Analysis of Single Server Queue with Modified Vacation...

Published online: 7 May 2019

Methodology and Computing in Applied Probability (2020) 22: 511–553https://doi.org/10.1007/s11009-019-09713-9

Analysis of Single Server Queue with ModifiedVacation Policy

Priyanka Kalita1 ·Gautam Choudhury1 ·Dharmaraja Selvamuthu2

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AbstractThis article deals with single server queue with modified vacation policy. The modifiedvacation policy captures the operation of a close down period, type 1 vacation period, type2 vacation period, a start-up period and a dormant period. Here, type 1 vacations takes ashort period of random duration and type 2 vacation take a long period of random duration.Explicit expressions have been obtained for steady state queue size distribution at servicecompletion point and steady state system size probabilities. The Laplace-Stieltjes transformof waiting time and its corresponding mean value have been obtained for the system. Finally,some numerical examples have been provided and use the parabolic method to search theoptimum value of the control parameter p.

Keywords M/G/1 queue · Modified vacation policy · Embedded Markov chain ·Markov regenerative process

Mathematics Subject Classification (2010) 60K25 · 90B22 · 60K20

1 Introduction

Queueing models with server vacations have been efficiently studied by many researchersin the last two decades and successfully applied in the various pratical problems such asmodern telecommunication systems, computer communication networks, production, man-ufacturing systems, etc. An excellent survey on the vacation queueing models have beendocumented in Doshi (1986, 1990), Takagi (1991), Tian and Zhang (2006) and others. Thereare two basic vacation queueing models namely, multiple vacation queueing model and sin-gle vacation queueing model. In multiple vacation queueing models, the server keeps ontaking sequential vacations until it finds some customers waiting in a queue at a vacationcompletion epoch; However, in single vacation queueing models, the server takes exactly

� Priyanka [email protected]

1 Mathematical Sciences Division, Institute of Advanced Study in Science and Technology,Guwahati 781035, Assam, India

2 Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

Received: 30 August 2017 /Accepted: 3 April 2019 /Springer Science+Business Media, LLC, part of Springer Nature 2019

http://crossmark.crossref.org/dialog/?doi=10.1007/s11009-019-09713-9&domain=pdfmailto: [email protected]

one vacation between two sequential busy periods. These two types of vacation models werefirst introduced by Levy and Yechiali (1975).

In this paper, we consider an M/G/1 queueing system with Modified vacation policy(MVP). The MVP captures the operation of a close down period, type 1 vacation period, atype 2 vacation period, a start-up period and a dormant period. Type 1 vacations takes a shortperiod of random duration and type 2 vacation take a long period of random duration.Whenthe server finishes the service to all arriving customers and no customer is holding in aqueue, the server goes for a close down period of a fixed duration. If any customer arrivesduring the close down period, the system resumes service immediately at that close downperiod completion epoch. On the other hand, if no arrival occurs during the close downperiod, the server leaves for a initial type 1 vacation. On return from the initial type 1vacation if the server finds no customer holding in a queue, it either remains dormant inthe system with probability p or goes for the second type 1 vacation with probability p̄.This procedure will be continued until either the server finds at least one customer at atype 1 vacation completion epoch or dormant period completion epoch or it has taken amaximum M number of type 1 vacations. After completing Mth type 1 vacation if thesystem still empty, the server either goes for a type 2 vacation with probability q or remainsdormant in the system with probability q̄. If an arrival occurs during the type 2 vacationor dormant period, it immediately reactivates at that type 2 vacation completion epoch ordormant period completion epoch and perform a start-up time before starting its service. Assoon as the server completes start-up period, it starts serving the holding customers.

There have been several models evolved in recent years for vacation queueing systemswith a close down period. The notion of a close down period was first introduced by Takagi(1991). Niu et al. (1998, 2003) and Niu and Takahashi (1999) investigated the performancemeasures of the switched virtual channel connection (SVCC) by a close down period, whichis corresponding to an inactive timer during which the SVCC resource is reserved to antic-ipate more customers (packets) from the same internet protocol flow. Sakai et al. (1998)studied the steady state queue size distribution and waiting time distribution with singlevacation, start-up and close down period for an M/G/1/k type queueing system and theirapplication to SVCC based ATM networks. Recently, Krishnakumar et al. (2015) investi-gated the performance analysis of an M/M/1 queue with maintanence state and close downperiod and its application to IEEE802.16e system.

Many researchers have been developed several models involving multiple vacations invacation queueing systems but a few models have been developed with two types of vaca-tions. The first study of two types of vacation queueing model was done by Zhang et al.(1997). They studied the optimal two threshold policies for an M/G/1 queueing systemwith two types of vacations. Later, Ke (2003) generalized this model for an M/G/1 queue-ing system with server start-up period and two types of vacations. He obtained two keyProbability generating functions (PGFs) for number of customers arrive during the two typesof vacations and some important system performance measures.

In recent years, there has been an increased interest in vacation queueing systems with adormant period. The concept of a dormant period was first introduced by Heyman (1968),where he assumed that the dormant period is a period in which the server is waiting for anumber of customers. Later, Choudhury (2002a) studied the dormant period, which arisesbetween vacation period and busy period. Moreover, Ke et al. (2010d) investigated sucha vacation queueing system with dormant period and its application to production andinventory systems.

Various queueing systems containing mechanical parts needs a start-up period to servethe customers. The server start-up period corresponding to the preparatory work of the

Methodology and Computing in Applied Probability (2020) 22: 511–553512

server begins before the service period begins. Examining queueing models which com-bine the N-policy with a start-up period, Baker (1973) first proposed the N-policy M/M/1queueing system with exponential start-up period. Borthakur et al. (1987) generalizedBaker’s model to the general start-up period and obtained the explicit expressions for steadystate probabilities, as well as system performance measures under the N-policy. Later, Minh(1988), Medhi and Templetion (1992), Takagi (1993), Lee and Park (1997) and Artalejoet al. (2005) investigated these type of queueing models with start-up period.

In the literature, the steady state performance measures are studied for various exhaus-tive service queueing systems with two types of vacations and server’s start-up /close downperiods. However, no work has been done in a queueing model taking into account a max-imum number of M type 1 vacations of short period of random duration, a type 2 vacationof long period of random duration, a dormant period, a start-up and a fixed length of closedown period together. This motivates us to study an M/G/1 queueing system in the formof modified vacation policy in this paper.

The rest of the paper is arranged as follows: The mathematical model is described inSection 2. In Section 3, the queue size distribution at service completion epoch and steadystate system size probabilities using embedded Markov chain (EMC) and Markov regen-erative process (MRGP) are derived. Some essential system performance measures of thismodel are obtained in Section 4. Some important particular cases of this model are demon-strated in Section 5. Some significant numerical results are presented in Section 6. Finally, inSection 7, the practical application of the queueing model under consideration is presented.Using the parabolic method, a cost optimization problem is considered.

2 Model Description

In this section, we consider an M/G/1 queue with modified vacation policy (MVP). TheMVP captures the operation of a close down period, a maximum number of M type 1 vaca-tion period, a type 2 vacation period, a start-up period and a dormant period. The detaileddescription of the model is stated below:

Customers arrive in the system according to a poisson process with rate λ > 0. The ser-vice time provided by a single server is an independently and identically distributed (i.i.d.)random variable (r.v) S with cumulative distribution function (CDF) FS(x), probability den-sity function (PDF) fS(x), Laplace stieltjes transform (LST) S∗(θ) = E(e−θS) and finitemoments E(Sr); r = 1, 2 . . .. Arriving customers in the system are served according tofirst-come first-served (FCFS) discipline. Once the server finishes the service to all cus-tomers, i.e., at the end of each service period, the server enters into a close down periodof a fixed duration � (a time out period of � at the end of a busy period). If any customerarrives during the close down period, the system resumes service (at that close down periodcompletion epoch) so that the busy period starts again, which is shown in Fig. 1. On theother hand, if no arrival occurs at the end of close down period, the server enters into atype 1 vacation period of short period of random duration ϑ1 (say). The duration of type1 vacation period ϑ1 is i.i.d. r.v with CDF Fϑ1(x), the corresponding PDF fϑ1(x), LSTϑ∗1 (θ) = E(e−θϑ1) and finite moments E(ϑr1 ); r = 1, 2 . . .. On return from the initial type1 vacation if the server finds no customer waiting in the queue, it either remains dormantin the system with probability p or goes for the second type 1 vacation with probabilityp̄. This procedure will be continued until either the server finds at least one customer ata type 1 vacation completion epoch or dormant period completion epoch or it has taken amaximum M number of type 1 vacations. In case of arrivals occured during a maximum M

Methodology and Computing in Applied Probability (2020) 22: 511–553 513

ClosedownBusy Busy

Idle period

Customerarrival

Regeneration Cycle

Fig. 1 The diagram of M/G/1 queueing system with modified vacation policy during type 1 cycle

number of type 1 vacations or a dormant period, the server immediately reactivates at thattype 1 vacation completion epoch or dormant period completion epoch and perform a start-up time with random duration U (say) before starting its service. The duration of start-upU is an i.i.d. r.v with CDF FU(x), the corresponding PDF fU(x), LST U∗(θ) = E(e−θU )and finite moments E(Ur); r = 1, 2 . . .. As soon as the server finishes start-up period, itstarts serving the holding customers and it is shown in Fig. 2. In case of no arrival occuredby the end of the Mth type 1 vacation, the server either goes for a type 2 vacation of longperiod of random duration ϑ2 (say) with probability q or remains dormant in the systemwith probability q̄. The duration of type 2 vacation ϑ2 is an i.i.d. r.v distributed with CDFFϑ2(x), the PDF fϑ2(x), LST ϑ

∗2 (θ) = E(e−θϑ2) and finite moments E(ϑr2 ); r = 1, 2 . . ..

If an arrival occurs during the type 2 vacation or dormant period, it immediately reactivatesat that type 2 vacation completion epoch or dormant period completion epoch and performa start-up time before starting its service. As soon as the server completes start-up period, itstarts serving the holding customers and it is shown in Fig. 3.

Figures 1, 2 and 3 show the diagram of M/G/1 queueing system with modified vacationpolicy during three different regeneration cycles namely, type 1 cycle, type 2 cycle and type3 cycle respectively. The regeneration cycle is defined as the elapsed time from the epochin which a close down period is enabled, until the epoch where the next close down periodis about to begin. Type 1 cycle represent that if any arrival occurs during the close downperiod, the server starts service immediately at that close down period completion epoch.On the other hand, type 2 cycle represent that if no arrival occurs during the close downperiod, the server goes for a initial type 1 vacation and upon returning from the initial type




1 vacation if the server finds no customer holding in a queue, it either remains dormant inthe system with probability p or goes for the second type 1 vacation with probability p̄.This procedure will be continued until either the server finds minimum one customer at atype 1 vacation completion epoch or a dormant period completion epoch or it has takena maximum M number of type 1 vacations. If any arrival occurs during the maximum Mnumber of type 1 vacations or a dormant period, the server immediately reactivates at thattype 1 vacation completion epoch or dormant period completion epoch and perform a start-up time before starting its service. As soon as the server stops start-up period, it startsserving the holding customers until the system becomes empty. Type 3 cycle represent thatif no arrival occurs by the end of the Mth type 1 vacation, the server either goes for a type 2vacation with probability q or remains dormant in the system with probability q̄ and if anyarrival occurs during this type 2 vacation or dormant period, it immediately reactivates atthat type 2 vacation completion epoch or dormant period completion epoch and perform astart-up time before starting its service. As soon as the server stops start-up period, it startsserving the holding customers until the system becomes empty.

Moreover, the system remains idle during a close down period, maximum M number oftype 1 vacations, type 2 vacation, start-up period and dormant period, i.e., in case of Fig. 1close down period is an idle period, in case of Fig. 2 close down period, maximum Mnumber of type 1 vacations, start-up period and dormant period together constitute an idleperiod and in case of Fig. 3 close down period, maximum M number of type 1 vacations,type 2 vacation, start-up period and dormant period together constitute an idle period.

Now, if we define ln, �1,n, �2,n and un as the probability that n individual customersarrive during the service time S, type 1 vacation time ϑ1, type 2 vacation time ϑ2 and start-uptime U , then for n = 1, 2 . . ., we have

ln =∫ ∞0

e−λx[

(λx)n

n!]

dFS(x), �1,n =∫ ∞0

e−λx[

(λx)n

n!]

dFϑ1(x),

�2,n =∫ ∞0

e−λx[

(λx)n

n!]

dFϑ2(x), un =∫ ∞0

e−λx[

(λx)n

n!]

dFU(x).

Again, we consider ϕ = ∫ �0 λe−λxdx is the probability that atleast one arrival occurswithin the � interval. Further, it is presumed that the arrival process, service time r.v, type 1vacation time r.v, type 2 vacation time r.v, close down time r.v and start-up time r.v all aremutually independent of each other.


3 Model Analysis

3.1 Queue Size Distribution at Service Completion Epoch

In this subsection, we derived the PGF of queue size distribution at service completionepoch. Let tm denote the service completion epoch of m-th customer. Then, the sequenceΞm = Z(t+m) (where Z(tm) represents the number of customers in the system at time epochtm) construct a discrete time Markov chain (DTMC), which is an embedded Markov chainin the underlying queueing system. The sequence {Ξm, m ≥ 0} is a homogeneous Markovchain and it is bacause of the transition

Ξm+1 ={

Ξm − 1 + Lm+1 while Ξm > 0Am+1 + Lm+1 − 1 while Ξm = 0 ,

where Lm represents the number of arrival occurs during the service time of m-th customerand Am represents the number of arrival occurs during the m-th idle period.

It is clear that {Ξm,m = 0, 1, . . .} is an irreducible and aperiodic DTMC. Therefore, theone- step transition probabilities of Ξm are given by

Ph,n =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

(1 − ϕ)∑n+1c=1 ∑n+1−cκ=0[

(1−(p̄�1,0)M)1−p̄�1,0 �1,c

+p�1,0(1−(p̄�1,0)M−1

1−p̄�1,0 +(p̄�1,0)

M−1p

)(q

�2,c1−�2,0 + q̄δc,1)

]uκ ln−c+1−κ

+ϕ∑n+1c=1 δc,1ln−c+1, if h = 0, n = 0, 1, . . . ,ln−h+1 if h ≥ 1, n = h − 1, h, h + 1, . . . ,0 if h ≥ 1, n = 0, 1, . . . , h − 2,

(1)

where

δm,n ={1, if m = n0, if m �= n

donetes Kronecker’s delta function.Now, we presume that the Markov chain {Ξm, m = 0, 1, . . .} is positive recurrent to

ensure that the following limiting probabilities

πn = limm→∞ Pr [Ξm = n], n = 0, 1, . . .

exist. Using Eq. 1, we can write the steady state equation as

πn = π0(1 − ϕ)n+1∑c=1

n+1−c∑κ=0

[(1 − (p̄�1,0)M)

1 − p̄�1,0 �1,c + p�1,0(1 − (p̄�1,0)M−1

1 − p̄�1,0

+ (p̄�1,0)M−1

p

)(q

�2,c

1 − �2,0 + q̄δc,1)]

uκ ln−c+1−κ + ϕn+1∑c=1

δc,1ln−c+1

+n+1∑c=1

πcln−c+1, n = 0, 1, . . . . (2)

Now, the probability generating functions for | z | ≤ 1 is defined as:PL(z) =

∑∞n=0z

nln, P�(z) =∑∞

n=0znπn.

Clearly, PL(z) = S∗(λ(1 − z)) and ddzPL(z)|z=1 = λE(S).


Now, multiplying both side of Eq. 2 by zn and then taking summation over n = 0, 1, . . .,we get on simplification

P�(z) =

⎧⎪⎨⎪⎩

π0[1− p̄ϑ∗1 (λ)(1−ϕz)−ϕz− (1−ϕ){(1− (p̄ϑ∗1 (λ))M)(PV1(z)−ϑ∗1 (λ))+(pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1) + (1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)(q(PV2 (z)−ϑ∗2 (λ))

1−ϑ∗2 (λ) + q̄z)}P(z)]PL(z)

⎫⎪⎬⎪⎭

(1 − p̄ϑ∗1 (λ))(PL(z) − z).

(3)

where

ϑ∗1 (λ) =∫ ∞0

e−λxdFϑ1(x) = �1,0, ϑ∗2 (λ) =∫ ∞0

e−λxdFϑ2(x) = �2,0,

P(z) =∫ ∞0

e−(λ−λz)dFU (x) = U∗(λ − λz), PV1(z) =∫ ∞0

e−(λ−λz)dFϑ1(x)

= ϑ∗1 (λ − λz) and PV2(z) =∫ ∞0

e−(λ−λz)dFϑ2(x) = ϑ∗2 (λ − λz).

Clearly, ddz

P(z)|z=1 = λE(U), ddzPV1(z)|z=1 = λE(ϑ1), ddzPV2(z)|z=1 = λE(ϑ2).Now, applying the normalizing condition P�(1) = 1, we get

π0 = (1 − λE(S))(1 − p̄ϑ∗1 (λ))⎧⎪⎪⎪⎨

⎪⎪⎪⎩

[(1 − ϕ)[(1 − (p̄ϑ∗1 (λ))M)λE(ϑ1) + (pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1)

+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)( qλE(ϑ2)1−ϑ∗2 (λ) + q̄)] + (1 − p̄ϑ∗1 (λ))(ϕ

+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(4)

Putting the value of π0 in Eq. 3, we finally obtain

P�(z) =

⎧⎪⎨⎪⎩

(1 − λE(S))[1 − p̄ϑ∗1 (λ)(1 − ϕz) − ϕz − (1 − ϕ){(1 − (p̄ϑ∗1 (λ))M)(PV1(z) − ϑ∗1 (λ)) + (pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1)+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)(

q(PV2 (z)−ϑ∗2 (λ))1−ϑ∗2 (λ) + q̄z)}P(z)]PL(z)

⎫⎪⎬⎪⎭

⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

,

(5)

which is the PGF of queue size distribution at service completion epoch.

Remark 1 It is noted from Eq. 4 that λE(S) < 1, which is a necessary and sufficientcondition for existance of a stable solution of our model. It is basically the stability conditionof our model. Since λE(S) < 1, therefore the equation PL(z)− z = 0 never vanishes insidethe region | z |≤ 1, which can be proved by using Rouche’s Theoram (see Takacs 1962)

Remark 2 It is essential to note that the stochastic decomposition property (see Fuhramnand Cooper 1985) can be exhibited for the underlying queueing system.


We can write the Eq. 5 as

P�(z) = Po(z)Pϒ(z), (6)

where Po(z) = (1−λE(S)(1−z)PL(z)(PL(z)−z) is the PGF of system size distribution of an ordinaryM/G/1 queue and

Pϒ(z) =

⎧⎪⎨⎪⎩

1 − p̄ϑ∗1 (λ)(1 − ϕz) − ϕz − (1 − ϕ){(1 − (p̄ϑ∗1 (λ))M)(PV1(z) − ϑ∗1 (λ))+(pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1) + (1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)(q(PV2 (z)−ϑ∗2 (λ))

1−ϑ∗2 (λ) + q̄z)}P(z)

⎫⎪⎬⎪⎭

⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

is the PGF of additional system size distribution when the system is idle.Thus, Eq. 6 assures that the stochastic decomposition property of Fuhrmann and Cooper

(1985) is also valid for the underlying queueing system under the steady state condition.

3.2 Steady State System Size Probabilities

To obtain the explicit expressions for steady state system size probabilities, we follow themethod of Markov regenerative process, which gives much quicker result than the morepopular method of supplementary variables techniques.

Let h ∈ {0, 1, . . .} denote the number of customers in the system (including the onebeing served if any) and s ∈ {0, 1, 2, 3, 4, 5, 6, . . . , M + 4} indicate whether the systemis in dormant period or close down period or busy period or type 2 vacation period orstart-up period or 1st type 1 vacation or 2nd type 1 vacation or . . . or Mth type 1 vaca-tion at time t. Thus, the state of the system can be described by a 2-tuple or ordered pair(h, s) ∈ {0, 1, . . .}×{0, 1, 2, 3, 4, 5, 6, . . . , M +4} ≡ Ω . Consider the system state process{ω(t), t ≥ 0}, where ω(t) denote the number of customers within the system at time t . Theprocess {ω(t), t ≥ 0} is not a continuous time Markov chain (CTMC) since the sojourn timein each state is not exponentially distributed. Moreover, ω(t) is not even a semi-Markovprocess since between two completion epochs (i.e., between two sequential regenerationepochs) system state can change due to customer arrivals in dormant period or close downperiod or type 2 vacation period or 1st type 1 vacation or 2nd type 1 vacation or . . . or Mthtype 1 vacation or start-up completion or service completion. Define {τn, n = 0, 1, . . .} asthe sequence of epochs at which either dormant period terminated or close down periodterminated or type 2 vacation terminated or 1st type 1 vacation completed or 2nd type 1vacation completed or . . . or Mth type 1 vacation terminated or start-up terminated or ser-vice terminated and let ζn ∈ Ω represents the state of the system at these epochs. Here, theelements of Ω are called regeneration epochs and they occur at service completion epoch,close down period completion epoch, 1st type 1 vacation completion epoch, 2nd type 1vacation completion epoch, . . ., Mth type 1 vacation completion epoch, dormant periodcompletion epoch, type 2 vacation period completion epoch and start-up period completionepoch. Hence, we can claim that {ζn, n = 0, 1, . . .} is a DTMC with one-step transitionprobability matrixK(∞). By observing stochastic process {ω(t), t ≥ 0} at these epochs, we


see that the sequence {ζn, τn} is an embedded Markov renewal sequence and {ω(t), t ≥ 0}is a MRGP (see Kulkarni 1995).

Our upcoming task is to establish the local kernel E(t) and global kernel K(t) of theMRGP. To simplify our analysis, we first draw the state transition diagram and it is shownin Fig. 4. In Fig. 4,

1. We have not considered the states (2, 0), (3, 0), . . . because when the server is in a dor-mant period, if at least one customer arrive during this dormant period, it immediatelygoes to the start-up period.

2. We have not considered the states (2, 1), (3, 1), . . . because when the server is in aclose down period, if atleast one arrival occurs during this fixed close down period, itimmediatly starts service (at the end of close down period).

Fig. 4 State transition diagram for M/G/1 system with modified vacation policy


3. We have not considered the state (0, 2) because when the server is in a busy state, thereis at least one customer for service and if no customer is holding in a queue, the systemmove to close down period.

4. We have not considered the state (0, 4) because the server will go to the start-up periodif at least one arrival occurs either during dormant period or type 2 vacation or 1st type1 vacation or 2nd type 1 vacation or . . . or Mth type 1 vacation.

In the state transition diagram, we apply an oval symbol with ordered pair (h, s) (as statedabove) to show the state and apply a directed line to show a state transition. There are twotypes of state transitions as follows:

– Transition with generally distributed sojourn time (corresponding to the service time,1st type 1 vacation time, 2nd type 1 vacation time, . . ., Mth type 1 vacation time, type 2vacation time and start-up time) and is denoted by FS(.), Fϑ1(.), Fϑ2(.) and FU(.) withdotted lines respectively.

– Transition with exponentially distributed sojourn time (corresponding to inter-arrivaltime) with transition rate σij (h, n), i ∈ {0, 1, 2, 3, 4, 5, 6, . . . , M + 4}, j ∈{0, 1, 2, 3, 4, 5, 6, . . . , M + 4}, h = 0, 1, . . . , n = 0, 1, . . . from state (h, i) to state(n, j) and is represented by the solid lines. The possible transitions and correspondingrates are given in the following table:

From To Rate

(0, i) (0, 0) σi0(0, 0) for i = 5, 6, . . . , M + 4(1, 0) (1, 4) σ04(1, 1)(h, i) (n, 4) σi4(h, n) for i = 3, 5, 6, . . . , M + 4; h = n = 1, 2, . . .(0,M + 4) (0, 3) σM+4 3(0, 0)(h, 4) (n, 2) σ42(h, n) for h = n = 1, 2, . . .(1, 2) (0, 1) σ21(1, 0)(h, 2) (n, 2) σ22(h, n) for h = 2, 3, . . . , n = h − 1(1, 1) (1, 2) σ12(1, 1)(0, 1) (0, 5) σ15(0, 0)(0, i) (0, j) σij (0, 0) for i = 5, 6, . . . , M + 3, j = i + 1

Now, to obtain the system state probabilities of MRGP, we find the closed form expressionsfor K(t) and E(t) as follows.

3.2.1 Closed Form Expression for K (t )

K(t) is a global kernel matrix. The interpretation for the elements of the global kernelmatrix K(t) is as follows: K(h,s);(n,s′)(t) denotes the probability that the system will bein state (n, s′), s′ ∈ {0, 1, 2, 3, 4, 5, 6, . . . , M + 4}, n = 0, 1, . . . at the time of the nextregeneration epoch which occur on or before time t , given that it was in state (h, s) just afterthe previous regeneration epoch. The evolution of MRGP between two sequential Markovregeneration epochs can be explained through a CTMC infinitesimal generator matrix whichinvolves only exponential transitions. Hence, by removing the general transitions FS(.),Fϑ1(.), Fϑ2(.) and FU(.) in Fig. 4 we can construct the CTMC infinitesimal generator matrix


with the ordered pair {(0,0), (0,1), (0,3), (0,5), . . ., (0,M + 4), (1,0), (1,1), . . ., (1,M + 4),(2,2), . . ., (2,M + 4), . . . } as follows:Q =

(0, 0) (0, 1) (0.3) (0, 5) · · · (0,M + 4) (1, 0) (1, 1) · · · (1,M + 4) (2, 2) · · · (2,M + 4) · · ·⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(0, 0) q00 0 0 0 0 0 λ 0 0 0 0 0 0 · · ·

(0, 1) 0 q01 0 σ1 0 0 0 λ 0 0 0 0 0 · · ·

(0, 3) 0 0 q03 0 0 0 0 0 0 0 0 0 0 · · ·(0, 5) σ 2 0 0 q05 · · · 0 0 0 · · · 0 0 0 0 · · ·

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. · · ·(0,M + 4) σ 3 0 σ 4 0 0 q0M+4 0 0 0 λ 0 0 0 · · ·

(1, 0) 0 0 0 0 0 0 q10 0 · · · 0 0 0 0 · · ·

(1, 1) 0 0 0 0 0 0 0 q11 · · · 0 0 0 0 · · ·

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. · · ·

(1,M + 4) 0 0 0 0 0 0 0 0 · · · q1M+4 0 0 λ · · ·

(2, 2) 0 0 0 0 0 0 0 0 · · · 0 q22 0 0 · · ·...

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. · · ·(2,M + 4) 0 0 0 0 0 0 0 0 0 0 0 · · · q2M+4 · · ·

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,

where q00 = −λ, σ 1 = σ15(0, 0), q01 = −(σ 1 + λ), q03 = −λ, σ 2 = σ50(0, 0), q05 =−(σ 2 + σ56(0, 0) + λ), σ 3 = σM+4 0(0, 0), σ 4 = σM+4 3(0, 0), q0M+4 = −(σ 3 + σ 4 + λ),q10 = −σ04(1, 1), q11 = −σ12(1, 1), q1M+4 = −(σM+4 4(1, 1)+ λ), q22 = −(σ22(2, 1)+ λ),q2M+4 = −(σM+4 4(2, 2) + λ).

Let N(t) ∈ {0, 1, . . .} represents the number of customers in the system during timeinterval (0, t] and D(t) ∈ {0, 1, 2, 3, 4, 5, 6, . . . , M + 4} be the state of the system. Thenwe define

Pij (h; n, t) = P {N(t) = n, D(t) = j |N(0) = h,D(0) = i}; n = h = 0, 1, . . . . (7)Pij (h; n, t) is the probability that the subordinate CTMC will be in state (n, j) during timet given that it was in state (h, i) initially. Let P(t) be the corresponding probability matrix.

Then

d

dtP(t) = P(t)Q, pii (h; h, 0) = 1, i ∈ {0, 1, 2, 3, 4, 5, 6, . . . , M + 4}. (8)

By solving Eq. 8, we obtain pij (h; n, t) and from the properties of transition probabili-ties, we get σij (h, n) = lim

t→0pij (h;n,t)

t< ∞, ∀ i ∈ {0, 1, 2, 3, 4, 5, 6, . . . , M + 4}, j ∈

{0, 1, 2, 3, 4, 5, 6, . . . , M + 4}, h = 0, 1, . . . , n = 0, 1, . . ..


Now, according to the defination of global kernel K(t) in Kulkarni (1995), we want tofind

K(h,s);(n,s′)(t)= Pr {ζ1=(n, s′), τ1 ≤ t | ζ0=(h, s)}, (s, s′) ∈ {0, 1, 2, 3, 4, 5, . . . , M +4},(9)

where τ1 is the interval between two sequential regeneration epochs.Now, we define the elements of global kernel for each transition as follows.

Transition from service period to close down period: Here, we presume that the cus-tomer service is the transition between two sequential regeneration epochs, i.e., τ1 isdistributed with CDF FS(.). When the server terminates the service to all customers andno customer is waiting in a queue, i.e., the leaving customer is the final customer (i.e.,s = 2, h = 1) then the server goes to the close down period (i.e., s′ = 1, n = 0).Therefore, we have

K(1,2);(0,1)(t) =∫ t0p21(1; 0, x)dFS(x). (10)

Transition from service period to service period: Here, we presume the general tran-sition of the system’s service period, i.e., τ1 is distributed with CDF FS(.). Clearly, theservice period is available only when at least one customer is holding in a queue. If theservice termination of each customer does not empty the queue, the system stays in theservice period (i.e., s = 2, s′ = 2). Hence, we have

K(h,2);(n,2)(t) =∫ t0p22(h; n, x)dFS(x), n = 1, 2, . . . , h = 1, 2, . . . . (11)

Transition from close down period to service period: Here, we presume the caseof the server’s close down period, i.e., after completion of service to all customers andno customer is holding in the queue, the server goes to the close down period. If anarrival occurs during the close down period, it immediately goes to a service period (i.e.,s = 1, s′ = 2) at that close down period completion epoch. Thus, we have

K(1,1);(1,2)(t) = p12(1; 1, t). (12)Transition from close down period to 1st type 1 vacation period:We suppose the caseof the server’s close down period. If no arrival occurs during the close down period, itimmediately goes to the 1st type 1 vacation ( i.e., s = 1, s′ = 5). Therefore, we have

K(0,1);(0,5)(t) = p15(0; 0, t). (13)Transition from ηth type 1 vacation period to dormant period, where η =2, . . . , M − 1: Here, we consider server’s ηth type 1 vacation is the transition betweentwo sequential regeneration epochs, i.e., τ1 is distributed with CDF Fϑ1(.). If no arrivaloccurs during the close down period, it immediately goes to the 1st type 1 vacation. Ifno arrival occurs during the 1st type 1 vacation period, the server can go for a dormantperiod (i.e., s = 5, s′ = 0) with probability p and this procedure will be continued untileither the server finds minimum one customer at a type 1 vacation completion epochor dormant period completion epoch or at the end of (M − 1)th type 1 vacation (i.e.,s = 5, 6, . . . , M + 3, s′ = 0). Hence, we have

K(0,s);(0,0)(t) =∫ t0ps0(0; 0, x)dFϑ1(x), s = 5, 6, . . . , M + 3. (14)

Transition from (η − 1)th type 1 vacation period to ηth type 1 vacation period,where η = 2, . . . , M: Here, we presume the case of the server’s type 1 vacation period,


i.e., after completion of 1st type 1 vacation if no customer is found holding in a queue,the server can go for the 2nd type 1 vacation with probability p̄ and this procedure willbe continued until either the server finds minimum one customer at a type 1 vacationcompletion epoch or it has taken maximum M number of type 1 vacations (i.e., s =5, 6, . . . , M + 3, s′ = s + 1). In that case, we have

K(0,s);(0,s′)(t) = ps s′(0; 0, t), s = 5, 6, . . . , M + 3, s′ = s + 1. (15)Transition from Mth type 1 vacation period to type 2 vacation period: We presumethe server’s Mth type 1 vacation is the transition between two sequential regenerationepochs, i.e., τ1 is distributed with CDF Fϑ1(.). After completing the Mth type 1 vacationif the server still empty, it can go for the type 2 vacation with probability q (i.e., s =M + 4, s′ = 3). In that case, we have

K(0,M+4);(0,3)(t) =∫ t0pM+4 3(0; 0, x)dFϑ1(x). (16)

Transition from Mth type 1 vacation period to dormant:We presume that the generaltransition of server Mth type 1 vacation, i.e., τ1 is distributed with CDF Fϑ1(.). Aftercompletion of Mth type 1 vacation if the server remains empty in the system, it can gofor the dormant period with probability q̄ (i.e., s = M + 4, s′ = 0). In that case, we have

K(0,M+4);(0,0)(t) =∫ t0pM+4 0(0; 0, x)dFϑ1(x). (17)

Transition from ηth type 1 vacation period to start-up period, η = 1, 2, . . . , M:Here, we presume the general transition of server ηth type 1 vacation, where η =1, 2, . . . , M , i.e., τ1 is distributed with CDF Fϑ1(.). If an arrival occurs during the ηthtype 1 vacation period, it directly goes to the start-up period (i.e., s = 5, 6, . . . , M +4, s′ = 4). Hence, we have

K(h,s);(n,4)(t) =∫ t0ps4(h; n, x)dFϑ1(x), n = h = 1, 2, . . . , s = 5, 6, . . . , M +4. (18)

Transition from type 2 vacation period to start-up period: We consider that the gen-eral transition of server type 2 vacation, i.e., τ1 is distributed with CDF Fϑ2(.). If anarrival occurs during the type 2 vacation period, it directly goes to a start-up period (i.e.,s = 3, s′ = 4). Thus, we have

K(h,3);(n,4)(t) =∫ t0p34(h; n, x)dFϑ2(x), n = h = 1, 2, . . . . (19)

Transition from dormant to start-up period: Here, we presume the case of dormantperiod. If an arrival occurs during the dormant period, it directly goes to the start-upperiod (i.e., s = 0, s′ = 4). Hence, we have

K(1,0);(1,4)(t) = p04(1; 1, t). (20)Transition from start-up period to service period: we presume the server’s start-up isthe transition between two sequential regeneration epochs, i.e., τ1 is distributed with CDFFU(.). When the server is in a start-up period, there is minimum one customer holdingfor service. Therefore, after completing the start-up period, the server directly goes for aservice period (i.e., s = 4, s′ = 2). Hence, we have

K(h,4);(n,2)(t) =∫ t0p42(h; n, x)dFU(x), n = h = 1, 2, . . . . (21)


In summary, the global kernel K(h,s);(n,s′)(t) of MRGP (see Eq. 9) is as follows:

K(h,s);(n,s′)(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∫ t0p21(1; 0, x)dFS(x),∫ t0p22(h; n, x)dFS(x), n = 1, 2, . . . , h = 1, 2, . . . ,p12(1; 1, t),p15(0; 0, t),∫ t0ps0(0; 0, x)dFϑ1(x), s = 5, 6, . . . , M + 3,ps s′(0; 0, t), s = 5, 6, . . . , M + 3, s′ = s + 1,∫ t0pM+4 3(0; 0, x)dFϑ1(x),∫ t0pM+4 0(0; 0, x)dFϑ1(x),∫ t0ps4(h; n, x)dFϑ1(x), n = h = 1, 2, . . . , s = 5, 6, . . . , M + 4,∫ t0p34(h; n, x)dFϑ2(x), n = h = 1, 2, . . . ,p04(1; 1, t),∫ t0p42(h; n, x)dFU(x), n = h = 1, 2, . . . .

3.2.2 Closed Form Expression for E (t )

The local kernel matrix E(t) describes the local behavior of the process between twosequential regeneration epochs. The description for the elements of local kernel matrix E(t)is as follows: E(h,s);(n,s′)(t) denotes the probability that the system will be in state (n, s′) attime t and the next regeneration epoch occurs after time t given that the system was in state(h, s). Hence, we can write the elements of E(t) as

E(h,s);(n,s′)(t) = Pr {ω(t) = (n, s′), τ1 ≥ t | ζ0 = (h, s))}, (22)where τ1 is the time period between two sequential regeneration epochs.

Following the similar procedure in obtaining the global kernel K(t), we find E(t) as

E(h,s);(n,s′)(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

∫ t0 p22(h; n, x)dFS(x), n = 1, 2, . . . , h = 1, 2, . . . ,p12(1; 1, t),∫ t0 ps4(h; n, x)dFϑ1(x), n = h = 1, 2, . . . , s = 5, 6, . . . , M + 4,∫ t0 p34(h; n, x)dFϑ2(x), n = h = 1, 2, . . . ,p04(1; 1, t),∫ t0 p42(h; n, x)dFU(x), n = h = 1, 2, . . . .0, otherwise.

3.2.3 System State Probabilities

To obtain the system state probabilities, we define a new variable αss′(h; n), which repre-sents the average amount of time the MRGP spends in state s′ with n customers betweentwo sequential regeneration epochs, given that it was in state s with h customers after lastregeneration:

αss′(h; n) =∫ ∞0

E(h,s);(n,s′)(t)dt .

It is easy to show that for s = 4, s′ = 2 and n = h = 1, 2, . . .α42(h; n) = ςn−h+1; n ≥ max(0, h − 1), (23)

where ςn =∫∞0 e

−λx[

(λx)n

n!](1 − FU(t))dt ; n = 0, 1, . . ..


For s = 3, s′ = 4 and h = n = 1, 2, . . .

α34(h; n) = λ∫ ∞0

e−λx[

(λx)n

n!]

(1 − Fϑ2(t))dt . (24)

Now, for s = 5, 6, . . . , M + 4, s′ = 4 and h = n = 1, 2, . . .

αs4(h; n) = (1 − (p̄ϑ∗1 (λ))

M)

1 − p̄ϑ∗1 (λ)[λ

∫ ∞0

e−λx[

(λx)n

n!]

(1 − Fϑ1(t))dt − (1 − ϑ∗1 (λ))]

+pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1

1 − p̄ϑ∗1 (λ)+ (p̄ϑ

∗1 (λ))

M−1

p

)− 1. (25)

For s = 1, s′ = 2 and h = n = 1, we have

α12(1; 1) = ϕλ. (26)

For s = 0, s′ = 4 and h = n = 1, we have

α04(1; 1) = 1λ. (27)

Lastly, for s = 2, s′ = 2, n = 1, 2, . . . and h = 1, 2, . . ., we haveα22(h; n) = gn−h+1; n ≥ max(0, h − 1), (28)

where gn =∫∞0 e

−λx[

(λx)n

n!](1 − FS(t))dt ; n = 0, 1, . . ..

Since, the system state is basically based on the value of h while s ∈{0, 1, 2, 3, 4, 5, 6, . . . , M + 4}, therefore, the system state probabilities are defined as

φ(n, s′) =∑∞

h=0 πhαss′(h; n)∑∞h=0 πhβh

, (29)

where πh’s are the solution of π = πK(∞) and∑∞h=0 πh = 1. Hereβh =

∑υ∈{0,1,2,3,4,5,6,...,M+4}αυs′(h; n); n = 0, 1, . . . and

∑∞h=0πhβh = (λ)

−1.

Since the arrival process is Poisson, therefore, by obeying Burke’s theorem (see Cooper1981, page 187–188) it is achieved that the steady state probabilities {φ(n, s′); (n, s′) ∈ Ω}are positive and exist under the similar condition of limiting probabilities {πn; n = 0, 1, . . .}of the Markov chain {Ξm,m = 0, 1, . . .}, i.e., iff λE(S) < 1.

First of all, we presume the case of stationary system size distribution when the server isin dormant period, i.e., s′ = 0, a probability argument with Eq. 29 gives rise to

φ(1, 0) = π0q̄(1 − ϕ)pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1

1 − p̄ϑ∗1 (λ)+ (p̄ϑ

∗1 (λ))

M−1

p

). (30)

Secondly, we consider the case of stationary system size distribution when the server stateis in close down period, i.e., s′ = 1, a probability argument with Eq. 29 gives rise to

φ(1, 1) = π0ϕ. (31)


Thirdly, we consider the case of stationary system size distribution when the server state isin busy, i.e., s′ = 2, a probability argument with Eq. 29 gives rise to

φ(n, 2) = λ[π0(1 − ϕ)n+1∑c=1

n+1−c∑κ=0

[(1 − (p̄ϑ∗1 (λ))M)

1 − p̄ϑ∗1 (λ)�1,c + pϑ∗1 (λ)

(1 − (p̄ϑ∗1 (λ))M−1

1 − p̄ϑ∗1 (λ)

+ (p̄ϑ∗1 (λ))

M−1

p

)(q

�2,c

1 − ϑ∗2 (λ)+ q̄δc,1

)]uκgn−c+1−κ

+ϕn+1∑c=1

δc,1gn−c+1 +n+1∑c=1

πcgn−c+1

], n = 0, 1, . . . . (32)

Fourthly, we consider the case of stationary system size distribution when the server state isin type 2 vacation period, i.e., s′ = 3, a probability argument with Eq. 29 gives rise to

φ(n, 3) = λqπ0(1 − ϕ)pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1

1 − p̄ϑ∗1 (λ)+ (p̄ϑ

∗1 (λ))

M−1

p

)1

1 − ϑ∗2 (λ)∫ ∞0

e−λx[

(λx)n

n!]

(1 − Fϑ2(t))dt, n = 0, 1, . . . . (33)

Fifthly, we consider the case of stationary system size distribution when the server state isin startup, i.e., s′ = 4, a probability argument with Eq. 29 gives rise to

φ(n, 4)=λπ0(1−ϕ)n+1∑c=1

[(1 − (p̄ϑ∗1 (λ))M)

1 − p̄ϑ∗1 (λ)�1,c

+ pϑ∗1 (λ)(1−(p̄ϑ∗1 (λ))M−1

1−p̄ϑ∗1 (λ)+ (p̄ϑ

∗1 (λ))

M−1

p

)(q

�2,c

1−ϑ∗2 (λ)+q̄δc,1

)]ςn−c+1,

n = 0, 1, . . . . (34)

Lastly, presume the case of stationary system size distribution when the server state is intype 1 vacation period, i.e., s′ = 5, 6, . . . , M + 4, a probability argument with Eq. 29 givesrise to

φ(n, s′)=π0(1−ϕ)[

(1−(p̄ϑ∗1 (λ))M)1−p̄ϑ∗1 (λ)

(λ

∫ ∞0

e−λx[

(λx)n

n!]

(1−Fϑ1(t))dt−(1−ϑ∗1 (λ)))

+ pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1

1 − p̄ϑ∗1 (λ)+ (p̄ϑ

∗1 (λ))

M−1

p

)− 1

], n = 0, 1, . . . . (35)


Now, we find the PGF of the above Eq. 32, 33, 34 and 35 which are stated in the followingtheorem.

Theorem 1 For λE(S) < 1, the PGF P2(z) of system size distribution when the system isbusy (i.e., s′ = 2) is found as

P2(z) =

⎧⎪⎨⎪⎩

(1 − λE(S))[(1 − p̄ϑ∗1 (λ))(ϕz − 1) + (1 − ϕ){(1 − (p̄ϑ∗1 (λ))M)(PV1(z) − ϑ∗1 (λ)) + (pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1) + (1 −

p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)(q(PV2 (z)−ϑ∗2 (λ))

1−ϑ∗2 (λ) + q̄z)}P(z)](1 − PL(z))

⎫⎪⎬⎪⎭

⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))](1 − z)

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(36)

The PGF P3(z) of system size distribution when the server state is in type 2 vacation period(i.e., s′ = 3) is found as

P3(z) =

{(1 − λE(S))(1 − ϕ) q1−ϑ∗2 (λ) (pϑ

∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1)

+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)(1 − PV2(z))

}

⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))](1 − z)

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(37)

The PGF P4(z) of system size distribution when the server state is in start-up period (i.e.,s′ = 4) is found as

P4(z) =

⎧⎪⎨⎪⎩

(1 − λE(S))[(1 − ϕ){(1 − (p̄ϑ∗1 (λ))M)(PV1(z) − ϑ∗1 (λ))+(pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1) + (1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)(q(PV2 (z)−ϑ∗2 (λ))

1−ϑ∗2 (λ) + q̄z)}(1 − P(z))]

⎫⎪⎬⎪⎭

⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))]z(1 − z)

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(38)


and the PGF Ps′(z) of system size distribution when the server state is in type 1 vacationperiod (i.e., s′ = 5, 6, . . . , M + 4) is found as

Ps′(z) =

{(1− λE(S))[(1− ϕ){(1− (p̄ϑ∗1 (λ))M)(ϑ∗1 (λ) − PV1(z)) + (pϑ∗1 (λ)(1−(p̄ϑ∗1 (λ))M−1) + (1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M) − (1 − p̄ϑ∗1 (λ))}]

}⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))](1 − z)

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(39)

Proof Multiplying Eq. 32, 33, 34 and 35 by zn in both sides and summing over n = 0, 1, . . .and then utilizing the established result of renewal theory∫ ∞

0e−θx(1 − F(x))dx = 1 − f

∗(θ)θ

,

where F(.) represents the CDF of a r.v and f ∗(.) its LST, we obtain the results Eqs. 36, 37,38 and 39 respectively.

Next, from the PGF of Theorem 1, we find the system state probabilities, where theserver can be in a dormant period (D), in a close down period (CL), in a busy period (S),in a type 2 vacation period (t2V ), in a start-up period (St), in a type 1 vacation (t1V ), or inan idle period (I ), which are summarized in Corollary 1.

Corollary 1 For λE(S) < 1, the probability that the server is in dormant period PD , theprobability that the server is in close down period PCL, the probability that the server isin busy period PS , the probability that the server is in type 2 vacation period Pt2V , theprobability that the server is in start-up period PSt , the probability that the server is in type1 vacation period Pt1V and in idle period PI are obtained as

PD =(1 − λE(S))(1 − ϕ)q̄(pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1)

+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(40)

PCL = (1 − λE(S))ϕ(1 − p̄ϑ∗1 (λ))⎧⎪⎪⎪⎨

⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(41)

PS = λE(S). (42)


Pt2V =

{(1 − λE(S))(1 − ϕ){pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1)

+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M } q1−ϑ∗2 (λ)λE(ϑ2)}

⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(43)

PSt = (1 − λE(S))(1 − ϕ)λE(U)(1 − p̄ϑ∗1 (λ))⎧⎪⎪⎪⎨

⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(44)

Pt1V = (1 − λE(S))(1 − ϕ)(1 − (p̄ϑ∗1 (λ))

M)λE(ϑ1)⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(45)

PI = 1 − λE(S). (46)

Proof From Eq. 36, 37, 38 and 39 by taking limit as z → 1 and after calculation, weobtain the results Eqs. 42, 43, 44 and 45, i.e., limz→1 P2(z) = PS , limz→1 P3(z) = Pt2V ,limz→1 P4(z) = PSt and limz→1 Ps′(z) = Pt1V . From Eq. refeq30 and 31, the resultsEqs. 40 and 41 are obtained. By adding Eqs. 40, 41, 43, 44 and 45 Eq. 46 is obtained.

Lastly, we find the PGF, mean and second moment of system size, which are given inCorollary 2.

Corollary 2 Let N denote the system size in steady state. Then under stability conditionλE(S) < 1, the corresponding PGF is given by

PN(z) =

⎧⎪⎨⎪⎩

(1−λE(S))[1− p̄ϑ∗1 (λ)(1−ϕz)−ϕz− (1−ϕ){(1− (p̄ϑ∗1 (λ))M)(PV1(z)-ϑ∗1 (λ)) + (pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1)+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)(

q(PV2 (z)−ϑ∗2 (λ))1−ϑ∗2 (λ) + q̄z)}P(z)]PL(z)

⎫⎪⎬⎪⎭

⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))](PL(z) − z)

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(47)


The mean of system size, E(N) is given by

E(N) = λE(S) + λ2E(S2)

2(1 − λE(S))

+

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(1 − ϕ){(1 − (p̄ϑ∗1 (λ))M)λ2E(ϑ21 ) + (pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1)+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)( qλ

2E(ϑ22 )

1−ϑ∗2 (λ) )} + 2(1 − ϕ){(1 − (p̄ϑ∗1 (λ))

M)

λE(ϑ1) + (pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1) + (1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)(

qλE(ϑ2)

1−ϑ∗2 (λ) + q̄)}λE(U) + (1 − p̄ϑ∗1 (λ))λ

2E(U2)(1 − ϕ)

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

⎧⎪⎪⎪⎨⎪⎪⎪⎩

2[(1 − ϕ)[(1 − (p̄ϑ∗1 (λ))M)λE(ϑ1) + (pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1)


+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(48)

and the second moment of system size, E(N2) is given by

E(N2)= λ2E(S2) + 2(1 − λE(S))

2(1 − λE(S)) +λ3E(S3)

3λ2E(S2)

+

{(1 − ϕ){(1 − (p̄ϑ∗1 (λ))M)λ2E(ϑ21 ) + (pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1)+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)( qλ

2E(ϑ22 )

1−ϑ∗2 (λ) )} + λ2E(U2)(1 − ϕ)

}

⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

+

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

(1−(p̄ϑ∗1 (λ))M){(1−ϕ)λ2E(ϑ1)E(U) + (1−ϕ)λ3E(ϑ31 )+3(1 − ϕ)λ3E(U)E(ϑ21 )+3(1−ϕ)λ3E(ϑ1)E(U2)} + (pϑ∗1 (λ)(1−(p̄ϑ∗1 (λ))M−1)+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)[(1 − ϕ)( qλE(ϑ2)1−ϑ∗2 (λ) + q̄)λE(U)+(1 − ϕ) q1−ϑ∗2 (λ)λ

3E(ϑ32 ) + 3(1 − ϕ)λ3 q1−ϑ∗2 (λ)E(U)E(ϑ22 )

+3(1 − ϕ)λ2E(U2)( qλE(ϑ2)1−ϑ∗2 (λ) + q̄)] + (1 − ϕ)λ3E(U3)

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(49)

Proof From Eq. 30, 31, 32, 33, 34 and 35, we get the probability of total system size insteady state as follows

ξn = δn,0(φ(1, 0) + φ(1, 1)) + (1 − δn,0)(φ(n, 4) + φ(n, 2)) + φ(n, 3) + φ(n, s′);for n = 0, 1, . . . , s′ = 5, 6, . . . , M + 4, (50)


where ξn = P {N = n}. Then multiplying Eq. 50 by zn and taking summation over n =0, 1, . . ., we obtain

PN(z) = φ(1, 0) + φ(1, 1) + z(P4(z) + P2(z)) + P3(z) + Ps′(z); for n = 0, 1, . . . ,s′ = 5, 6, . . . , M + 4, (51)

which represents the connection between the two probabilities and 5 PGFs namely,φ(1, 0), φ(1, 1), P2(z), P3(z), P4(z), Ps′(z) and PN(z). After simplifying Eq. 51, we obtainthe result Eq. 47, which is equal to the PGF of queue size distribution at service completionepoch .

The result Eq. 48 follows directly from Eq. 47 on differentiation with respect to z andassigning z = 1. Again, calculating the second differentiation of Eq. 47 with respect to zand assigning z = 1 and then adding the result with Eq. 48, after simplification we obtainthe formula (49).

Remark 3 It is necessary to note that, the result Eq. 47 can be rearranged for the underlyingqueueing model and for that we need to define some other general descriptors which arelisted below:

Let TSt denote the length of start-up period during a regeneration cycle. Then the averageresidence time say, E(TSt ), of the system in the start-up period during a regeneration cycleis stated by

E(TSt ) =

{(1−ϕ)E(U)[(1−(p̄ϑ∗1 (λ))M)(1−ϑ∗1 (λ))+(pϑ∗1 (λ)

(1−(p̄ϑ∗1 (λ))M−1)+(1−p̄ϑ∗1 (λ))(1−p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)]}

(1 − p̄ϑ∗1 (λ)).

(52)

Consider Tt2V as the length of type 2 vacation period during a regeneration cycle. Then theaverage residence time say, E(Tt2V ), of the system in the type 2 vacation period during aregeneration cycle is given by

E(Tt2V ) = (1 − ϕ)[(pϑ∗1 (λ) − (p̄ϑ∗1 (λ))M) + (1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M ]×qE(ϑ2)((1 − p̄ϑ∗1 (λ))(1 − ϑ∗2 (λ)))−1. (53)

Similarly, consider Tt1V as the length of type 1 vacation period during a regeneration cycle.Then the average residence time say, E(Tt1V ), of the system in the type 1 vacation periodduring a regeneration cycle is given by

E(Tt1V ) = (1 − ϕ)(1 − (p̄ϑ∗1 (λ))M)E(ϑ1)(1 − p̄ϑ∗1 (λ))−1. (54)Assume that TCL as the length of close down period during a regeneration cycle. Thenthe average residence time say, E(TCL), of the system in the close down period during aregeneration cycle is given by

E(TCL) = ϕλ−1. (55)Consider TD as the length of dormant period during a regeneration cycle. Then the averageresidence time say, E(TD), of the system in the dormant period during a regeneration cycleis given by

E(TD) = (1 − ϕ)[pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1) + (1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M ]×q̄((1 − p̄ϑ∗1 (λ))λ)−1. (56)


Finally, if TI as the length of an idle period during a regeneration cycle, then the averageresidence time say, E(TI ), of the system in the idle period during a regeneration cycle isgiven by

E(TI ) = E(TSt ) + E(Tt2V ) + E(Tt1V ) + E(TCL) + E(TD)

=

{(1−ϕ)[(1−(pϑ∗1 (λ))M)E(ϑ1)+(pϑ∗1 (λ)(1−(p̄ϑ∗1 (λ))M−1)+(1−p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)(

q̄λ

+ qE(ϑ2)1−ϑ∗2 (λ) )] + (1 − p̄ϑ∗1 (λ))(

ϕλ

+ E(U)(1 − ϕ))}

(1 − p̄ϑ∗1 (λ)).

(57)

Now, by utilizing the theory of regenerative process, we get the following conditional timeaverage probabilities

P̄St = P(server is in start-up period/server is idle) = E(TSt )E(TI )

=

{(1−ϕ)λE(U)[(1−(p̄ϑ∗1 (λ))M)(1−ϑ∗1 (λ))+(pϑ∗1 (λ)(1−(p̄ϑ∗1 (λ))M−1)+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)]

}⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(58)

P̄t2V = P(server is in type 2 vacation period/server is idle) = E(Tt2V )E(TI )

= (1−ϕ)[pϑ∗1 (λ)(1−(p̄ϑ∗1 (λ))M−1)+(1− p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M ]qλE(ϑ2)⎧⎪⎪⎪⎨

⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(59)

P̄t1V = P(server is in type 1 vacation period/server is idle) = E(Tt1V )E(TI )

= (1 − ϕ)(1 − (p̄ϑ∗1 (λ))

M)λE(ϑ1)⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(60)


P̄CL = P(server is in close down period/server is idle) = E(TCL)E(TI )

= ϕ(1 − p̄ϑ∗1 (λ))⎧⎪⎪⎪⎨

⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭(61)

and

P̄D = P(server is in dormant period/server is idle) = E(TD)E(TI )

= (1 − ϕ)(pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1) + (1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)q̄⎧⎪⎪⎪⎨

⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

.

(62)

Using Eqs. 58, 59, 60, 61 and 62, we can rearrange the Eq. 47 as

PN(z) =P̄D(1 − λE(S))(z − 1q̄ )P(z)PL(z)

(z − PL(z))+ P̄t1V (1 − λE(S))(PV1(z) − 1)P(z)PL(z)

λE(ϑ1)(z − PL(z)) +P̄St (1 − λE(S))P(z)PL(z)

(z − PL(z))λE(U)

+P̄t2V (1 − λE(S)) (PV2 (z)−ϑ

∗2 (λ))

1−ϑ∗2 (λ) P(z)PL(z)

λE(ϑ2)(z − PL(z)) +P̄CL(z − 1ϕ )(1 − λE(S))PL(z)

(z − PL(z)) .(63)

Now, by considering P̄D = 0, P̄St = 0, P̄t2V = 0, P̄CL = 0 and P̄t1V = 1 and P(z) = 1in Eq. 63, we get

PN(z) = (1 − λE(S))(PV1(z) − 1)PL(z)λE(ϑ1)(z − PL(z)) , (64)

which is the expression for the PGF of system size for an ordinary M/G/1 queue withmultiple vacations (see Takagi 1991, Equation (2.12c), page 122).

Similarly, by assigning P̄D = 1, q̄ = 1, P̄St = 0, P̄t2V = 0, P̄CL = 0 and P̄t1V = 0 andP(z) = 1 in Eq. 63, we get

PN(z) = (1 − λE(S))(z − 1)PL(z)(z − PL(z)) , (65)

which is the expression for the PGF of system size for an ordinary M/G/1 queue withoutvacation (see Takagi 1991, Equation (1.40), page 13).


4 System PerformanceMeasures

In this section, we derive two essential system performance measures namely, distributionof a number of customers served during the busy period and waiting time distribution.

4.1 Distribution of a Number of Customers Served During the Busy Period

Here, we derive the PGF and mean number of customers served during the busy period ofthis model under stability condition λE(S) < 1, which are given in the following theorem.

Theorem 2 Let ψ denote the number of customers served during the busy period. Then,under stability condition λE(S) < 1, the corresponding PGF Pψ(z) and mean E(ψ) aredetermined as

Pψ(z) = (1 − ϕ) (1 − (p̄ϑ∗1 (λ))

M)(PV1(P (z)) − ϑ∗1 (λ))1 − p̄ϑ∗1 (λ)

P(P (z))

+(1 − ϕ)pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1

1 − p̄ϑ∗1 (λ)+ (p̄ϑ

∗1 (λ))

M−1

p

)

(q(PV2(P (z)) − ϑ∗2 (λ))

1 − ϑ∗2 (λ)+ q̄P (z)

)P(P (z)) + ϕP (z), (66)

and

E(ψ) = 11 − λE(S)

[(1 − ϕ) (1 − (p̄ϑ

∗1 (λ))

M)(λE(ϑ1) + (1 − ϑ∗1 (λ))λE(U))1 − p̄ϑ∗1 (λ)

P(P (z)) + (1 − ϕ)pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1

1 − p̄ϑ∗1 (λ)+ (p̄ϑ

∗1 (λ))

M−1

p

)

(qλE(ϑ2)

1 − ϑ∗2 (λ)+ q̄ + λE(U)

)+ ϕ

], (67)

Proof The idle period will be terminated if and only if at least one customer arrived eitherduring close down period or 1st type 1 vacation or 2nd type 1 vacation ... or Mth type1 vacation or dormant period or type 2 vacation period or start up period. Therefore, theprobability that n customers arrived in the system at the idle period completion epoch willbe

an = (1 − ϕ)n+1∑c=1

[(1 − (p̄�1,0)M)

1 − p̄�1,0 �1,c + p�1,0(1 − (p̄�1,0)M−1

1 − p̄�1,0 +(p̄�1,0)

M−1

p

)

(q

�2,c

1 − �2,0 + q̄δc,1)]

un−c+1 + ϕδn,1, n = 1, 2, . . . . (68)


Now, if there are n customers in the system at the idle period completion epoch, thenutilizing the argument of Takagi 1991, we have

Pψ(z) =∞∑

n=1an(P (z))

n (69)

where P(z) follows the following function equation under stability

P(z) = zS∗(λ − λP (z)).Utilizing Eq. 68 in Eq. 69 we obtain the result Eq. 66. Then differentiating Eq. 66 withrespect to z and assigning z = 1, we obtain the result Eq. 67.

4.2 Waiting Time Distribution

In this subsection, we derive the waiting time distribution of our model. Let W be the r.vrepresenting the waiting time of a customer in the queue and W ∗(θ) be its correspondingLST.

Theorem 3 If λE(S) < 1, then the LST of waiting time distribution, W ∗(θ), the aver-age waiting time, E(W) and the average sojourn time, E(Ws), of the underlying queueingsystem are obtained as

W ∗(θ) =

⎧⎪⎪⎨⎪⎪⎩

(1 − λE(S))[1 − p̄ϑ∗1 (λ)(1 − ϕ(1 − θλ )) − ϕ(1 − θλ ) − (1 − ϕ){(1− (p̄ϑ∗1 (λ))M)(PV1(1− θλ )−ϑ∗1 (λ))+ (pϑ∗1 (λ)(1− (p̄ϑ∗1 (λ))M−1)+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)(

q(PV2 (1− θλ )−ϑ∗2 (λ))1−ϑ∗2 (λ) q̄(1 −

θλ))}P(1 − θλ )]

⎫⎪⎪⎬⎪⎪⎭

⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))](PL(1 − θλ ) − 1 + θλ )

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(70)

E(W) = λE(S2)

2(1 − λE(S))

+

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

λ[(1 − ϕ){(1 − (p̄ϑ∗1 (λ))M)E(ϑ21 ) + (pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1)

+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)( qE(ϑ22 )

1−ϑ∗2 (λ) )}+2(1 − ϕ){(1 − (p̄ϑ∗1 (λ))M)λE(ϑ1)(pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1)+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)( qλE(ϑ2)1−ϑ∗2 (λ) q̄)}E(U) + E(U

2)(1 − ϕ)]

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(71)


E(Ws)= λE(S2) + 2E(S) − 2λ(E(S))2

2(1 − λE(S))

+

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

λ[(1 − ϕ){(1 − (p̄ϑ∗1 (λ))M)E(ϑ21 ) + (pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1)

+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)( qE(ϑ22 )

1−ϑ∗2 (λ) )}+2(1 − ϕ){(1 − (p̄ϑ∗1 (λ))M)λE(ϑ1)(pϑ∗1 (λ)(1 − (p̄ϑ∗1 (λ))M−1)+(1 − p̄ϑ∗1 (λ))p̄M−1ϑ∗1 (λ)M)( qλE(ϑ2)1−ϑ∗2 (λ) q̄)}E(U) + E(U

2)(1 − ϕ)]

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

⎧⎪⎪⎪⎨⎪⎪⎪⎩



+λE(U)(1 − ϕ))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(72)

Proof As the customers are provided in FCFS discipline, the holding time of a customer inthe queue and its service time are independent. Therefore, we have

W ∗(θ) = P�(1 − θ

λ

)(S∗(θ))−1. (73)

By utilizing Eq. 5 in 73, we obtain the result Eq. 70. The result Eq. 71 is obtained directlyfrom Eq. 70 on differentiation with regard to θ and putting θ = 0. Adding E(S) with Eq. 71,we obtain Eq. 72, which represents the mean waiting time of a customer in the system, i.e.,the average sojourn time.

5 Some Particular Cases

Here, we discuss momentarily some particular cases of our model, which are consistentwith the existing literature.

Case 1: If q = 0, ϕ = 0, un = 1, then the expression (47) yields

PN(z)=0

⎧⎨⎩

(1 − λE(S))[(1 − (p̄ϑ∗1 (λ))M)(PV1(z) − 1)+ϑ∗1 (λ)(z − 1){(p̄ϑ∗1 (λ))M−1(1 − p̄ϑ∗1 (λ)) + p(1 −

(p̄ϑ∗1 (λ))M−1)}]PL(z)(z − 1)

⎫⎬⎭

⎧⎪⎪⎨⎪⎪⎩

[(1−(p̄ϑ∗1 (λ))M)λE(ϑ1) + ϑ∗1 (λ){(p̄ϑ∗1 (λ))M−1(1−(p̄ϑ∗1 (λ))) + p(1 −

(p̄ϑ∗1 (λ))M−1)}]

(z − PL(z))(z − 1)

⎫⎪⎪⎬⎪⎪⎭

,

(74)

which is the PGF of system size of an M/G/1 queueing system under randomized vaca-tion policy with atmost M vacations. Note that for M tends to ∞ and p = 0, the aboveEq. 74 is in accordance with the result obtained by Choudhury (2002b) (for single arrivalcase). Also, we note that the Eq. 74 is consistent with Ke et al. (2010d) for single arrivalcase.


Case 2: If q = 0, p = 0, M tends to ∞, then the expressions (47) yields

PN(z)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

PL(z){(1 − ϕ)ϑ∗1 (λ)(1 − λE(S))P(z)(1 − PV1(z))+(1 − ϑ∗1 (λ))(1 − λE(S))[1 − (ϕz+(1 − ϕ)PV1(z)P(z))]} × {(1 − ϕ)ϑ∗1 (λ)[λ(E(S) + E(U)-E(ϑ1))−λE(U)−1]+ (1−ϑ∗1 (λ))[(ϕ −1)λ(E(U)+E(V )−

E(S)) − 1]}

⎫⎪⎪⎪⎬⎪⎪⎪⎭

⎧⎪⎪⎪⎨⎪⎪⎪⎩

[(1 − ϑ∗1 (λ)){[ϕ + (1 − ϕ)λ(E(U) + E(V ))] + [(1 − ϕ)(1 − λE(S))]}+ϑ∗1 (λ)(1 − ϕ)[1 − λE(S) + λE(ϑ1)]

](PL(z) − z)

{[(1 − ϑ∗1 (λ))(ϕ −

1)λ(E(U)E(V )) − ϕ] − (1 − ϕ)ϑ∗1 (λ)E(V )}

⎫⎪⎪⎪⎬⎪⎪⎪⎭

,

(75)

which represent the PGF of system size of an M/G/1 queueing system with serverstart-up period, close down period and multiple vacations. Note that, the above Eq. 75is consistent with Krishnakumar et al. (2013) without considering Bernoulli vacationschedule.

6 Numerical Illustration

In this section, we present the numerical results of the effects of system parameters on thepreviously obtained key performance measures in the form of graphs. In the previous sec-tions, we have obtained the probability PD of a server is in dormant period, the probabilityPCL of a server is in close down period, the probability Pt2V of a server is in type 2 vacationperiod, the probability PSt of a server is in start-up period, the probability Pt1V of a serveris in type 1 vacation period and the mean system size E(N).

For the purpose of numerical illustration, we consider that the type 1 vacation time, type2 vacation time and start-up time follow exponential distributions with parameters ν1, ν2and γ , then we have the PDFs fϑ1(x) = ν1e−ν1x, x > 0, fϑ2(x) = ν2e−ν2x, x > 0 andfU(x) = γ e−γ x, x > 0, LSTs ϑ∗1 (θ) = ν1ν1+θ , ϑ∗2 (θ) =

ν2ν2+θ , U

∗(θ) = γγ+θ , means

E(ϑ1) = 1ν1 , E(ϑ2) = 1ν2 , E(U) = 1γ and second moments E(ϑ21 ) = 2ν21 , E(ϑ22 ) = 2ν22 and

E(U2) = 2γ 2

respectively. Again, we presume that the service time conforms to:

– 2-stage Erlang distribution (denoted by 2-stage Erlang) with parameter μ, then we havethe PDF fS(x) = μ2xe−μx, x > 0, LST S∗(θ) = 2μ(2μ+θ)2 , meanE(S) = 2μ and secondmoment E(S2) = 2(2+1)

μ2respectively.

– Exponential distribution (denoted by Exp) with parameter μ, then we have the PDFfS(x) = μe−μx, x > 0, LST S∗(θ) = μμ+θ , mean E(S) = 1μ and second momentE(S2) = 2

μ2respectively.

– Hyper-Exponential distribution (denoted by Hyp-Exp) with parameter μ, then we have

the PDF fS(x) = 0.5μe−μx + 0.5μ2e−μ2x, x > 0, LST S∗(θ) = 0.5μμ+θ + 0.5μ2

μ2+θ , meanE(S) = 0.5(μ+1)

μ2and second moment E(S2) = μ+1

μ2respectively.

Considering the parameteric values as λ = 0.6, ν1 = 10, ν2 = 20, γ = 3, μ = 5, � = 1.5,M = 4, we display the effect of p and q on mean system size E(N) for three service timedistributions in Fig. 5. In Fig. 5, we notice that for a fixed value of q, as the value of p


00.2

0.40.6

0.81

00.2

0.40.6

0.81

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

probability p

probability q

E(N

)

2−stage Erlang

Exp Hyp−Exp

when λ=0.6, μ=5, ν1=10, ν

2=20, γ=3, ε=1.5, M=4

Fig. 5 Effect of p and q on E(N) for different service time distributions

increases E(N) decreases. This is due to the fact that for any value of q as the probabilityof server goes to dormant period, (i.e., p) increases waiting time of customer decreases (ifany customer arrive to the system) which gives rise to decrease in mean system size E(N).On the other hand, as the values of q increases E(N) increases. However, it is observedthat 2-stage Erlang distribution provides highest mean system size E(N), hyper-exponentialdistribution provides lowest mean system size E(N) and exponenetial distribution remainsin between 2-stage Erlang and hyper-exponential distribution.

Presuming the model parameters as λ = 0.6, ν2 = 20, p = 0.5, q = 0.5 γ = 3, μ = 5,� = 1.5, we display the effect of maximum number of type 1 vacationM and ν1 onE(N) forthree service time distributions namely, 2-stage Erlang, exponential and hyper-exponentialdistribution in Fig. 6.

In Fig. 6, we observe that as the number of type 1 vacation M increases the mean E(N)also increases upto a fixed number of type 1 vacation then it becomes stable whereas asthe value of ν1 increases E(N) decreases. It is clear that for large type 1 vacation rate ν1,shorter the type 1 vacation period and the probability that the customer is served through aservice rate increases which gives rise to decrease in mean system size E(N). However, itis noticed that 2-stage Erlang distribution provides highest E(N) values, hyper-exponentialdistribution provides lowest E(N) values and exponenetial distribution remains in between2-stage Erlang and hyper-exponential distribution.

Considering the model parameters as λ = 0.6, ν1 = 10, ν2 = 20, p = 0.5, q = 0.5,γ = 3, � = 1.5, M = 4, we exhibit the behaviour of service rate μ on E(N) due tothree service time distributions namely, 2-stage Erlang, exponential and hyper-exponentialin Fig. 7. From Fig. 7, we see that as the service rate μ increases the mean system size E(N)


02

46

8

0.5

1

1.5

0.4

0.5

0.6

0.7

0.8

0.9

1

Mν1

E(N

)

when λ=0.6, ν2=20, μ=5, γ=3, ε=1.5, p=0.5, q=0.5

2−stage Erlang

Hyp−Exp

Exp

Fig. 6 Effect of M and ν1 on E(N) for different service time distributions

decreases before arrives stability. However, we notice that E(N) is larger for 2-stage Erlangdistribution as compared to exponential and hyper-exponenetial distribution.

In Fig. 8, we exhibit the behaviour of close down period � on E(N) due to three servicetime distributions as mention above with parameter μ = 0.9. The other parameters are

2 4 6 80

0.2

0.4

0.6

0.8

1

1.2

1.4

μ

E(N

)

Hyp−ExpExp2−stage Erlang

Fig. 7 Effect of μ on E(N) for different service time distributions


2 4 6 80.75

0.8

0.85

0.9

0.95

1

E(N

)

ε

Hyp−ExpExp2−stage Erlang

Fig. 8 Effect of � on E(N) for different service time distributions

same as in Fig. 7. From Fig. 8, we observe that as the close down period � increases meansystem size E(N) decreases before attains stability. However, we see that hyper-exponentialdistribution provides highest mean system size E(N) whereas 2-stage Erlang provides thelowest and exponenetial stays in between hyper-exponential and 2-stage Erlang. It is clearfrom the fact that a higher coefficient of variation (or variance) of distribution gives thehigher E(N) values.

0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

p, when S ∼ Hyp−Exp

Sys

tem

sta

te P

roba

bilit

ies

PSt

PCL

Pt1V

Pt2V

PD

Fig. 9 Effect of p on system state probabilities


0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

p, when S ∼ Exp

Sys

tem

sta

te P

roba

bilit

ies

PSt

PCL

Pt1V

Pt2V

PD


Considering the model parameters as λ = 0.6, ν1 = 0.3, ν2 = 0.3, q = 0.5, γ = 3, μ =5, � = 1.5, M = 4, we show the effect of p on system state probabilities for three servicetime distributions namely, 2-stage Erlang, exponential and hyper-exponential distribution inFigs. 9, 10 and 11.

While the other model parameters are fixed, Figs. 9, 10 and 11 show that the systemstate probabilities PD , PCL, Pt2V and Pt1V are increases with the increasing value of p.However, the system state probability PSt decreases slightly as p increases. This is because

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

p, when S ∼ 2−stage Erlang

Sys

tem

sta

te P

roba

bilit

ies

PSt

PCL

Pt1V

Pt2V

PD



as the values of p increases the mean system size decreases, which gives rise to increasein probability that the server goes to a dormant period, probability that the server goes to aclose down period and probability that the server goes to a type 1 vacation as well as type2 vacation. However, for a fixed value of p, it is observed that the values of system stateprobabilities are larger for hyper-exponenetial distribution as compared to exponential and2-stage Erlang service time distribution.

Assuming the model parameters are λ = 0.6, ν1 = 0.3, ν2 = 0.3, p = 0.5, γ = 3,μ = 5, � = 1.5 and M = 4, we display the effect of q on system state probabilities forthree service time distributions namely, 2-stage Erlang, exponential and hyper-exponentialdistribution in Figs. 12, 13 and 14.

From Figs. 12, 13 and 14, we see that as the value of q increases the system state proba-bilities PCL, Pt1V , PSt and PD decreases but Pt2V increases as it should be. However, it isalso noticed that for a fixed value of q, the values of system state probabilities is larger forhyper-exponenetial distribution as compared to exponential and 2-stage Erlang service timedistribution.

Presuming the model parameters as λ = 0.6, ν1 = 0.3, ν2 = 0.3, p = 0.5, q = 0.5,γ = 3, μ = 5 and M = 4, we exhibit the behavior of close down period � on system stateprobabilities for three service time distributions as above mention in Figs. 15, 16 and 17.

From Figs. 15, 16 and 17, we observe that as the values of � increases the system stateprobabilities Pt1V , Pt2V , PSt and PD decreases whereas the system state probability PCLincreases with the increasing values of � as it should be. Also it is noticed from Figs. 15,16 and 17 that for a fixed value of �, hyper-exponential distribution provides highest sys-tem state probabilities whereas 2-stage Erlang provides lowest and exponenetial remains inbetween hyper-exponential and 2-stage Erlang.

Thus, as shown in our analysis, we can conclude that the fixed close down period, Maxi-mum M fixed number of type 1 vacations and type 2 vacation plays an important role in ouranalysis. Although we only display the three different service time distributions, i.e., three

0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

q, when S ∼ Hyp−Exp

Sys

tem

sta

te P

roba

bilit

ies

PSt

PCL

Pt1V

Pt2V

PD

Fig. 12 Effect of q on system state probabilities


0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

q, when S ∼ Exp

Sys

tem

sta

te P

roba

bilit

ies

PSt

PCL

Pt1V

Pt2V

PD


different special queues namely, M/E2/1 (when the service time follows 2-stage Erlang),M/M/1 (when the service time conforms exponential) andM/Hyp−Exp/1 (when the ser-vice time conforms hyper-exponential), our model can also handle the other general servicetime distributions.

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Sys

tem

sta

te P

roba

bilit

ies

q, when S ∼ 2−stage Erlang

PSt

PCL

Pt1V

Pt2V

PD



0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ε, when S ∼ Hyp−Exp

Sys

tem

sta

te P

roba

bilit

ies

PSt

PCL

Pt1V

Pt2V

PD

Fig. 15 Effect of � on system state probabilities

7 Practical Application of the ProposedModel

The model we presume can represent many problems in real life. Especially, The proposedmodel can be effectively used in some stochastic production and inventory control systems.In that type of systems, the demand for the product is modeled as a Poisson process. The

0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ε, when S ∼ Exp

Sys

tem

sta

te P

roba

bilit

ies

PSt

PCL

Pt1V

Pt2V

PD



0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

ε, when S ∼ 2−stage Erlang

Sys

tem

sta

te P

roba

bilit

ies

PSt

PCL

Pt1V

Pt2V

PD


production time of each product is generally distributed r.v. A practical situation connectedto a production to order system is shown for explanatory purpose. Production to Order isbasically a production strategy in production planning and management. For example, thecustomer orders for a product arrive conforming to a poisson process. As soon as all ordersare terminated and no new orders in the queue, the production will be stopped and facilityhas performed a fixed closed down task (close down period) and waits for the new orderarrives. If any order arrives during the close down task, the management policy is to startthe facility and begin the production at that close down task completion epoch. On the otherhand, if no order arrives after completing a fixed close down task, the facility performs afirst optional vacation (can be referred to an essential type 1 vacation). This optional jobmay be a succession of finite maintenance or other additional work. Upon completion ofthe first optional job, the manager checks the order and decides whether or not to restart themajor production. If the manager finds no major orders waiting in a queue after completingthe first optional job, a decision is taken for another optional job (type 1 vacation) to beperformed next with probability p̄(1 − p), which means the facility may not perform thedormant time with probability p and vice versa, i.e., if the decision is taken for dormanttime to perform next with probability p it means the facility may not perform the optionaljob with probability p̄(1−p). This pattern will continue till the facility takes the maximumnumber of M optional jobs. Upon termination of the Mth optional job if the manager findsno major order waiting in a queue, a decision is taken for other optional job of differentrandom length (type 2 vacation) to be performed next with probability q, which means thefacility may not perform dormant time with probability q̄(1 − q) and vice versa. If anyorder arrives during the other optional job (type 2 vacation) or dormant time, the managerimmediately resume the start-up time of the facility before starting the major production.In this scenario, the facility, optional jobs, other optional job (different length), close down


task and dormant times correspond to the server, type 1 vacation, type 2 vacation, closedown period and dormant period respectively in the queueing terminology.

In practice, the queueing managers are always interested in optimizing the system oper-ating cost per unit time. Here, we develop a steady state expected system operating costfunction per unit time for the underlying queueing model, where probability p is a deci-sion variable. Our main purpose is to determine the optimum value of the probability p inorder to minimize the expected system operating cost function per unit time which is statedbelow. Now we define

CSt = Per unit time cost while the server is in start-up period. Thus, in long run theaverage cost owing to server is in start-up period is CStPSt .Ch = Holding cost per unit time for each arrival available in the system. Therefore, inlong run the average holding cost is ChE(N)CCL = Per unit time cost while the server is in close down period. Therefore, in long runthe average cost owing to server is in close down period is CCLPCL.Ct1V = Per unit time cost while the server is in type 1 vacation period. Thus, in long runthe average cost owing to server is in type 1 vacation period is Ct1V Pt1V .Ct2V = Per unit time cost while the server is in type 2 vacation period. Thus, in long runthe average cost owing to server is in type 2 vacation period is Ct2V Pt2V .CD = Per unit time cost while the server is in dormant period. Thus, in long run theaverage cost owing to server is in dormant period is CDPD .CS = Per unit time cost for holding the server busy. Therefore, in long run the averagecost owing to server busy is CSλE(S).

Depending on the above cost parameters, the expected cost function per unit time can bedemonstrated mathematically as

min:p

f (p) = CStPSt + ChE(N) + CCLPCL + Ct1V Pt1V + Ct2V Pt2V + CDPD+CSλE(S). (76)

Since the expected cost function is non-linear and complex, therefore, it would be a difficultto obtain the optimum

Analysis of Single Server Queue with Modified Vacation...

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