν̂−(0, t−) be the 1×m vector whose i-th component ν̂−i (0, t−) is the probability that the serveris in the dormant state and the arrival process is in phase i (1 ≤ i ≤ m) just prior to an arrival of a customer andlet C A(t−) represent the event that a customer arrives in the queue with phase change at time t−. Applying Bayes’theorem, we have for 1 ≤ i ≤ m

12 U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19

π̂−i (n, u, t−) = P{Nt− = n, Jt− = i, Ut− = u, ξt− = 2 | C A(t−)},

=

m∑j=1

π j (n, u, t−)d j i{N−1∑n=0

{π(n, t−) + ω(n, t−)} + (1 − δM )ν(0, t−)

}De

, 0 ≤ n ≤ N − 1.

Taking the limit as t− → ∞, using matrix and vector notations, we have

π̂−(n, u) =

π(n, u)D{N−1∑n=0

{π(n) + ω(n)} + (1 − δM )ν(0)

}De

=π(n, u)D

(1 − PBL)λ∗.

The v.g.f. of π̂−(n, u) is given by

π̂∗−

(n, z) =∞∑

u=0

π̂−(n, u)zu =

π∗(n, z)D(1 − PBL)λ∗

. (28)

Similarly, we obtain

ω̂∗−

(n, z) =ω∗(n, z)D

(1 − PBL)λ∗and ν̂−(0) =

ν(0)D(1 − PBL)λ∗

.

Let Wq(z) = [Wq1(z), Wq2(z), . . . , Wqm(z)] be the probability vector-generating function of actual waiting timein the queue, where Wq j (z) is the p.g.f. of actual waiting time of an arrived customer and arrival process in phase j .Note that an arrived customer may be either

(i) served immediately if he sees the system in the dormant state, or(ii) served after the customer under service and all other waiting customers in front of him depart if he sees the

system in the busy state, or(iii) served after a vacation period ends and all other waiting customers in front of him depart if he sees the system in

the vacation state.

Therefore, the actual waiting time v.g.f. is given by

Wq(z) = (1 − δM )ν̂−(0) +

N−1∑n=0

π̂∗−

(n, z){S(z)}n +N−1∑n=0

ω̂∗−

(n, z){S(z)}n,

=1

(1 − PBL)λ∗

{(1 − δM )ν(0)D +

N−1∑n=0

π∗(n, z)D{S(z)}n +N−1∑n=0

ω∗(n, z)D{S(z)}n}

.

Thus, the expected actual waiting time is given by

Wq =1

(1 − PBL)λ∗

{N−1∑n=0

n{π(n) + ω(n)}DeE(S) +N−1∑n=0

{π∗(1)(n, 1) + ω∗(1)(n, 1)}De

}where π∗(1)(n, 1) and ω∗(1)(n, 1), (0 ≤ n ≤ N − 1) are obtained by differentiating Eqs. (9)–(11), (13) and (14) w.r.t.z and setting z = 1. This leads to

π∗(1)(0, 1) =[{(1 − δM )ν(0)D + σ(π+(1) + ω+(1))}E(S) − π(0)

](I − C)−1,

π∗(1)(n, 1) =[σ {π+(n + 1) + ω+(n + 1)}E(S) + π∗(1)(n − 1, 1)D − π(n)

](I − C)−1, 1 ≤ n ≤ N − 1,

ω∗(1)(0, 1) =[σ {π+(0) + δMω+(0)}E(V ) − ω(0)

](I − C)−1,

ω∗(1)(n, 1) =[ω∗(1)(n − 1, 1)D − ω(n)

](I − C)−1, 1 ≤ n ≤ N − 1.

Again, we can also obtain the average waiting time in the queue (Wq) of a customer using Little’s rule, Wq = Lq/λ′,

U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19 13

Table 1Queue-length distributions at various epochs in the case of a single vacation when service and vacation times are deterministic with ρ = 1.6667,λ∗ = 0.3333, s5 = 1, v3 = 1, N = 10, m = 2

n p+k (n) π+

k (n) ω+

k (n)

k = 1 k = 2∑m

k=1 k = 1 k = 2∑m

k=1 k = 1 k = 2∑m

k=1

0 0.0079 0.0009 0.0087 0.0078 0.0009 0.0087 0.0057 0.0006 0.00631 0.0116 0.0020 0.0137 0.0115 0.0020 0.0135 0.0004 0.0006 0.00102 0.0164 0.0036 0.0200 0.0162 0.0035 0.0198 0.0003 0.0005 0.00083 0.0225 0.0055 0.0281 0.0223 0.0055 0.0278 0.0001 0.0004 0.00064 0.0305 0.0080 0.0386 0.0303 0.0080 0.0382 0.0000 0.0000 0.00005 0.0411 0.0131 0.0542 0.0407 0.0130 0.0537 0.0000 0.0000 0.00006 0.0547 0.0155 0.0702 0.0542 0.0154 0.0696 0.0000 0.0000 0.00007 0.0734 0.0210 0.0945 0.0728 0.0208 0.0936 0.0000 0.0000 0.00008 0.0985 0.0282 0.1267 0.0976 0.0280 0.1256 0.0000 0.0000 0.00009 0.1320 0.0385 0.1705 0.1309 0.0382 0.1691 0.0000 0.0000 0.0000

10 0.1769 0.1981 0.3749 0.1753 0.1963 0.3717 0.0000 0.0000 0.0000Sum 1.0000 0.9913 0.0087

n πk (n) ωk (n) pk (n)

k = 1 k = 2∑m

k=1 k = 1 k = 2∑m

k=1 k = 1 k = 2∑m

k=1

0 0.0107 0.0024 0.0131 0.0042 0.0005 0.0047 0.0262 0.0041 0.03031 0.0147 0.0038 0.0185 0.0001 0.0003 0.0004 0.0148 0.0041 0.01892 0.0199 0.0055 0.0254 0.0000 0.0001 0.0001 0.0199 0.0056 0.02553 0.0268 0.0076 0.0343 0.0000 0.0000 0.0000 0.0268 0.0076 0.03434 0.0359 0.0106 0.0465 0.0000 0.0000 0.0000 0.0359 0.0106 0.04655 0.0480 0.0138 0.0618 0.0000 0.0000 0.0000 0.0480 0.0138 0.06186 0.0644 0.0186 0.0830 0.0000 0.0000 0.0000 0.0644 0.0186 0.08307 0.0863 0.0249 0.1112 0.0000 0.0000 0.0000 0.0863 0.0249 0.11128 0.1157 0.0335 0.1492 0.0000 0.0000 0.0000 0.1157 0.0335 0.14929 0.1551 0.0737 0.2287 0.0000 0.0000 0.0000 0.1551 0.0737 0.2287

10 0.2104 0.0000 0.0736 0.1369 0.2104Sum 0.9823 0.0051 1.0000

ν(0) = [0.0113 0.0013], ν(0)e = 0.0125.

n π−k (n) ω−

k (n) p−

k (n)

k = 1 k = 2∑m

k=1 k = 1 k = 2∑m

k=1 k = 1 k = 2∑m

k=1

0 0.0015 0.0058 0.0073 0.0003 0.0011 0.0014 0.0025 0.0099 0.01241 0.0023 0.0091 0.0114 0.0002 0.0006 0.0008 0.0024 0.0098 0.01222 0.0033 0.0132 0.0165 0.0001 0.0003 0.0003 0.0034 0.0135 0.01693 0.0045 0.0182 0.0227 0.0000 0.0000 0.0000 0.0045 0.0182 0.02274 0.0064 0.0255 0.0319 0.0000 0.0000 0.0000 0.0064 0.0255 0.03195 0.0083 0.0331 0.0414 0.0000 0.0000 0.0000 0.0083 0.0331 0.04146 0.0111 0.0445 0.0557 0.0000 0.0000 0.0000 0.0111 0.0445 0.05577 0.0149 0.0597 0.0747 0.0000 0.0000 0.0000 0.0149 0.0597 0.07478 0.0201 0.0804 0.1005 0.0000 0.0000 0.0000 0.0201 0.0804 0.10059 0.0442 0.1768 0.2210 0.0000 0.0000 0.0000 0.0442 0.1768 0.2210

10 0.0821 0.3285 0.4106Sum 0.5831 0.0025 1.0000

ν−(0) = [0.0008 0.0030], ν−(0)e = 0.0038.ρ′ = 0.9823, PBL = 0.4106, Lq = 7.3010, Wq = 37.1635.

where Lq =∑N

n=1 np(n)e is the average queue length at arbitrary epoch and λ′= λ∗(1−PBL) is the effective arrival

rate.

Remark. Setting z = 1 in (28) and using Lemma 12, we obtain π̂−(n)(1 − PBL) = π−(n) and similarly we can getω̂

−(n)(1 − PBL) = ω−(n) and ν̂−(0)(1 − PBL) = ν−(0), 0 ≤ n ≤ N − 1.

14 U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19

Table 2Queue-length distributions at various epochs in the case of a single vacation when service and vacation times are geometric with ρ = 0.8399,λ∗ = 0.0924, E(S) = 0.9090, E(V ) = 5.0000, N = 60, m = 7

n∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1

p+k (n) π+

k (n) ω+

k (n) πk (n) ωk (n) pk (n) π−

k (n) ω−

k (n) p−

k (n)

0 0.0840 0.0775 0.0607 0.0806 0.0321 0.2341 0.1092 0.0182 0.19321 0.1223 0.1128 0.0047 0.1010 0.0019 0.1029 0.0911 0.0131 0.10422 0.0934 0.0862 0.0100 0.0782 0.0040 0.0822 0.0883 0.0023 0.09063 0.0900 0.0830 0.0006 0.0744 0.0002 0.0746 0.0754 0.0017 0.07714 0.0757 0.0699 0.0013 0.0629 0.0005 0.0634 0.0675 0.0003 0.06785 0.0678 0.0625 0.0001 0.0561 0.0000 0.0561 0.0586 0.0002 0.0588

10 0.0343 0.0317 0.0000 0.0284 0.0000 0.0284 0.0300 0.0000 0.030020 0.0089 0.0082 0.0000 0.0074 0.0000 0.0074 0.0078 0.0000 0.007850 0.0002 0.0001 0.0000 0.0001 0.0000 0.0001 0.0001 0.0000 0.000159 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.000060 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Sum 1.0000 0.9225 0.0775 0.8398 0.0388 1.0000 0.8983 0.0358 1.0000

ν(0)e = 0.1214, ν−(0)e = 0.0658, ρ′ = 0.8398, PBL = 0.0000, Lq = 6.0149, Wq = 65.1109.

4. Numerical results

In this section, we present some numerical results in the form of self-explanatory tables and graphs for the modelsunder discussion. Various performance measures such as probability of blocking, probability that the server is busy,probability that the server is in the dormant state (for single vacation), average queue-length and average waiting timein the queue are given at the bottom of the tables. In Tables 1 and 2, the results are given for a single vacation, andin Tables 3 and 4, the results are given for multiple vacations. One may note here that in Table 1, we have presentedqueue-length distributions at various epochs for each phase whereas in Tables 2–4 the results are presented withoutphases. It can be seen from Table 2 that ρ and ρ′ are approximately equal when N is chosen sufficiently large andρ < 1, as the model tends to an infinite-buffer queue. The average waiting time in the queue obtained through p.g.f.of actual waiting time in the queue (given at the bottom of the tables) is exactly the same as the one obtained usingLittle’s rule. It is interesting to note that for v1 = 1 and vi = 0, i ≥ 2, i.e., when the server never takes any vacationbut remains idle, the results exactly match with the non-vacation queue given in [7]. The following input matrices Cand D of the arrival process have been taken from Johnson and Narayana [13, p. 121].

Case 1. m = 2

C =[

0.9 0.10 0

], D =

[0 0

0.2 0.8

], π =

[0.6667 0.3333

].

Case 2. m = 7

C =

0 1 0 0 0 0 00 0 1 0 0 0 00 0 0 0 0 0 00 0 0 0.95 0 0 00 0 0 0 0 0.05 00 0 0 0 0 0.95 0.050 0 0 0 0 0 0

, D =

0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 1 0 0 00 0 0 0 0.05 0 00 0 0 0.95 0 0 00 0 0 0 0 0 0

0.8 0.2 0 0 0 0 0

,

π =[0.0018 0.0023 0.0023 0.9013 0.0451 0.0451 0.0023

].

It may be remarked that since all the results reported here were rounded to four decimal places, the sum of theelements of probability vectors may not add to one in some cases.

U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19 15

Table 3Queue-length distributions at various epochs in the case of multiple vacations when service and vacation times are arbitrary with ρ = 0.6097,λ∗ = 0.0924, s5 = 0.6, s9 = 0.4, v2 = 0.2, v5 = 0.3, v7 = 0.5, N = 10, m = 7

n∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1

p+k (n) π+

k (n) ω+

k (n) πk (n) ωk (n) pk (n) π−

k (n) ω−

k (n) p−

k (n)

0 0.1872 0.1048 0.3352 0.1414 0.3453 0.4867 0.1180 0.1869 0.30491 0.2635 0.1475 0.0235 0.1856 0.0170 0.2027 0.1693 0.1450 0.31432 0.1845 0.1033 0.0730 0.1063 0.0274 0.1336 0.1438 0.0148 0.15863 0.1532 0.0858 0.0032 0.0758 0.0009 0.0767 0.0863 0.0091 0.09544 0.0869 0.0486 0.0049 0.0426 0.0008 0.0433 0.0547 0.0004 0.05515 0.0549 0.0308 0.0001 0.0253 0.0000 0.0253 0.0313 0.0002 0.03146 0.0313 0.0175 0.0001 0.0146 0.0000 0.0146 0.0182 0.0000 0.01827 0.0183 0.0102 0.0000 0.0084 0.0000 0.0084 0.0105 0.0000 0.01058 0.0105 0.0059 0.0000 0.0049 0.0000 0.0049 0.0061 0.0000 0.00619 0.0061 0.0034 0.0000 0.0028 0.0000 0.0028 0.0036 0.0000 0.0036

10 0.0036 0.0020 0.0000 0.0010 0.0000 0.0010 0.0018Sum 1.0000 0.5599 0.4401 0.6086 0.3914 1.0000 0.6418 0.3564 1.0000

ρ′ = 0.6086, PBL = 0.0018, Lq = 1.2198, Wq = 13.2275.

Table 4Queue-length distributions at various epochs in the case of multiple vacations when service and vacation times are geometric with ρ = 1.1111,λ∗ = 0.3333, E(S) = 3.3333, E(V ) = 5.0000, N = 10, m = 2

n∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1∑m

k=1

p+k (n) π+

k (n) ω+

k (n) πk (n) ωk (n) pk (n) π−

k (n) ω−

k (n) p−

k (n)

0 0.0383 0.0346 0.0622 0.0398 0.0989 0.1387 0.0204 0.0297 0.05001 0.0525 0.0474 0.0109 0.0502 0.0124 0.0627 0.0353 0.0204 0.05572 0.0635 0.0574 0.0075 0.0585 0.0085 0.0670 0.0466 0.0140 0.06053 0.0724 0.0654 0.0051 0.0653 0.0058 0.0712 0.0554 0.0096 0.06504 0.0799 0.0722 0.0035 0.0712 0.0040 0.0753 0.0626 0.0066 0.06915 0.0865 0.0781 0.0024 0.0766 0.0027 0.0793 0.0687 0.0045 0.07326 0.0926 0.0836 0.0016 0.0816 0.0019 0.0834 0.0741 0.0031 0.07727 0.0983 0.0888 0.0011 0.0864 0.0013 0.0877 0.0792 0.0021 0.08138 0.1040 0.0939 0.0008 0.0912 0.0009 0.0921 0.0840 0.0015 0.08549 0.1096 0.0990 0.0005 0.1187 0.0006 0.1193 0.1567 0.0010 0.1579

10 0.2023 0.1827 0.0012 0.1220 0.0013 0.1233 0.2247Sum 1.0000 0.9032 0.0968 0.8615 0.1385 1.0000 0.6830 0.0923 1.0000

ρ′ = 0.4673, PBL = 0.2247, Lq = 5.2661, Wq = 20.3761.

In Fig. 2, we have plotted the effect of buffer size (N ) on the probability that the server is busy (ρ′) by consideringthe arrival process of Case 2. It can be observed that for moderately small values of N , ρ′ in the case of a single vaca-tion is greater than the one in the case of multiple vacations, implying that the server becomes more busy in the case ofsingle vacation than in the case of multiple vacations. Further, ρ′ increases as N increases in both cases and finally ittends to ρ. This is due to the fact that the models behave as infinite-buffer queues and hence ρ and ρ′ become identical.

Fig. 3 compares the loss probability versus buffer size with and without considering the correlated nature of arrivals.The mean arrival rate for D-MAP with input matrices given in Case 1 as well as for the Bernoulli arrival process isset at λ∗ = 0.3333. One may note here that the geometric distribution has no correlation among arrivals. It can beobserved that for all values of N , D-MAP yields higher loss probability than Bernoulli arrivals. We further observethat the loss probability in the case of multiple vacations is a little higher than the one obtained using a single vacationand they become identical as buffer size increases.

In Fig. 4, the effect of mean vacation time on average waiting time in the queue is shown by considering the arrivalprocess of Case 1. From this figure we see that as the mean vacation time increases, the corresponding average waitingtime increases linearly. Further, the average waiting time in the case of multiple vacations is slightly higher than theone obtained using a single vacation when mean vacation time is comparatively small, and for longer vacation durationthe difference between them becomes negligible.

16 U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19

Fig. 2. Effect of buffer size (N ) on the probability that the server is busy (ρ′), when service time is deterministic with mean E(S) = 5 and vacationtime is geometric with mean E(V ) = 20.

Fig. 3. Effect of buffer size (N ) on the blocking probability (PBL), when service time is geometric with mean E(S) = 3.3333 and vacation timeis geometric with mean E(V ) = 5.

The effect of buffer size on the loss probability for various coefficients of correlation (ccorr) with lag 3 is shownin Fig. 5. This figure displays results for the single vacation model. The mean arrival rate is set at λ∗ = 0.2364 byadjusting the elements of the input matrices C and D of the arrival process with phase m = 2. Using the formula givenin Blondia [4], various coefficients of correlation between arrivals are calculated. Fig. 5 shows that, for fixed N , theloss probability increases with the increase of coefficient of correlation. Further, it can be seen that for fixed coefficientof correlation the loss probability decreases as N increases and finally reaches its minimum value of zero. Besidesother revelations, this study also reveals that not only the service-, vacation-times and buffer size play an importantrole in queueing processes, but also the correlation among arrivals plays a major role. Similar observations also holdtrue in the case of multiple vacations.

5. Conclusion

This paper presents a comprehensive analysis of a discrete-time single-server finite-buffer exhaustive (single-and multiple-) vacation queueing system with a discrete-time Markovian arrival process (D-MAP). The analysis of

U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19 17

Fig. 4. Effect of average vacation time (E(V )) on the average waiting time (Wq ), when service time is geometric with mean E(S) = 3.3333,N = 10 and vacation time is deterministic.

Fig. 5. Effect of buffer size (N ) on the blocking probability (PBL), when service time is deterministic with mean E(S) = 4 and vacation time isdeterministic with mean E(V ) = 7.

a discrete-time batch Markovian arrival process (D-BMAP) and a server on vacation with partial and total batchrejection can also be investigated using the procedure discussed in this paper.

Acknowledgment

The authors are grateful to Mr. Sudhir Kumar Singh, M.Sc. (Mathematics and Computing), IIT Kharagpur, India,for his programming support. We are also grateful to the referees for their valuable comments and suggestions whichhave helped in improving the quality of the presentation of this paper.

References

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Academic Publishers, Boston, 2000, pp. 365–407.[7] M.L. Chaudhry, U.C. Gupta, Queue length distributions at various epochs in discrete-time D-MAP/G/1/N queue and their numerical

evaluations, International Journal of Information and Management Sciences 14 (3) (2003) 67–83.[8] M.L. Chaudhry, U.C. Gupta, Analysis of a finite-buffer bulk-service queue with discrete-Markovian arrival process: D-MAP/Ga,b/1/N ,

Naval Research Logistics 50 (2003) 345–363.[9] D. Fiems, H. Bruneel, Analysis of a discrete-time queueing system with timed vacations, Queueing Systems 42 (2002) 243–254.

[10] D. Fiems, S.D. Vuyst, H. Bruneel, The combined gated-exhaustive vacation system in discrete time, Performance Evaluation 49 (2002)227–239.

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Press, New York, 1983.[13] M.A. Johnson, S. Narayana, Descriptors of arrival process burstiness with application to the discrete time Markovian arrival process, Queueing

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U.C. Gupta is currently Professor in the Department of Mathematics at Indian Institute of Technology, Kharagpur. Hedid his masters in Statistics from Banaras Hindu University, Varanasi, in the year 1978. Further he completed his Ph.D.from Indian Institute of Technology, Delhi, in 1982. For a short stint (December 1982–October 1985) Dr. Gupta was inIndian Statistical Services (ISS) and worked with Govt. of India in various capacities. Since November 1985, Dr. Guptahas been teaching various courses at IIT Kharagpur. Dr. Gupta has contributed significantly in the area of modeling andanalysis of continuous- and discrete-time queueing systems. He has published several research articles in various journalssuch as Stochastic Processes and Their Applications, Queueing Systems, European Journal of Operational Research,Performance Evaluation, Journal of Applied Probability, Probability in the Engineering and Informational Sciences,Operational Research Letters, Informs Journal of Computing.

S.K. Samanta was born in West Bengal, India, in 1977. He received the B.Sc. and M.Sc. degrees in Mathematics fromthe Vidyasagar University, West Bengal, India, in 1998 and 2000, respectively. Since July 2001, he is working as a Ph.D.student at the Department of Mathematics, Indian Institute of Technology, Kharagpur, India. His main research interestsinclude discrete-time queueing theory and its applications. He has published research articles in Journal of the OperationalResearch Society, Computers and Mathematics with Applications.

R.K. Sharma is currently Associate Professor in the Department of Mathematics at Indian Institute of Technology, NewDelhi, India. He received Masters degree in Mathematics in the year 1978 and Ph.D. from Indian Institute of Technology,New Delhi, India, in 1984. From July 1988 to November 2003, Dr. Sharma worked at IIT Kharagpur as Assistant andAssociate Professor. His research interests include Algebra, Linear Algebra and its application in various fields such asQueueing, Applied Stochastic Process, Cryptography and their Applications. He has published a number of papers inreputed international journals such as Proceedings of American Mathematical Society, Communication in Algebra.

U.C. Gupta et al. / Performance Evaluation 64 (2007) 1–19 19

M.L. Chaudhry, a Professor in the Department of Mathematics and Computer Science at the Royal Military College ofCanada, Kingston, Ont., and a distinguished member of the Alumni of Kurukshetra University has served the college invarious capacities including as Head of the Department. He has also served and continues to serve in various professionalsocieties as member and/or Associate Editor of Editorial Boards. He has won national and international awards and has heldappointments at universities in India, Canada, USA, South Korea (distinguished professorship), and Belgium (distinguishedprofessorship), in addition to having held adjunct Professorship at several Canadian universities. Another benchmark ofhis research impact is his success at obtaining and retaining grants from various funding agencies. Having published over12 dozen articles in numerous international operations research and statistics journals, he has co-authored two books (i) AFirst Course in Bulk Queues, Wiley, New York, 1983 and (ii) An Introduction to Queueing Theory, A & A Publications,395 Carrie Crescent, Kingston, Ont., Canada K7M 5X7 (Tel./fax: 613-389-7697). In addition, he has edited or co-editedspecial issues of several journals and has prepared software packages for several queueing models.

Discrete-time single-server finite-buffer queues under discrete Markovian arrival process with vacationsIntroductionAssumptions and notationsBasic equations and analysisQueue-length distributions at various epochsQueue-length distribution at departure epochQueue-length distribution at service completion and vacation termination epochsQueue-length distribution at an arbitrary epochQueue-length distribution at prearrival epoch

Waiting time analysis

Numerical resultsConclusionAcknowledgmentReferences