1 queue with multiple vacations

Applied Mathematical Modelling 36 (2012) 5964–5975

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Stationary analysis of a discrete-time GI/D-MSP/1 queue with multiple vacations S.K. Samanta a,⇑, Zhe G. Zhang b,c a

LIA/CERI, University of Avignon, Agroparc BP 1228, Avignon 84911, Cedex 9, France Department of Decision Sciences, Western Washington University, College of Business and Economics, Bellingham, WA 98225, USA c Beedie School of Business, Simon Fraser University, Burnaby, BC, Canada V5A 1S6 b

a r t i c l e

i n f o

Article history: Received 25 February 2011 Received in revised form 23 December 2011 Accepted 17 January 2012 Available online 28 January 2012 Keywords: Discrete-time Markovian service process (DMSP) Matrix-geometric method Multiple vacations Queue Waiting time

a b s t r a c t This paper analyzes the steady-state behavior of a discrete-time single-server queueing system with correlated service times and server vacations. The vacation times of the server are independent and geometrically distributed, and their durations are integral multiples of slot duration. The customers are served one at a time under discrete-time Markovian service process. The new service process starts with the initial phase distribution independent of the path followed by the previous service process when the server returns from a vacation and finds at least one waiting customer. The matrix-geometric method is used to obtain the probability distribution of system-length at prearrival epoch. With the help of Markov renewal theory approach, we also derive the system-length distribution at an arbitrary epoch. The analysis of actual-waiting-time distribution in the queue measured in slots has also been carried out. In addition, computational experiences with a variety of numerical results are discussed to display the effect of the system parameters on the performance measures. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Queueing models with non-renewal service processes are often used to model complex computer and communication systems. Several connections (data, voice, video, etc.) generate traffic streams with very different characteristics (required bandwidth, burstiness, correlation, etc.). Traditional teletraffic analysis based on exponential/geometric service time distribution is not powerful enough to capture this correlative and bursty feature of traffic streams in high-speed packet/cell based networks. These correlated and bursty non-renewal service processes in queueing systems have been shown empirically and theoretically to have a significant impact on queueing behavior. The discrete-time Markovian service process (DMSP) is a versatile service process and can capture the correlation among successive service times. Note that the D-MSP is independent of the arrival process and it generates real service completions only when the server is busy. This service process forms a rich class of point processes that include several known processes such as Markov-modulated Bernoulli process (MMBP), discrete-time phase type renewal process, and superposition of such processes. In the past decades, many researchers have analyzed several queueing models with various types of service processes and are available in the literature. Alfa et al. [1] discussed the asymptotic behavior of the GI/MSP/1 queue using perturbation theory. The analysis of finite- and infinite-buffer G/MSP/1 queue has been carried out by Bocharov et al. [2], wherein they derived stationary characteristics of system performance by considering that the service phase does not change in an idle period. Gupta and Banik [3] have analyzed GI/MSP/1 queue with finite as well as infinite buffer using a combination of

⇑ Corresponding author. Tel.: +33 4 90 84 35 62. E-mail addresses: [email protected] (S.K. Samanta), [email protected] (Z.G. Zhang). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2012.01.049

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the matrix-geometric method, the imbedded Markov chain and the supplementary variable techniques. Samanta et al. [4] carried out discrete-time GI/D-MSP/1 queue with finite and infinite buffers, where the phase of the Markovian service process is not frozen, that is, service phase keeps changing when the server is idle. Wang et al. [5] analyzed the packet loss pattern of the finite-buffer D-BMAP/D-MSP/1 queueing system using matrix-geometric approach. In these connections, see also Alfa [6] and Horváth et al. [7]. Queueing models with server vacations are characterized by the fact that the idle time of the server may be utilized for some other jobs. During the past decades, queueing systems with different types of vacation policies have been extensively studied by numerous researchers and applied to flexible manufacturing environments, production, computers, communication networks, telecommunication systems, traffic concentrators and other related areas. Some discrete-time vacation queueing models’ applications has been reported by Alfa [8]. A variety of vacation policies as well as a vast amount of related references are available in Takagi [9], and Tian and Zhang [10]. Samanta [11] carried out analysis of discrete-time GI/Geo/1 queue with single vacation. Ozawa [12] first investigated different vacation policies queueing model under Markovian service process, and obtained a matrix-type factorization of the vector generating function for the stationary queue length in MAP/MSP/1 queues. In this paper, we analyze an infinite-buffer discrete-time GI/D-MSP/1 multiple vacations and exhaustive service queueing system, where the input follows a discrete-time renewal process and the departures form a discrete-time Markovian service process. The new service process starts with the initial phase distribution independent of the path followed by the previous service process when the server returns from a vacation and finds at least one waiting customer. The matrix-geometric method is applied to obtain prearrival epoch probability and the Markov renewal theory approach is used to develop a relation between prearrival and arbitrary epoch probabilities. The analysis of actual-waiting-time distribution in the queue measured in slots has also been investigated. In addition, computational experiences with a variety of numerical results are discussed to display the effect of the system parameters on the performance measures. The model presented in this paper may be useful for the performance evaluation of an energy-aware medium access control (MAC)/physical (PHY) layer protocol in view of bursty traffic service patterns (modeled as D-MSP). The MAC/PHY layer in a node is modeled as a server and a vacation queueing model is utilized to represent the sleep and wakeup mechanism of the server. Energy efficiency is a major concern in traditional wireless networks due to the limited battery power of the nodes. In one direction to save energy in such a network is to bring into play an efficient sleep and wakeup mechanism to turn off the radio transceiver irregularly in order that the desired trade-off between the node energy savings and the network performance can be achieved. This paper is organized as follows. In Section 2, we give the description of the model and introduce the notations to describe the model parameters. The steady-state system-length distributions at various epochs and waiting-time distribution of an arriving customer are analyzed in Section 3. A variety of numerical results are presented in Section 4. Section 5 concludes the paper. 2. Model description and notations We consider a single-server infinite-buffer queueing system wherein customers are served according to a discrete-time Markovian service process (D-MSP). Formally, D-MSP is characterized by the services which are governed by an underlying m-state Markov chain having probability (L0)ij, 1 6 i, j 6 m, of a transition from state i to j without service completion and probability (L1)ij, 1 6 i, j 6 m, of a transition from state i to j with service completion. Let (L0)ij and (L1)ij be the (i, j)-th entry of the m  m non-negative matrices L0 and L1, respectively, with L1 having at least one positive entry such that (L0 + L1)e = e, where e is a column vector of ones with an appropriate dimension. The sum (L0 + L1) is a stochastic matrix corresponding to the transition probability matrix of an irreducible Markov chain underlying D-MSP. We call the actual state of this chain the ‘‘phase’’ of the service process. Let p = ½p1 ; p2 ;    ; pm  be the stationary probability vector of the Markov process with the stochastic matrix ðL0 þ L1 Þ, where pj denotes the steady-state probability of service process being in phase j. The stationary probability row-vector p can be calculated from pðL0 þ L1 Þ = p with pe = 1. Let Y(t) denote the number of customers served in the first t time slots when the server is busy and I(t) the state of the underlying Markov chain (called service phase) after the same amount of time. Then {Y(t), I(t)}tP0 is a two-dimensional discrete-time Markov process with state space {(n, i) : n P 0, 1 6 i 6 m} and the state transition matrix

0

1

L0

L1

0

0



B0 B Q DMSP ¼ B B0 @ .. .

L0

L1

0

0 .. .

L0 .. .

L1 .. .

C C C: C A .. .

Let {P(n, t):n P 0,t P 1} denote the matrix of order m  m whose (i, j)-th element (Pi, j(n, t)) represents the probability that n customers are served in (0, t] with the service process being in phase j at time t, provided initially there were at least (n + 1) customers (including the one in service) in the system and the service process was in phase i. Then, the probabilities

Pij ðn; tÞ ¼ PrfYðtÞ ¼ n;

IðtÞ ¼ jjYð0Þ ¼ 0; Ið0Þ ¼ ig;

lead to the following equations (in matrix notation)

1 6 i; j 6 m;

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Pð0; tÞ ¼ Pð0; t  1ÞL0 ;

t P 1;

ð1Þ

Pðn; tÞ ¼ Pðn; t  1ÞL0 þ Pðn  1; t  1ÞL1 ;

n P 1; t P 1;

ð2Þ

with P(0, 0) = Im and P(n, t) = 0, n > t P 0, where Im is the identity matrix of order m  m. Let us define the matrix-generating function (MGF) P⁄(z, t) as

P ðz; tÞ ¼

1 X

Pðn; tÞzn ;

jzj 6 1:

ð3Þ

n¼0

Multiplying (1) by z0 and (2) by zn and summing over n = 0 to 1, after using (3), we get

P ðz; tÞ ¼ P ðz; t  1ÞðL0 þ L1 zÞ;

t P 1;

⁄

with P (z, 0) = Im. Solving the above matrix-difference equations, we get

P ðz; tÞ ¼ ðL0 þ L1 zÞt ;

jzj 6 1;

t P 0:

ð4Þ

The first moment in matrix form [differentiation of (4) w.r.t. z and setting z = 1] is given by

M1 ðtÞ ¼

t1 X ðL0 þ L1 Þt1k L1 ðL0 þ L1 Þk : k¼0

The mean number of service completions during a time of length t is

lI ðtÞ ¼ pM1 ðtÞe ¼ tpL1 e; where we are assuming that Ið0Þ has distribution p. The average service rate lI (the so called fundamental service rate) of I the stationary D-MSP is given by lI =l tðtÞ=pL1 e. Here we discuss the model for the late arrival system with delayed access (LAS-DA), and the early arrival system (EAS). Assume that the time axis is slotted into intervals of equal length with the length of a slot being unity, and it is marked as 0,1, 2, . . . , t, . . .. In LAS-DA, a potential arrival occurs in (t  , t) and a potential departure occurs in (t, t+), whereas in EAS a potential arrival occurs in (t, t+) and a potential departure occurs in (t  , t). The concepts concerning discrete-time queueing systems have been explained in the past at several places by many researchers and are available in the literature. The server takes a vacation as soon as the system-length becomes empty at the completion of service. According to the late arrival system with delayed access, the vacation can only start or end at discrete-time instant t + ; whereas in the early arrival system, it can only start or end at instant t, that is, just after service completion. On return from a vacation if the server finds one or more customers waiting, he takes them for service on a one-by-one basis until the system empties, after which time he takes another vacation. However, if, on return from a vacation, the server finds no customer waiting, then, he immediately proceeds for another vacation and continues in this manner until he finds at least one waiting customer upon return from a vacation. The times between successive arrivals are independent and identically-distributed (i.i.d.) random variables (r.v.’s) A1, A2, . . . , An, . . . , with common probability mass function (p.m.f.) P k ak = P(An = k), k P 0(a0 = 0), corresponding probability generating function (p.g.f.) AðzÞ ¼ 1 k¼0 ak z ; jzj 6 1 and they are indeP1 pendent of the service process. The mean interarrival time is 1=k ¼ k¼1 kak . The traffic intensity is given by q = k/lq < 1. The vacation time is an independent and identically distributed (i.i.d.) random variable and is denoted by V. In this analysis, we assume that V follows a geometric distribution with probability mass function (p.m.f.) P(V = k) = (1  h)k1h, k P 1(0 < h < 1) and mean vacation time 1/h. Note that when the server is on vacation, the phase of the Markovian service process is not frozen, i.e., service phase of the underlying m-state Markov chain keeps changing. The case when the server returns from a vacation and finds at least one waiting customer then the service process starts with the initial phase distribution P fj ; j ¼ 1; 2; . . . ; m; m j¼1 fj ¼ 1, independent of the path followed by the previous service process. Therefore, D-MSP is characterized by the matrices L0, L1 along with the vector f ¼ ðf1 ; f2 ; . . . ; fm Þ. In order to obtain the system-length distributions at various epochs, first we derive a few basic results which are used later. Let Sn, n P 0, denote the matrix of order m  m whose (i, j)-th element represents the probability that n customers are served during an interarrival period of the arrival process with the service process being in phase j, given that there were at least (n + 1) customers in the system and the service process was in phase i at the beginning of the interarrival period. Then, we have

Sn ¼

1 X

ak Pðn; kÞ;

n P 0:

k¼1

If S(z) is the MGF of Sn, using (4), then

SðzÞ ¼

1 X n¼0

Sn zn ¼

1 X 1 X n¼0 k¼1

ak Pðn; kÞzn ¼

1 X

ak ½L0 þ L1 zk :

ð5Þ

k¼1

P k Setting z = 1 in the above equation, we get S  Sð1Þ ¼ 1 k¼1 ak ½L 0 þ L 1  : We note that the matrix S is stochastic and that the  defined earlier satisfies p S ¼ p ; p  e ¼ 1. The matrix S also represents the number of customers served durstationary vector p ing an interarrival time and the phase changes of the underlying Markov chain during a busy period.

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Let b S n ; n P 1, denote the matrix of order m  m whose (i, j)-th element represents the probability that at least (n + 1) customers are served during an interarrival period of the arrival process with the service process being in phase j, given that the service process was in phase i at the beginning of the interarrival period. Then, we have 1 X

b Sn 

Sr ¼ S 

n X

r¼nþ1

Sr ;

n P 1:

r¼0

Let Kn, n P 0, denote the matrix of order m  m whose (i, j)-th element represents the conditional probability that n customers are served during an interarrival period with the service process being in phase j, assuming that a vacation period has elapsed and the service process was in phase i. In LAS-DA, if an interarrival period consisting of k slots and a vacation period of length r, (0 6 r 6 k  1  n), slots has elapsed then during this interarrival period there are (k  1  r) possible service positions available for n departures. Hence, we obtain 1 X

Kn ¼

ak Tnðk1Þ ;

n P 0;

k¼nþ1

where

TnðkÞ ¼

kn X ð1  hÞr hPðn; k  rÞ;

n P 0;

k P 0:

ð6Þ

r¼0

Now, using (1) and (2) in (6), we obtain

"

ðkÞ

kþ1 # 1 L0 L0 Im  ð1  hÞk h; k P 0 1h 1h     1 h L0 ðk1Þ Pð1; kÞL0 Im  ¼ T0 L1  ; kP1 1h 1h       1 L1 h L0 ðkÞ Pðn; k þ 1Þ Im  ¼ Tn1 ; k P n P 2:  1h 1h 1h

T0 ¼ Im  ðkÞ

T1

TnðkÞ



Further, since in EAS system customers may depart in vacation termination slot, if an interarrival period consisting of k slots and a vacation period of length r + 1, (0 6 r 6 min(k  1, k  n)), slots has elapsed then during this interarrival period there are (k  r) possible service positions available for n departures. Hence, we obtain

K0 ¼

1 X

ak

k¼1

Kn ¼

1 X

   1 k1 X hL0 L0 ð1  hÞr hPð0; k  rÞ ¼ ½Að1  hÞIm  AðL0 Þ Im  ; 1h 1h r¼0

ak TnðkÞ ;

n P 1:

k¼n

b n ; n P 0, denote the matrix of order m  m whose (i, j)-th element represents the conditional probability that at least Let K (n + 1) customers are served during an interarrival period of the arrival process with the service process being in phase j, assuming that a vacation period has elapsed and the service process was in phase i. Then, we have

bn ¼ K

1 X

Kr ;

n P 0:

r¼nþ1

Further, let M denote the conditional probability that interarrival period is smaller than a vacation period. Thus, we have

M

1 X

ak ð1  hÞk ¼ Að1  hÞ:

k¼1

3. Analysis of the model In this section, we carry out the distributions of system-length at prearrival and arbitrary epochs, and waiting time for an arrival customer. The matrix-geometric method is used to obtain prearrival epoch probability and the classical argument based on Markov renewal theory approach is applied to develop a relation between prearrival and arbitrary epoch probabilities. 3.1. System-length distribution at prearrival epoch We derive here the system-length distribution at prearrival epoch. Let the k-th customer arrive at time instant sk, k = 0, 1, . . . , with s0 = 0, and let s k denote the prearrival epoch, i.e., the time epoch just before the arrival instant sk. Then

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n o the state of the system at s N sk ; J s is a Markov chain, where N sk represents the number of customers in the k defined as k system and J s the state of the server at the embedded point s k . Then we define the state of the server as k

J s ¼



0;

the k  th arrival occurs when the server is on vacation;

ði; 1Þ; the k  th arrival occurs when the server being busy in phase i:

k

n o Then N sk ; J s ; k P 1 is a Markov chain with state space X = {(0, 0)} [ {(n, i, 1), (n, 0) : 1 6 i 6 m, n P 1}. Using the lexigraphk ical sequence to order the states as (0, 0),{(n, i, 1), (n, 0) : 1 6 i 6 m, n P 1}, the transition probability matrix associated with this Markov chain in the case of LAS-DA can be written as

0

B0 B B B 1 B B Q ¼ B B2 B B B3 @ .. .

A0

0

0

A1

A0

0

A2

A1

A0

A3 .. .

A2 .. .

A1 .. .



1

C C C C C; C C A .. .

where the block matrices of Q are given by

b 0 e; B0 ¼ f K

 A0 ¼ fK0



M ;

Bn ¼

! b Sn e ; f K b ne

n P 1;

A0 ¼



S0 fK0

0 M

 ;

An ¼



Sn fKn

0 0

 ;

n P 1:

Note that in the case of EAS, at the time of numerical computation, the expression of Kn defined at the end of Section 2 is only changed in the above block matrices. Let w(n), n P 0, denote the probability of n customers in the system at prearrival epoch when the server is on vacation,

   and p ðnÞ ¼ p 1 ðnÞ; p2 ðnÞ; . . . ; pm ðnÞ ; n P 1, is the 1  m vector whose i-th component pi ðnÞ denotes the probability of n customers in the system at prearrival epoch when the server being busy in phase i. Based on the matrix-geometric solution method given in Neuts [13], we have

½p ðnÞ; w ðnÞ ¼ ½p ð1Þ; w ð1ÞRn1 ;

n P 1;

with R0 ¼ Imþ1 ;

where the square matrix R of order (m + 1) is the minimal non-negative solution to the matrix polynomial equation

R¼

1 X

Rk Ak

k¼0

and all eigenvalues of R lie inside the unit disk. This matrix R is evaluated numerically by a simple iterative scheme (Alfa et al. [14]) as follows:

Rðn þ 1Þ ¼

A0 þ

1 X

! k

R ðnÞAk ðI  A1 Þ1 ;

with Rð0Þ ¼ 0;

k¼2

where R(n) is the value of R at the nth iteration. In order to obtain w(0), p(1) and w(1), we evaluate [u(0), v(1), u(1)] which is the left eigenvector corresponding to the eigenvalue 1 of the matrix B[R], where

vð1Þ ¼ ½v1 ð1Þ; v2 ð1Þ; . . . ; vm ð1Þ; 0 B½R ¼ @

A0

B0 1 P k¼1

Rk1 Bk

1 P

Rk1 Ak

1 A:

k¼1

Note that the vector [w(0), p(1), w(1)] is proportional to the invariant probability vector [u(0), v(1), u(1)] and therefore the normalizing condition w(0) + [p(1), w(1)](I  R)1e = 1 yields w(0) = d1u(0), p(1) = d1v(1) and w(1) = d1u(1), where d = u(0) + [v(1), u(1)](I  R)1e. 3.2. System-length distribution at arbitrary epoch We now derive explicit expressions for the steady-state system-length distribution w(n), n P 0 and p(n) = [p1(n), p2(n), . . . , pm(n)],n P 1 at an arbitrary epoch using the classical argument based on Markov renewal theory, which relates the steadystate system-length distribution at an arbitrary epoch to that at the corresponding prearrival epoch. Note that w(n) denotes the probability of n customers in the system at an arbitrary epoch when the server is on vacation, and p(n) is the 1  m vector whose i-th component pi(n) denotes the probability of n customers in the system at an arbitrary epoch when the server being busy in phase i.

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In order to obtain arbitrary epoch probabilities, we introduce the matrix Gn of order m  m whose (i, j)-th element represents the limiting probability that n customers are served during an elapsed interarrival time of the arrival process with the service process being in phase j, given that there were at least (n + 1) customers in the system and the service process was in phase i at the beginning of the interarrival period. Then, we have 1 X

Gn ¼ k

Pðn; kÞ

k¼1

1 X

al ;

n P 0:

ð7Þ

l¼k

Now, using (1) in (7), for n = 0, we have 1 X

G0 ¼ k

Lk0

k¼1

1 X

al ¼ kðIm  S0 ÞðIm  L0 Þ1 L0 :

l¼k

Using (2) in (7), for n = 1, we have 1 X

G1 ¼ k

Pð1; k  1ÞL0

1 X

k¼2

al þ k

l¼k

1 X

Pð0; k  1ÞL1

k¼1

1 X

al ¼ k

1 X

l¼k

Pð1; kÞL0

k¼1

1 X

! al  ak

þk

l¼k

1 X k¼1

Lk1 0 L1

1 X

al

l¼k

¼ G1 L0  kS1 L0 þ kðIm  S0 ÞðIm  L0 Þ1 L1 ; which yields

h i G1 ¼ k ðIm  S0 ÞðIm  L0 Þ1 L1  S1 L0 ðIm  L0 Þ1 : Using (2) in (7), for n P 2, we have 1 X

Gn ¼ k

Pðn; kÞL0

k¼n

1 X

! al  ak

þk

l¼k

1 X

Pðn  1; kÞL1

k¼n1

1 X

! al  ak ;

l¼k

which leads to

Gn ¼ ½Gn1 L1  kSn1 L1  kSn L0 ðIm  L0 Þ1 ;

n P 2:

Further, let H denote the limiting probability that the server is on vacation. Thus, we have

H¼k

1 1 X X kð1  hÞð1  MÞ ð1  hÞk al ¼ : h k¼1 l¼k

Let Fn denote the matrix of order m  m whose (i, j)-th element represents the limiting probability that n customers are served during an elapsed interarrival time of the arrival process with the service process being in phase j, assuming that a vacation period has elapsed and the service process was in phase i. In LAS-DA, we have 1 k1n 1 X X X ð1  hÞr hPðn; k  1  rÞ al ;

Fn ¼ k

k¼nþ1

r¼0

n P 0:

l¼k

Now, using (1) in (8), for n = 0, we have

 F0 ¼

  1 h L0 ðHIm  G0 Þ Im  : 1h 1h

Using (2) in (8), for n = 1, we have

k1r  X 1 X k3 1 1 k1  1 X X X X L0 hL1 ð1  hÞr hPð1; k  2  rÞL0 al þ k ð1  hÞk al 1  h 1  h r¼0 k¼3 r¼0 l¼k k¼1 l¼kþ1 !  1   1 X k2 1 X X L0 hL1 ¼k ð1  hÞr hPð1; k  1  rÞL0 al  ak þ ½ðH  kMÞIm  G0 þ kS0  Im  1h 1h k¼2 r¼0 l¼k  1   L0 hL1 ; ¼ F1 L0  kK1 L0 þ ½ðH  kMÞIm  G0 þ kS0  Im  1h 1h

F1 ¼ k

which yields

"

#  1   L0 hL1  kK1 L0 ðIm  L0 Þ1 : F1 ¼ ½ðH  kMÞIm  G0 þ kS0  Im  1h 1h Using (2) in (8), for n P 2, we have

ð8Þ

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S.K. Samanta, Z.G. Zhang / Applied Mathematical Modelling 36 (2012) 5964–5975

!

1 k1n 1 X X X ð1  hÞr hPðn; k  1  rÞL0 al  ak

Fn ¼ k

k¼nþ1

r¼0

þk

1 X kn 1 X X ð1  hÞr hPðn  1; k  1  rÞL1 al  ak k¼n r¼0

l¼k

!

l¼k

which leads to

Fn ¼ ½Fn1 L1  kKn L0  kKn1 L1 ðIm  L0 Þ1 ;

n P 2:

Similarly, in EAS, we have 1 X

minðk1;knÞ X

k¼maxð1;nÞ

r¼0

Fn ¼ k

ð1  hÞr hPðn; k  rÞ

1 X

al ;

n P 0:

ð9Þ

l¼k

Now, using (1) and (2) in (9), after simplification, we have

 1   L0 hL0 ; F0 ¼ ½HIm  G0  Im  1h 1h " #  1   L0 hL1 F1 ¼ ½HIm  G0  Im   kK1 L0 ðIm  L0 Þ1 ; 1h 1h Fn ¼ ½Fn1 L1  kKn L0  kKn1 L1 ðIm  L0 Þ1 ;

n P 2:

Using the results of Markov renewal theory and semi-regenerative processes, see Neuts [13, pp. 147-150], for both the cases (LAS-DA and EAS), we obtain

wðnÞ ¼ w ðn  1ÞH; n P 1; 1 1 X X pð1Þ ¼ w ðnÞfFn þ p ðnÞGn ;

pðnÞ ¼

n¼0 1 X

ð10Þ ð11Þ

n¼1

w ðrÞfFrnþ1 þ

r¼n1

1 X

p ðrÞGrnþ1 ; n P 2:

ð12Þ

r¼n1

Finally, w(0) is obtained using the condition

P1

n¼1

pðnÞe þ

P1

n¼0 wðnÞ

¼ 1, which is based on the definition of joint distribution.

3.3. Waiting-time distribution In this section, we obtain the actual-waiting-time distribution in the queue measured in slots of an arriving customer under the first-come-first-served queueing discipline. Let us define the random variable Tq as the total amount of time measured in slots that an arrival spends in the queue and the corresponding p.m.f. wk = P(Tq = k), k P 0. Further, let P Wq ¼ 1 k¼1 kwk denote the average waiting time in the queue of an arriving customer. In LAS-DA, an arriving customer may observe the system in any one of the following two cases. Case 1. If an arrival finds no customer in the system and the server is about to return from a vacation or one customer in the system who is about to depart, the service on the new arrival starts immediately. Thus, the probability that an arriving customer does not wait is given by

w0 ¼ w ð0Þh þ p ð1ÞL1 e: Case 2. If an arrival finds (i) no customer in the system in vacation state and the server returns from a vacation of length (k + 1) slots, (ii) n customers in the system in vacation state such that (n  1) departures occur during (k  1) slots, assuming that a vacation period has elapsed and one departure must occur in the k-th slot after his arrival, (iii) n customers in the system in busy state such that (n  1) departures occur during k slots and one departure must occur in the (k + 1)-th slot after his arrival, then the arriving customer will have to wait exactly k slots in the queue. Hence, the probability that an arriving customer will have to wait exactly k slots is given by

wk ¼ w ð0Þð1  hÞk h þ

k X n¼1

ðk1Þ w ðnÞfTn1 L1 e þ

kþ1 X

p ðnÞPðn  1; kÞL1 e; k P 1:

n¼1

Similarly, in EAS, an arriving customer also may observe the system in any one of the following two cases. Case 1. If an arrival finds no customer in the system and the server is about to return from a vacation, the service on the new arrival starts immediately. Thus, the probability that an arriving customer does not wait is given by

w0 ¼ w ð0Þh: Case 2. If an arrival finds (i) no customer in the system in vacation state and the server returns from a vacation of length (k + 1) slots, (ii) n customers in the system in vacation state such that (n  1) departures occur during (k  1) slots, assuming that a vacation period has elapsed and one departure must occur in the k-th slot after his arrival, (iii)n customers in the system in busy state such that (n  1) departures occur during (k  1) slots and one departure must occur in the k-th slot after

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his arrival, then the arriving customer will have to wait exactly k slots in the queue. Hence, the probability that an arriving customer will have to wait exactly k slots is given by

wk ¼ w ð0Þð1  hÞk h þ

k X

ðk1Þ w ðnÞfTn1 L1 e þ

n¼1

k X

p ðnÞPðn  1; k  1ÞL1 e; k P 1:

n¼1

4. Discussion of numerical results To demonstrate the applicability of the analytical results obtained in this paper, we present some numerical results in the form of tables and graphs. It is hoped that they will be useful to other researchers who would like to check their results against ours when they use other methods. Numerical results presented in this paper were performed using C++ programming language on Pentium IV PC in double precision but in tables they are given up to six decimal places. Of course, the values of the parameters are chosen so as to satisfy the stability condition q < 1. In the case of LAS-DA, the system length distributions at various epochs for all phases and waiting time distribution are given in Table 1 with arbitrary interarrival time distribution: a2 = 0.7, a7 = 0.3 and h = 0.2. Similar analysis has also been carried out in the case of EAS system when interarrival time is geometrically distributed with k = 0.4 and h = 0.6, and the results are presented in Table 2. The following matrices L0 and L1 of order 2 of the service process (D-MSP) along with the vector f ¼ ½0:3 0:7 are used for Tables 1 and 2.

L0 ¼



0:30 0:20 0:20 0:30

 ;

L1 ¼



0:20 0:30 0:30 0:20

 :

 ðL0 þ L1 Þ ¼ p  and p  e ¼ 1 give p  ¼ ½0:5 0:5 with l = 0.5. We also present at the bottom of the Tables 1 and 2, Now equations p the average waiting time in the queue (Wq) of a customer using Little’s rule, Wq = Lq/k, where P P1 Lq ¼ 1 n¼1 nwðnÞ þ n¼1 ðn  1ÞpðnÞe is the average queue length at arbitrary epoch. It can be seen in the case of LAS-DA system that the Wq using Little’s rule in Table 1 matches with the result obtained from waiting time distribution, but it is not for the case of EAS system as in this system the Little’s rule holds only at outside observer’s observation epoch. However, intuitively it is observed that Wq in LAS-DA and EAS systems are the same. Further, it can be seen from Table 2 that prearrival and arbitrary epoch probabilities are the same due to the memoryless property of Bernoulli arrivals. The effect of traffic load (q) on the average waiting time in the queue (Wq) in the case of EAS system is shown in Fig. 1 for different values of h and geometric interarrival time distribution. The following matrices L0 and L1 of order 2 of the service process (D-MSP) along with the vector f ¼ ½0:6 0:4 are used to obtain this figure.

Table 1 System-length and waiting-time distributions for LAS-DA case, when interarrival time distribution is arbitrary with a2 = 0.7, a7 = 0.3, k = 0.285714, q = 0.571429, h = 0.2. n

i=1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 sum

w(n)

pi ðnÞ

0.086054 0.073595 0.047180 0.027197 0.014894 0.007932 0.004157 0.002157 0.001113 0.000572 0.000293 0.000150 0.000077 0.000039 0.000020 0.000010 0.000005 0.000003 0.000001 0.000001 0.000000 0.265452

i=2 0.097739 0.079646 0.050271 0.028777 0.015701 0.008344 0.004367 0.002265 0.001168 0.000600 0.000308 0.000158 0.000081 0.000041 0.000021 0.000011 0.000006 0.000003 0.000001 0.000001 0.000000 0.289509

0.217662 0.111207 0.056817 0.029029 0.014831 0.007577 0.003871 0.001978 0.001011 0.000516 0.000264 0.000135 0.000069 0.000035 0.000018 0.000009 0.000005 0.000002 0.000001 0.000001 0.000000 0.000000 0.445038

pi(n) i=1

i=2

0.082329 0.076613 0.050355 0.029348 0.016163 0.008635 0.004534 0.002356 0.001216 0.000626 0.000321 0.000164 0.000084 0.000043 0.000022 0.000011 0.000006 0.000003 0.000002 0.000001 0.000000 0.272832 Lq = 1.415309 Wq = 4.953580

0.094562 0.083229 0.053738 0.031076 0.017046 0.009086 0.004764 0.002473 0.001276 0.000656 0.000337 0.000172 0.000088 0.000045 0.000023 0.000012 0.000006 0.000003 0.000002 0.000001 0.000000 0.298597

w(n)

k

0.179815 0.121663 0.062159 0.031758 0.016226 0.008290 0.004235 0.002164 0.001106 0.000565 0.000289 0.000147 0.000075 0.000038 0.000020 0.000010 0.000005 0.000003 0.000001 0.000001 0.000000 0.000000 0.428571

0 1 2 3 4 5 6 7 8 9 10 15 20 25 30 35 40 45 50 55 59 60

wk

0.135429 0.130205 0.118624 0.104464 0.089897 0.076102 0.063649 0.052750 0.043411 0.035529 0.028953 0.009996 0.003340 0.001103 0.000362 0.000119 0.000039 0.000013 0.000004 0.000001 0.000001 0.000000 1.000000 Wq = 4.953580

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Table 2 System-length and waiting-time distributions for EAS case, when interarrival time distribution is geometric with k = 0.4, q = 0.8, h = 0.6.

w(n)

pi ðnÞ

n

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 31 32 sum

i=1

i=2

0.091961 0.081391 0.058519 0.039910 0.026796 0.017904 0.011944 0.007964 0.005310 0.003540 0.002360 0.001573 0.001049 0.000699 0.000466 0.000311 0.000207 0.000138 0.000092 0.000061 0.000008 0.000001 0.000001 0.000000 0.352319

0.094274 0.081972 0.058645 0.039937 0.026801 0.017905 0.011944 0.007965 0.005310 0.003540 0.002360 0.001573 0.001049 0.000699 0.000466 0.000311 0.000207 0.000138 0.000092 0.000061 0.000008 0.000001 0.000001 0.000000 0.355373

pi(n)

0.230769 0.048583 0.010228 0.002153 0.000453 0.000095 0.000020 0.000004 0.000001 0.000000

0.292308

i=1

i=2

0.091961 0.081391 0.058519 0.039910 0.026796 0.017904 0.011944 0.007964 0.005310 0.003540 0.002360 0.001573 0.001049 0.000699 0.000466 0.000311 0.000207 0.000138 0.000092 0.000061 0.000008 0.000001 0.000001 0.000000 0.352319

0.094274 0.081972 0.058645 0.039937 0.026801 0.017905 0.011944 0.007965 0.005310 0.003540 0.002360 0.001573 0.001049 0.000699 0.000466 0.000311 0.000207 0.000138 0.000092 0.000061 0.000008 0.000001 0.000001 0.000000 0.355373 Lq = 1.682051 Wq = 4.205128

w(n)

k

0.230769 0.048583 0.010228 0.002153 0.000453 0.000095 0.000020 0.000004 0.000001 0.000000

0 1 2 3 4 5 6 7 8 9 10 11 15 20 25 30 35 40 45 50 55 60 65 69 70

wk

0.138462 0.163077 0.124205 0.098827 0.080485 0.066323 0.054970 0.045688 0.038026 0.031669 0.026383 0.021983 0.010600 0.004260 0.001712 0.000688 0.000276 0.000111 0.000045 0.000018 0.000007 0.000003 0.000001 0.000001 0.000000 1.000000 Wq = 4.974359

0.292308

10

Average waiting time in the queue

9 8

θ=0.1

7

θ=0.3 θ=0.5

6

θ=0.7

5

θ=0.9

4 3 2 1 0

0.1

0.2

0.3

0.4

0.5

ρ

0.6

0.7

0.8

0.9

1

Fig. 1. Effect of q on Wq for different values of h.

 L0 ¼

0:20 0:10 0:10 0:10

 ;

 L1 ¼

0:30 0:40 0:50 0:30

 ;

with l = 0.745455. This figure shows that the average waiting time in the queue increases slowly when the traffic load q increases except for h = 0.1. The average waiting time for h = 0.1 is longer than for other values of h and it is slowly decreasing

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S.K. Samanta, Z.G. Zhang / Applied Mathematical Modelling 36 (2012) 5964–5975

as q increasing. This is due to the longer mean vacation time of the server. Further, it is observed that for low traffic, Wq decreases with increasing value of h but for high traffic closer to one, they converge to the same value as the server is always busy. In Fig. 2, we analyze the effect of the vacation on average queue length (Lq) in the case of LAS-DA system for various interarrival time distributions with the same mean 5 and different variances. The interarrival time distributions are taken as geometric (k = 0.2), deterministic (a5 = 1) and arbitrary (a2 = 0.2, a3 = 0.7, a25 = 0.1), where the variances of these distributions are 20, 0, and 44.6, respectively. The following matrices L0 and L1 of order 2 of the service process (D-MSP) along with the vector f ¼ ½0:3 0:7 are used to obtain this figure.

5 4.5 Deterministic Arbitrary Geometric

Average queue length

4 3.5 3 2.5 2 1.5 1 0.5 0

0.1

0.2

0.3

0.4

0.5

θ

0.6

0.7

0.8

0.9

1

0.9

1

Fig. 2. Effect of h on Lq for various interarrival time distributions.

Average waiting time in the queue

20

Corr=0.114020 Corr=0.003385 Corr=0.000000

15

10

5

0 0.1

0.2

0.3

0.4

0.5

θ

0.6

0.7

0.8

Fig. 3. Effect of h on Wq for various values of coefficients of correlation.

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S.K. Samanta, Z.G. Zhang / Applied Mathematical Modelling 36 (2012) 5964–5975

 L0 ¼

0:34 0:25 0:30 0:45

 ;

 L1 ¼

0:25 0:16 0:10 0:15

 ;

with l = 0.329012. It can be seen that the average queue length in the case of arbitrary interarrival time distribution with higher variance is larger than the one with lower variances. Again deterministic distribution with zero variance yields the lowest average queue length. We further observe that for all distributions considered here, Lq decreases as h increases. This is due to the fact that mean vacation time decreases and the server is available with the shorter break. When the value of h is closer to one (smaller mean vacation time of the server), as expected, the average queue length is lesser. The effect of the vacation on the average waiting time in the queue (Wq) for various coefficients of correlation (Corr) with lag-2 is shown in Fig. 3 This figure displays the results for the LAS-DA system, where interarrival time distribution is taken as deterministic with a3 = 1. Using the formula given in Samanta et al. [4], various coefficients of correlation between interdeparture intervals are calculated. Three types of service time distributions are assumed, viz., geometric having correlation (Corr) equal to 0.0000 along with f ¼ 1, and the following two D-MSP representations along with the same vector f ¼ ½0:8 0:2.

L0 ¼



 L0 ¼

0:40

0:00

0:00

0:68

 ;

L1 ¼

0:4197 0:1060 0:1340 0:3210

 ;



0:59 0:01 0:02 0:30 

L1 ¼

 ;

0:4610 0:0133 0:0073 0:5377

 ;

with Corr equal to 0.114020 and 0.003385, respectively. They all have equal mean service rate l = 0.506666. Fig. 3 shows that, for fixed h, the average waiting time in the queue is longer when the D-MSP is highly correlated. Further, it can be seen that for fixed value of coefficient of correlation the average waiting time in the queue decreases as h increases. This is due to the fact that mean vacation time decreases and the server is available with the shorter break. When the value of h is closer to one (smaller mean vacation time of the server), as expected, the average waiting time in the queue is shorter. Besides other revelations, this study also reveals that not only interarrival times and vacation duration play an important role in queueing processes, but also the correlation among services plays a major role. 5. Conclusion In this paper, we have carried out an analysis of a discrete-time GI/D-MSP/1/1 queue with multiple vacations and exhaustive service for the late arrival system with delayed access, and the early arrival system. The new service process starts with the initial phase distribution independent of the path followed by the previous service process when the server returns from a vacation and finds at least one waiting customer. With the help of matrix-geometric method and the classical argument based on Markov renewal theory approach, we obtain the distributions of system-length at prearrival and arbitrary epochs, and waiting time for an arrival customer. In addition, computational experiences with a variety of numerical results are discussed to display the effect of the system parameters on the performance measures. The analytical approach used here would be interesting to analyze discrete-time GI/D-MSP/1 queue with single vacation and other vacation policies, and are left for future investigations. Acknowledgments A part of this work was done while the first author was a post doctoral fellow at LIA/CERI, University of Avignon, France and was supported by Dr. Rachid El-Azouzi’s ANR Winem Project grant. The second author wishes to thank the support from NSERC Grant 316 RGPIN197319 of Canada. References [1] A.S. Alfa, J. Xue, Q. Ye, Perturbation theory for the asymptotic decay rates in the queues with Markovian arrival process and/or Markovian service process, Queueing Systems 36 (4) (2000) 287–301. [2] P.P. Bocharov, C. D’Apice, A.V. Pechinkin, S. Salerno, The stationary characteristics of the G/MSP/1/r queueing system, Automation and Remote Control 64 (2) (2003) 127–142. [3] U.C. Gupta, A.D. Banik, Complete analysis of finite and infinite buffer GI/MSP/1 queue – a computational approach, Operations Research Letters 35 (2) (2007) 273–280. [4] S.K. Samanta, U.C. Gupta, M.L. Chaudhry, Analysis of stationary discrete-time GI/D-MSP/1 queue with finite and infinite buffers, 4OR: A Quarterly Journal of Operations Research 7 (2009) 337–361. [5] Y.C. Wang, J.H. Chou, S.Y. Wang, Loss pattern of DBMAP/DMSP/1/K queue and its application in wireless local communications, Applied Mathematical Modelling 35 (2011) 1782–1797. [6] A.S. Alfa, Markov chain representations of discrete distributions applied to queueing models, Computers and Operations Research 31 (2004) 2365– 2385. [7] A. Horváth, G. Horváth, M. Telek, A joint moments based analysis of networks of MAP/MAP/1 queues, Performance Evaluation 67 (9) (2010) 759–778. [8] A.S. Alfa, Vacation models in discrete time, Queueing Systems 44 (2003) 5–30. [9] H. Takagi, Queueing analysis – a foundation of performance evaluation, Discrete-time Systems, vol. 3, North-Holland, Amsterdam, 1993. [10] N. Tian, Z.G. Zhang, Vacation Queueing Models: Theory and Applications, Springer-Verlag, New York, 2006.

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