An unreliable Geo/G/1 queue with startup and closedown times under randomized finite vacations

Applied Mathematical Modelling 39 (2015) 1383–1399

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Applied Mathematical Modelling

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An unreliable Geo/G/1 queue with startup and closedown timesunder randomized finite vacations

http://dx.doi.org/10.1016/j.apm.2014.09.0060307-904X/� 2014 Elsevier Inc. All rights reserved.

Tsung-Yin WangDepartment of Accounting Information, National Taichung University of Science and Technology, Taichung 404, Taiwan, ROC

a r t i c l e i n f o

Article history:Received 1 June 2012Received in revised form 18 August 2014Accepted 3 September 2014Available online 16 September 2014

Keywords:Unreliable serverStartup timeClosedown timeVacationWaiting time

a b s t r a c t

A Bernoulli-schedule randomized vacation policy Geo/G/1 queueing system of an unreli-able server with startup time and closedown time is studied. The vacation time, repairtime, startup time and closedown time are generally distributed. After one basic vacationthe server may take a finite vacation policy with Bernoulli successful probability p up toJ � 1 vacations. Based on the generating function and supplementary variable technique,the analytical expressions for the steady-state distributions of system size at various statesand server state are obtained. Also by the concept of generalized service time, the steady-state length distributions of various state periods are derived. Furthermore, the customer’swaiting time in the system is also obtained.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Over the past several decades, continuous-time queueing systems with vacations have been studied by many pioneersand now are progressing. Doshi’s survey paper [1], Takagi’s monograph [2] and Tian and Zhang’s monograph [3] point tonumerous contributions on vacation systems. Recent developments in the continuous time queueing models related withBernoulli-schedule vacation policy have Ke and Huang [4], Ke et al. [5,6] and Ghoudhury et al. [7,8].

In contrast to continuous time queues, the discrete-time queues are worked few. The classical monographs on discrete-time queueing models see Takagi [9] and Woodward [10]. Recent years, the study of the discrete-time queueing systems hasreceived more attention and become very important due to the progress of computer and communication technology. Thereason is that discrete-time queues, the inter-arrival time and service time are random positive integers, have been found tobe more appropriate for modeling and analyzing digital communication systems than the continuous-time counterparts.Zhang and Tian [11] studied a Geo/G/1 queue with taking a random maximum number of vacations. Samanta [12] analyzeda GI/Geo/1 queue in which the server takes exactly one Bernoulli vacation after each empty system. Wang et al. [13] studiedthe discrete-time Geo/G/1 queue with randomized vacations and at most J vacations.

On the other hand, Queueing systems with server breakdowns during service are a natural and valuable abstraction oftemporary server unavailability caused by external factors. Similarly, the server needs some time to start up before begin-ning the first service in each cycle, and it also needs some time to shut down after completing a service which leaves nocustomers in the system. Like these events often occur in the real world. So far, very few authors dealt with the discrete-timequeues about these issues. Atencia and Moreno [14] examined a discrete-time Geo/G/1 retrial queue with server subject tobreakdowns. Moreno [15] analyzed a Geo/G/1 system with a generalized N-policy and setup/closedown times. Later Moreno[16] discussed the system performances and presented the numerical study of the cost function for a Geo/G/1 system undermultiple vacations and setup/closedown times. Wang and Zhang [17] treated a discrete-time single server retrial queue withgeometrical arrivals of both positive and negative customers in which the server is subject to breakdowns and repairs.

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1384 T.-Y. Wang / Applied Mathematical Modelling 39 (2015) 1383–1399

Recently Wang [18] studied the extended N-Policy Geo/G/1 queue with startup and closedown times in which the thresholdN is a random variable and is newly determined every time when a new cycle begins. To operate the N policy, the server mustbe continuously monitoring the queue for an arrival when the server is inactive. However, the continuous monitor may notbe performed in some situations or to keep monitoring may result in an expensive cost. Hence the vacation policy could beconsidered. To the best of my knowledge, the existing studies have never covered cases that an unreliable server with ran-domized finite vacation policy and startup/closedown times. Besides, the suggested model has potentially application inpractical Voice over Internet Protocol (VoIP) system. In VoIP system, the potential customers and users in different offices,different faculties or research centers, send a voice mail or voice connection across the network through the office automa-tion system. Channel requests, grants, data transmissions, and receptions all proceed in fixed time intervals. Once the VoIPclient sends connection request to the VoIP server, the received user request is identified and processed if the server is idle.The time required to process a user request is assumed to have a general distributed process time because the time requiredhandling a voice mail or voice connection is random and directly influenced by the voice mail or voice connection type. If theserver is busy, the received voice mail or voice connection is placed in the queue according to a first in first out discipline.After all user requests being served completely, the VoIP server need a period to record the system status and then take oneof maintenance activities, such as virus scan, disk cleaning, and disk defragment to keep the VoIP server functioning well andefficiently. Once the maintenance activity is finished, the VoIP server may take another maintenance activity or will stay dor-mant in the system if there are no received user requests waiting in the queue. Otherwise, it starts to serve user requestswith some preparing time (startup time). Moreover, the VoIP server may be interrupted when machine encounters unpre-dicted breakdowns. When this situation occurs (breakdowns), the server is emergently recovered with a random time. Aninteresting issue raised by this case is to investigate how the effects of the changes of the system parameters influencethe cost function. The operational scheme of a VoIP system is depicted in Fig. 1.

Although both [18] and the presented model consider the server with startup and closedown times, the differencesbetween the two papers are: (i) for the feature of server, the former is reliable and the latter is unreliable which is more real-istic in the real world; and (ii) for control policy, the former is the threshold control policy where the server starts its servicedepending on the number N of customers in the queue for per regenerative cycle and the latter is vacation policy where theserver provides its service after a random length of vacation time finitely repeated until at least one customer is in the queuewhen every cycle begins; and (iii) for practical applications, the former has the practical use in the streaming player byremote viewers and the latter has the practical use in VoIP system; and (iv) for the numerical analysis, the former has nocost analysis and the latter not only investigates how the effects of the changes of the system parameters influence the totalexpected cost function but also provides the numerical study of the application example’s managerial insights on optimaldecision variables p and J .

The main contributions of this work are twofold. We present a mathematical model for the VoIP system and solve itsimportant system performances. The second contribution is to provide numerical study of the VoIP system from the viewof the cost. The remainder of this paper is structured as follows. The following section presents the model description.Section 3 derives the probability generating functions of the important system performance characteristics and their mainexpectations. Section 4 discusses the turned-off, startup, generalized service, and closedown periods of the system. Section 5

Fig. 1. The operational scheme of a VoIP system.

T.-Y. Wang / Applied Mathematical Modelling 39 (2015) 1383–1399 1385

derives the waiting times in the queue. Section 6 presents some illustrative examples to investigate how the effects of thechanges of the system parameters influence the cost. Conclusions are given in Section 7.

2. Model description and assumptions

We consider a discrete-time single server queue with startup and closedown times in which the server can take Bernoulli-schedule vacations after one basic vacation and also the server can break down during the times when services are offered. Inthis article, the time axis is divided into constant length intervals (called slots) and all queueing activities occur only at theboundaries of slots. Let the time axis be marked by 0, 1,. . ., n,. . .. We assume that a potential arrival only occurs just beforethe end of a slot and a potential departure occurs just after the beginning of a slot. It is well-known Late Arrival System (LAS)in the discrete-time queueing systems. LAS has two variants depending on that an arrival can or cannot commence serviceduring his arrival slot while the server is idle. The former case is referred to as LAS with immediate access (LAS-IA) and thelatter case is referred to as LAS with delayed access (LAS-DA). In this study, we adopt the LAS-DA and denote �x ¼ 1� x for anyreal number x e [0, 1] .

In the investigated model, customers arrive according to a Bernoulli process with rate k in each slot. Therefore, the inter-arrival times of customers fAn;n P 1g are independent and identically distributed (i.i.d.) random variables and have geomet-rical distribution

PðAn ¼ iÞ ¼ �ki�1k; i ¼ 1;2;3; ::::

Service times fBn;n P 1g are i.i.d. random variables with general distribution

PðBn ¼ iÞ ¼ bi; i ¼ 1;2;3; ::::; BðxÞ ¼X1i¼1

bixi; E½Bn� ¼ bð1Þ; E½BnðBn � 1Þ� ¼ bð2Þ:

The server is subject to breakdowns and we assume that the breakdown distribution is geometrically distributed withparameter a, when he is working. Whenever the server fails, it is immediately repaired at a repair facility, where repair timesfRn; n P 1g are i.i.d. random variables with arbitrary distribution

PðRn ¼ iÞ ¼ ri; i ¼ 1;2;3; ::::; RðxÞ ¼X1i¼1

rixi; E½Rn� ¼ rð1Þ; E½RnðRn � 1Þ� ¼ rð2Þ:

After each departure leaving an empty system, the server is shut down by a closedown time where closedown timesfCn; n P 1g are i.i.d. random variables and have general distribution

PðCn ¼ iÞ ¼ ci; i ¼ 1;2;3; ::::; CðxÞ ¼X1i¼1

cixi; E½Cn� ¼ cð1Þ; E½CnðCn � 1Þ� ¼ cð2Þ:

If customers arrive at the system during the closedown time, the server immediately enters the busy state to serve thecustomers; otherwise, the server takes one basic vacation. Upon returning from the basic vacation, if no customers are inthe system, the server may take another vacation with probability p or go into idle state with probability �p to wait one arri-val. This pattern continues until the number of vacations reaches J. At the end of a vacation or idleness, before the servercomes back to start service, the server requires for pre-service work (i.e. begin startup). Vacation times fVn;n P 1g andstartup times fSn;n P 1g are all i.i.d. random variables and have the respective general distributions as below

PðVn ¼ iÞ ¼ v i; i ¼ 1;2;3; ::::; VðxÞ ¼X1i¼1

v ixi; E½Vn� ¼ v ð1Þ; E½VnðVn � 1Þ� ¼ v ð2Þ;

PðSn ¼ iÞ ¼ si; i ¼ 1;2;3; ::::; SðxÞ ¼X1i¼1

sixi; E½Sn� ¼ sð1Þ; E½SnðSn � 1Þ� ¼ sð2Þ:

The service discipline is based on their arrival order and the server can serve at most only one customer at a time. Whencustomers arrive and find that the server is busy or broken-down, they must queue in the waiting line until the server isavailable. Although no service occurs during the repair period of the server, customers continue to arrive following a Ber-noulli process. All customers arriving to the system are assumed to be eventually served. Furthermore, various stochasticprocesses in the system are independent. Fig. 2 depicts time epochs in LAS-DA.

It should be noted that in the LAS-DA the arrival can not enter the free service facility in the nth slot for a departure beingcompleted service in the nth slot and the arrival is blocked until the servicing slot terminates. Fig. 3 gives one cycle samplepath for the studied model with LAS-DA. In the first cycle, arrivals occur at times 1, 3, 6, 8, 9, 15 (with inter-arrival times 2, 3,2, 1, 6), with service times 1, 1, 2 (with passing repair times 3), 1, 1, 3 (with passing repair times 2).

Fig. 2. Time epochs in the LAS with delay access.

Fig. 3. One-cycle sample path for an unreliable Geo/G/1 queue with startup and closedown times under randomized finite vacations.


3. Model formulation and stationary distribution

We begin the analysis by defining the state of the server at n+, C(n), as

CðnÞ ¼

m; if the sever is on the mth vacation;m ¼ 1;2; . . . J;

J þ 1; if the sever is id1e;J þ 2; if the sever is under setup;J þ 3; if the sever is busy;J þ 4; if the sever is under repair;J þ 5; if the sever is under closedown:

8>>>>>>>><>>>>>>>>:

Let nðnÞ1 and nðnÞ2 be the supplementary random variables as follows:

nðnÞ1 ¼

the remaining vacation time at nþ; if CðnÞ ¼ 1;2; . . . ; J;

the remaining setup time at nþ; if CðnÞ ¼ J þ 2;the remaining service time at nþ; if CðnÞ ¼ J þ 3 ; J þ 4;the remaining closedown time at nþ; if CðnÞ ¼ J þ 5;

8>>><>>>:

nðnÞ2 ¼ remaining repair time at nþ; if CðnÞ ¼ J þ 4:

L(n) denotes the number of customers in the system at n+. The process ðCðnÞ; LðnÞ; nðnÞ1 ; nðnÞ2 Þ; n P 1n o

forms a Markov chainwhose state space is

fðm; k; iÞ : m ¼ 1;2; . . . J; J þ 2; J þ 3; J þ 5; k P 0; i P 1g [ fðJ þ 1;0Þg [ fðJ þ 4; k; i; jÞ : k P 1; i P 1; j P 1g:

Under stability condition qH ¼ q½1þ ða=�aÞrð1Þ� ðq ¼ kbð1Þ), the steady-state probabilities are as follows:

pm;k;i ¼ limn!1

Pr½CðnÞ ¼ m; LðnÞ ¼ k; nðnÞ1 ¼ i�; m ¼ 1;2; . . . J; J þ 2; J þ 3; J þ 5; k P 0; i P 1;

pJþ1 ¼ limn!1

Pr½CðnÞ ¼ J þ 1; LðnÞ ¼ 0�;

pJþ2;k;i ¼ limn!1

Pr½CðnÞ ¼ J þ 2; LðnÞ ¼ k; nðnÞ1 ¼ i�; k P 1; i P 1;


pJþ3;k;i ¼ limn!1

Pr½CðnÞ ¼ J þ 3; LðnÞ ¼ k; nðnÞ1 ¼ i�; k P 1; i P 1;

pJþ4;k;i;j ¼ limn!1

Pr½CðnÞ ¼ J þ 4; LðnÞ ¼ k; nðnÞ1 ¼ i; nðnÞ2 ¼ j�; k P 1; i P 1; j P 1;

pJþ5;k;i ¼ limn!1

Pr½CðnÞ ¼ J þ 5; LðnÞ ¼ k; nðnÞ1 ¼ i�; k P 0; i P 1; j P 1:

The steady state transition diagram is shown in Fig. 4. In the vacation phase, node k denotes that k customers in the sys-tem and the remaining vacation time of server. Similarly, in the busy phase, node k denotes k customers in the system andthe remaining service time of a customer being serving. In the idle phase, node 0 denotes no customers in the system. In thebreakdown phase, node k denotes k customers in the system, the remaining repair time of server and the remaining servicetime while breakdown occurring. In the startup (closedown) phase, node k denotes k customers in the system and theremaining startup (closedown) time of server. The arrows in the diagrams show only possible transitions in the state ofthe system, and the entry for each arrow gives the transition probability for that transition when the system is in the stateat the base of the arrow. In the busy phase: (i) the arrow from node k to itself represents that the transition probability fromthe current state (J + 3, k, i + 1) to the next state (J + 3, k, i) is �k�a and the transition probability from the current state (J + 3, k,1) to the next state (J + 3, k, i) is kbi �a; and (ii) the arrow from node k to node k of the breakdown state represents that thetransition probability from the current state (J + 3, k, i + 1) to the next state (J + 4, k, i, j) is �karj and the transition probabilityfrom the current state (J + 3, k, 1) to the next state (J + 4, k, i, j) is kbiarj. In the breakdown phase: (i) the arrow from node k to

Fig. 4. Steady-state transition diagram.


itself represents that the transition probability from the current state (J + 4, k, i, 1) to the next state (J + 4, k + 1, i, j) is karj andthe transition probability from the current (J + 4, k, i, j + 1) to the next state (J + 4, k + 1, i, j) is �k; and (ii) the arrow from node kto node k + 1 represents that the transition probability from the current state (J + 4, k, i, 1) to the next state (J + 4, k + 1, i, j) iskarj and the transition probability from the current state (j + 4, k, i, j + 1) to the next state (j + 4, k + 1, i, j) is k.

The Kolmogorov equations for the stationary distribution are given by

p1;k;i ¼ �kp1;k;iþ1 þ d0;k�kpJþ5;0;1v i þ ð1� d0;kÞp1;k�1;iþ1k; k P 0; i P 1; ð1Þ

pm;k;i ¼ �kpm;k;iþ1 þ d0;k�kppm�1;k;1v i þ ð1� d0;kÞpm;k�1;iþ1k; 2 6 m 6 J; k P 0; i P 1; ð2Þ

pJþ1 ¼ �pXJ�1

m¼1

pm;0;1�kþ pJ;0;1

�kþ pJþ1�k; ð3Þ

pJþ2;k;i ¼ si d1;kpJþ1kþXJ

m¼1

kpm;k�1;1 þXJ

m¼1

�kpm;k;1

!þ �kpJþ2;k;iþ1 þ kð1� d1;kÞpJþ2;k�1;iþ1; k P 1; i P 1; ð4Þ

pJþ3;k;i ¼ �a �kbiðpJþ2;k;1 þ pJþ3;kþ1;1Þ þ �kðpJþ3;k;iþ1 þ pJþ4;k;i;1Þ þ kbiðpJþ3;k;1 þ pJþ5;k�1;1Þ þ pJþ5;k;1�kbi

�þð1� d1;kÞkðpJþ2;k�1;1bi þ pJþ3;k�1;iþ1 þ pJþ4;k�1;i;1Þ

�; k P 1; i P 1; ð5Þ

pJþ4;k;i;j ¼ kð1� d1;kÞ½ðpJþ2;k�1;1bi þ pJþ3;k�1;iþ1bi þ pJþ4;k�1;i;1Þarj þ pJþ4;k�1;i;jþ1� þ ½�kðpJþ2;k;1 þ pJþ3;kþ1;1 þ pJþ5;k;1Þþ kðpJþ3;k;1 þ pJþ5;k�1;1Þ�biarj þ �k½pJþ4;k;i;jþ1 þ ðpJþ3;k;iþ1 þ pJþ4;k;i;1Þarj�; k P 1; i P 1; j P 1; ð6Þ

pJþ5;k;i ¼ d0;kpJþ3;1;1�kci þ ð1� d0;kÞpJþ5;k�1;iþ1kþ pJþ5;k;iþ1

�k; k P 0; i P 1 ð7Þ

where da,b denotes Kronecker’s delta.Define the following generating functions of the Markov chain ðCðnÞ; LðnÞ; nðnÞ1 ; nðnÞ2 Þ; n P 1

n o,

Gmðx; zÞ ¼X1k¼0

X1i¼1

zkxipm;k;i; m ¼ 1;2 . . . ; J; J þ 5

Gmðx; zÞ ¼X1k¼1

X1i¼1

zkxipm;k;i; m ¼ J þ 2; J þ 3

GJþ4ðx; y; zÞ ¼X1k¼1

X1i¼1

X1j¼1

zkxiyjpJþ4;k;i;j

jxj 6 1; jyj 6 1; jzj 6 1:

Theorem 1. When qH < 1, the generating functions of the steady-state distribution of the Markov chain ðCðnÞ;LðnÞ;nðnÞ1 ;nðnÞ2 Þ;n

n P 1o

where C(n) = 1, 2, ..., J, J + 2, ..., J + 5, are given by

Gmðx; zÞ ¼x�kjm�1ðVðxÞ � VðsÞÞ

x� spJþ5;0;1; 1 6 m 6 J;

GJþ2ðx; zÞ ¼x�kðSðxÞ � SðsÞÞ#ðzÞ

ðx� sÞ pJþ5;0;1;

GJþ3ðx; zÞ ¼xz�k�aðBðxÞ � BðrÞÞ #ðzÞSðsÞ þ CðsÞ�1

Cð�kÞ � 1� �

ðz� BðrÞÞ½xð1� aRðsÞÞ � �as� pJþ5;0;1;

GJþ4ðx; y; zÞ ¼xyz�kaðBðxÞ � BðrÞÞðRðyÞ � RðsÞÞ #ðzÞSðsÞ þ CðsÞ�1

Cð�kÞ � 1� �

ðy� sÞðz� BðrÞÞ½xð1� aRðsÞÞ � �as� pJþ5;0;1;

GJþ5ðx; zÞ ¼x�kðCðxÞ � CðsÞÞðx� sÞCð�kÞ

pJþ5;0;1;

and


pJþ1 ¼�kð�pjþ pjJ � jJþ1Þ

pkð1� jÞ pJþ5;0;1;

Where qH ¼ 1þ a�a rð1Þ

� �q ðq ¼ kbð1ÞÞ; s ¼ �kþ kz;j ¼ pVð�kÞ;r ¼ �as

1�aRðsÞ and #ðzÞ ¼ ðVðsÞ�ð1�zÞVð�kÞÞð1�jJ Þ�zðj�jJÞð1�jÞ .

Proof. See Appendix A. h

Corollary 1. When qH < 1,

(a) The PGF of the number of customers in the system is

LSðzÞ ¼�kB �as

1�aRðsÞ

� �#ðzÞSðsÞ þ CðsÞ�1

Cð�kÞ � 1� �

k z� B �as1�aRðsÞ

� �� pJþ5;0;1;

(b) The PGF of the number of customers in the queue is

LQ ðzÞ ¼�k #ðzÞSðsÞ þ CðsÞ�1

Cð�kÞ � 1� �

k z� B �as1�aRðsÞ

� �� pJþ5;0;1:

Corollary 2. When qH < 1, the probability of pJ+5,0,1 is given by pJþ5;0;1 ¼ 1�qH

�k Xþsð1Þþcð1ÞCð�kÞ

� �where X ¼ ðkvð1ÞþVð�kÞÞð1�jJÞ�ðj�jJÞ

kð1�jÞ .

Proof. Using the normalization condition,PJ

m¼1Gmð1;1Þ þ pJþ1 þPJþ3

m¼Jþ2Gmð1;1Þ þ GJþ4ð1;1;1Þ þ GJþ5ð1;1Þ ¼ 1; the resultcan be easily obtained from corollary 1. h

Corollary 3. When qH < 1, the steady–state probabilities of server states are as follows:

(a) The probability that the server is on the jth vacation:

PVj¼

�kv ð1Þjj�1ð1� qHÞXþ sð1Þ þ cð1Þ

Cð�kÞ

� � ; 1 6 j 6 J:

(b) The probability that the server is on vacation:

PV ¼ð1� jJÞð1� qHÞv ð1Þ

ð1� jÞ Xþ sð1Þ þ cð1ÞCð�kÞ

� � :
(c) The probability that the server is idle:
PI ¼Vð�kÞð�pþ pjJ�1 � jJÞð1� qHÞ

kð1� jÞ Xþ sð1Þ þ cð1ÞCð�kÞ

� � :

(d) The probability that the server is start-up:

PST ¼ GJþ2ð1;1Þ ¼sð1Þð1� qHÞXþ sð1Þ þ cð1Þ

Cð�kÞ

� � :
(e) The probability that the server is busy:
PB ¼ q

(f) The probability that the server is broken down:

PD ¼ GJþ4ð1;1;1Þ ¼arð1Þq

�a:

(g) The probability that the server is closedown:


PCD ¼ GJþ5ð1;1Þ ¼cð1Þð1� qHÞ

Cð�kÞ Xþ sð1Þ þ cð1ÞCð�kÞ

� � :

Corollary 4. When qH < 1, the probability of the system is empty is given by

P0 ¼XJ

m¼1

X1i¼1

pm;0;i þ pJþ1 þX1i¼1

pJþ5;0;i ¼ð1� qHÞ

kCð�kÞ Xþ sð1Þ þ cð1ÞCð�kÞ

� � :
The proofs of Corollaries 1–4 can be obtained easily from Theorem 1.
Corollary 5. When qH < 1, the mean numbers of customers in the system and in the queue are given by

E½LS� ¼ qH þk ð1�jJÞvð2Þ

1�j þ 2Xsð1Þ þ sð2Þ þ cð2ÞCð�kÞ

� �2 Xþ sð1Þ þ cð1Þ

Cð�kÞ

� � þk2 1þ a

�a rð2Þ� �2

bð2Þ þ k a�a ðqrð2Þ þ 2qHrð1ÞÞ

2ð1� qHÞ

and

E½LQ � ¼E½LS�

k� bð1Þ 1þ a

�arð1Þ

� �:

Proof. Differentiating LS(z) and LQ(z) in Corollary 1 and using Corollary 2, we apply L’Hospital’s and finally the mean num-bers of customers in the system and in the queue are obtained. h

4. The turned-off, startup, generalized service, and closedown periods

This section studies the turned-off, startup, generalized service, and closedown periods. The turned-off period consistsvacation period and idle period. The startup period begins at the end of an idle period or a vacation period, and ends atthe beginning of a service. The generalized service period consists busy period and breakdown period. Therefore, the general-ized service period starts at the completion of a startup time and terminates when no customers are in the system at thecompletion of a generalized service. A closedown period starts from an empty system at a service completion and ends whenthere are no customers in the system at a closedown completion.

4.1. The turned-off period

The turned-off period is composed of vacation period and idle period. A vacation period starts at the end of a closedowntime without customers in the system and terminates when the following jth ð1 6 j 6 JÞ vacation concludes with at least onecustomer accumulated in the system or the consecutive J vacations with no one accumulated in the system. An idle periodstarts at the end of a vacation in which the server is just dormant in the system and terminates at the end of the succeedingslots that a customer arrives. According to the definition, we have

� The PGF of the length of an idle period for the classical Geo/G/1 queue is given by
IðxÞ ¼ kx
1��kx.

� The joint PGF for the length of the successive j vacations and the probability that no customers arrive during these vaca-

tions is given by hj

p ð1 6 j 6 JÞ; where h ¼ pVð�kxÞ.� The joint PGF for the length of a vacation and the probability that at least one arrives during that vacation is given by½VðxÞ � Vð�kxÞ�.

Hence the PGF for the length of the turned-off period and the mean length of the turned-off period are given by

Ioff ðxÞ ¼hpþ h2

pþ � � � þ hJ�1

p

!�pIðxÞ þ hJ

pIðxÞ þ

XJ�1

k¼0

hk½VðxÞ � Vð�kxÞ� ¼ ðphJ � hJþ1 þ �phÞIðxÞ þ ½pVðxÞ � h�ð1� hJÞpð1� hÞ

and

E½Ioff � ¼ðkv ð1Þ þ Vð�kÞð1� jJÞ � ðj� jJÞ

kð1� jÞ : ð8Þ


4.2. The startup period

Next, we derive the startup period. A startup period begins when the server performs to start up the server at the end of avacation or idleness and terminates when the server starts actually providing the service. The PGF and mean length for thestartup period are given by IS(x) = S(x) and

E½IS� ¼ sð1Þ: ð9Þ

4.3. The generalized service period

The generalized service time is immediately followed by a random startup time and ends at instant that complete theservice. That is, the generalized service time includes actual service time and possible repair time. The length of actual ser-vice time for a customer consists of the passed service time before interruption by server failed and the remaining servicedue to interruption by server failed. Let hk be the probability that the generalized service time to serve a customer is k slots.From Atencia et al. [14], the generalized service times are independent and fhkg1k¼1 is given by

hk ¼�akbk; k ¼ 1

�akbk þXk�1

i¼1

Xk�i

‘¼1

Ciþ‘�1‘

�aia‘biPrfR1 þ R2 þ � � � þ R‘ ¼ k� ig; k P 2

8><>:

which leads to the PGF of fhkg1k¼1

HðzÞ ¼X1k¼1

hkzk ¼X1k¼1

�akbkzk þX1k¼2

Xk�1

i¼1

Xk�i

‘¼1

Ciþ‘�1l

�aia‘biPrfR1 þ R2 þ � � � þ R‘ ¼ k� igzk

¼X1k¼1

�akbkzk þX1i¼1

X1l¼1

Ciþl�1l bið�azÞial

X1k¼l

PrfR1 þ R2 þ � � � þ Rl ¼ kgzk ¼X1k¼1

�akbkzk þX1i¼1

X1l¼1

Ciþl�1l bið�azÞial½RðzÞ�l

¼ B�az

1� aRðzÞ

� :

Let U be the generalized service period of the classical Geo/G/1 queue with unreliable server. Applying the busy period ofthe classical Geo/G/1 queue with reliable server, WðzÞ ¼ BðkzWðzÞ þ �kzÞ, we obtain

UðzÞ ¼ HðkzUðzÞ þ �kzÞ ¼ B�aðkzUðzÞ þ �kzÞ

1� aRðkzUðzÞ þ �kzÞ

� :

To derive the PGF of the mean length of the generalized service period for the present model, we consider the follows:

Case 1: The joint PGF for at least one customer arriving at the jth vacation, 1 6 j 6 J, and the probability of no customerarrives at the preceding vacations (if having vacations) is jj-1 ½VðxÞ � Vð�kxÞ�: Similarly, the joint PGF for no cus-tomer arriving at the jth vacation and the probability that the server opts to remain dormant in the system at

the end of the jth vacation to wait one customer arriving at the idle period is given by �p1�dj;J jjzp ; 1 6 j 6 J.

Therefore, the PGF for the number xVI of customers that arrive during the vacation and idle period with arrivals isgiven by

xVIðzÞ ¼XJ

j¼1

jj�1ðVð�kþ kzÞ � Vð�kÞÞ þXJ

j¼1

�p1�dj;J jjzp

¼ ð1� jJÞðVð�kþ kzÞ � Vð�kÞÞ þ ½Vð�kÞð1� jJÞ � ðj� jJÞ�z1� j

:

Case 2: The PGF for the number of arrivals in startup time is xSðzÞ ¼ Sð�kþ kzÞ .Case 3: The PGF for the number of customers that arrive during a shutdown time with arrivals is given by Cð�kþkzÞ�Cð�kÞ

1�Cð�kÞ andthe random variable D, the number of closedowns in a cycle, obeys the geometric distribution with parameterCð�kÞ. Therefore, we can obtain the PGF for the number xC of customers that arrive during the shutdown periodwith arrivals,

xCðzÞ ¼X1k¼1

Cð�kþ kzÞ � Cð�kÞ1� Cð�kÞ

� k�1

PfD ¼ kg ¼ Cð�kÞ1� Cð�kþ kzÞ þ Cð�kÞ

:

Now we define xT = xVI + xS + xC. The number of arrivals during the vacation and idle, startup and closedown periods areindependent. Therefore, we have xT(z) = xVI(z) �xS(z) �xC(z).

Following Takagi [9, p. 95 (3.17a)], the PGF for the length of the generalized service period is given by


UðzÞ ¼ xTðUðzÞÞ ¼ xVIðUðzÞÞ �xSðUðzÞÞ �xCðUðzÞÞ

¼ ð1� jJÞðVð�kþ kUðzÞÞ � Vð�kÞÞ þ ðVð�kÞð1� jJÞ � ðj� jJÞÞUðzÞð1� jÞ � Sð�kþ kUðzÞÞ � Cð�kÞ

1� Cð�kþ kUðzÞÞ þ Cð�kÞ

which yields the mean length

E½IBD� ¼ U0ð1Þ ¼qH Xþ sð1Þ þ cð1Þ

Cð�kÞ

� �ð1� qHÞ

: ð10Þ

Eq. (10) yields the mean length of busy period and the mean length of breakdown period as

E½IB� ¼Xþ sð1Þ þ cð1Þ

Cð�kÞ

� �q

ð1� qHÞ

and

E½ID� ¼Xþ sð1Þ þ cð1Þ

Cð�kÞ

h ia�a rð1Þq

ð1� qHÞ:

4.4. The closedown period

To obtain the closedown period distribution, we first derive the number of shutdowns in a cycle, say D. The probabilitythat the sever has finished closedown and no customer is in the system is Cð�kÞ and the probability that the sever has finishedshutdown and at least one customer in the system is 1� Cð�kÞ. Therefore, D follows the geometric distribution with param-eter Cð�kÞ. Besides, the PGF for the length of a closedown time is C(x). Consequently, the PGF for the length of a closedownperiod is given by

ICðxÞ ¼X1k¼1

ðCðxÞÞkPfD ¼ kg ¼ Cð�kÞCðxÞ1� CðxÞð1� Cð�kÞÞ

and the mean length is

E½IC � ¼cð1Þ

Cð�kÞ: ð11Þ

From (8)–(11), the mean length of the service cycle is given by

E½IT � ¼ Xþ sð1Þ þ cð1Þ

Cð�kÞ:

5. Waiting time

When a customer considers entering a service system, one of the most critical concerns is how much time he will spend inthe queue (system). In this section, we provide the waiting time distributions in the queue and in the system. The waitingtime in the queue for a customer arriving at an arbitrary slot n includes possible residual vacation time, startup time, close-down time or generalized service time of the customers in the system in front of him. Thus, the waiting time in the systemfor a customer arriving at slot n is the sum of his waiting time in the queue and his generalized service time.

Theorem 2. When qH < 1, the PGFs of the waiting time in the queue and in the system are given by

WQ ðzÞ ¼½VðzÞð1� jJÞ � ðj� jJÞ�SðzÞ

ð1� jÞ � ð1� zÞ½Vð�kÞð1� jJÞ � ðj� jJÞ�SðzÞkð1� jÞ þ CðzÞ � 1� Cð�kÞ

Cð�kÞ

� �kpJþ5;0;1

ðz� ð�kþ kHðzÞÞÞ

and

WSðzÞ ¼WQ ðzÞHðzÞ:

Proof. See Appendix B. h

Corollary 6. When qH < 1, the mean waiting times of a customer in the queue and in the system are given by

E½WQ � ¼ð1�jJÞð1�jÞ v

ð2Þ þ 2Xsð1Þ þ sð2Þ þ cð2ÞCð�kÞ

2 Xþ sð1Þ þ cð1ÞCð�kÞ

� � þk 1þ a

�a rð1Þ� �2

bð2Þ þ a�a ðqrð2Þ þ 2qHrð1ÞÞ

2ð1� qHÞ;


and

Table 1

F(p,J

p = 0p = 0p = 0p = 0p = 0p = 0p = 1

E½WS� ¼ E½WQ � þqH

k:

Proof. The proof is obtained from Theorem 2. h

6. Numerical examples

In this section, we first construct a long-run average cost function per customer per unit time for an unreliable Geo/G/1queue with startup and closedown times under randomized finite vacations. We consider the following cost elements:

Ch : holding cost per unit time for each customer present in the system;Cs : fixed cost per regenerative cycle;Cv : cost per unit time for the server on vacation;Ci : cost per unit time for keeping the server idle;Cst :cost per unit time for the preparatory work of the server before starting the service;Cb : cost per unit time for keeping the server on and in operation;Cd : breakdown cost per unit time for a failed server;Ccd: closedown cost per unit time for the closedown time of the server.Employing the definition of each cost element and its corresponding system performance, the long-run average cost func-

tion is given by

F ¼ ChE½LS� þ Cs1

E½IT �þ CvPV þ CstPST þ CiPI þ CbPB þ CdPD þ CcdPCD:

For illustrative purposes, we present some numerical examples to evaluate the effects of changes in the system parametervalues on the cost function. Throughout the numerical analysis, we assume that the vacation time, startup time, service time,repair time and closedown time distributions are all assumed geometric distributions with rates -V, -ST, -B, -R and -C,respectively. To perform the numerical investigation, the following cost elements are setting:

Ch = 25, Cs = 800, Cv = 15, Ci = 40, Cst = 20, Cb = 30, Cd = 50, Ccd = 20 and we consider the following cases.

Case 1: p = 0, 0.1, 0.3, 0.5, 0.7, 0.9, 1, J = 1, 3, 6, 10, 14, 18, 20, k ¼ 0:5, a = 0.03, -V = 0.6, -ST = 0.5, -B = 0.7, -R = 0.9 and-C = 0.5. The numerical results are shown in Table 1.

Case 2: p = 0.1, 0.3, 0.5 0.7, 0.8, 0.9, 1, -V = 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, k = 0.5, a = 0.03, -ST = 0.5, -B = 0.7, -R = 0.9,-C = 0.5 and J = 4. The numerical results are shown in Table 2.

Case 3: p = 0.1, 0.3, 0.5 0.7, 0.8, 0.9, 1, -ST = 0.15, 0.25, 0.35, 0.45, 0.55, 0.65, k = 0.5, a = 0.03, -V = 0.6, -B = 0.7, -R = 0.9,-C = 0.5 and J = 4. The numerical results are shown in Table 3.

Case 4: p = 0.1, 0.3, 0.5 0.7, 0.8, 0.9, 1, a = 0.01, 0.04, 0.07, 0.10, 0.13, 0.16, k ¼ 0:5, -V = 0.6, -ST = 0.5, -B = 0.7, -R = 0.9,-C = 0.5 and J = 4. The numerical results are shown in Table 4.

Case 5: p = 0.1, 0.3, 0.5 0.7, 0.8, 0.9, 1, -R = 0.25, 0.35, 0.45, 0.55, 0.65, 0.75, k ¼ 0:5, a = 0.03, -V = 0.6, -ST = 0.5, -B = 0.7,-C = 0.5 and J = 4. The numerical results are shown in Table 5.

Case 6: p = 0.1, 0.3, 0.5 0.7, 0.8, 0.9, 1, -C = 0.1, 0.3, 0.5, 0.7, 0.8, 0.9, k ¼ 0:5, a = 0.03, -V = 0.6, -ST = 0.5, -B = 0.7, -R = 0.9,and J = 4. The numerical results are shown in Table 6.

Case 7: J = 2, 3, 4, 5, 6, 7, 8 -V = 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, p = 0.6, k = 0.5, a = 0.03, -ST = 0.5 -B = 0.7, -R = 0.9and -C = 0.5. The numerical results are shown in Table 7.

Case 8: J = 2, 3, 4, 5, 6, 7, 8, -ST = 0.15, 0.25, 0.35, 0.45, 0.55, 0.65, p = 0.6, k = 0.5, a = 0.03, -V = 0.6 -B = 0.7, -R = 0.9 and-C = 0.5. The numerical results are shown in Table 8.

Case 9: J = 2, 3, 4, 5, 6, 7, 8, a = 0.01, 0.04, 0.07, 0.10, 0.13, 0.16, p = 0.6, k = 0.5, -V = 0.6, -ST = 0.5, -B = 0.7, -R = 0.9 and-C = 0.5. The numerical results are shown in Table 9.

) J = 1 J = 3 J = 6 J = 10 J = 14 J = 18 J = 20

143.80538.1 143.70503 143.70489 143.70489 143.70489 143.70489 143.70489.3 143.48368 143.47959 143.47958 143.47958 143.47958 143.47958.5 143.23529 143.21484 143.21470 143.21470 143.21470 143.21470.7 142.96048 142.89995 142.89884 142.89883 142.89883 142.89883.9 142.65998 142.52121 142.51575 142.51568 142.51568 142.51568

142.50032 142.30274 142.29199 142.29178 142.29177 142.29177


Case 10: J = 2, 3, 4, 5, 6, 7, 8, -R = 0.25, 0.35, 0.45, 0.55, 0.65, 0.75, p = 0.6, k=0.5, a = 0.03, -V = 0.6, -ST = 0.5, -B = 0.7, and-C = 0.5. The numerical results are shown in Table 10.

Case 11: J = 2, 3, 4, 5, 6, 7, 8, -C = 0.1, 0.3, 0.5, 0.7, 0.8, 0.9, p = 0.6, k=0.5, a = 0.03, -V = 0.6, -ST = 0.5, -B = 0.7, -R = 0.9 and-C = 0.5. The numerical results are shown in Table 11.

Table 2

Fðp;-V Þ p = 0.1 p = 0.3 p = 0.5 p = 0.7 p = 0.8 p = 0.9 p = 1

-V =0.20 142.03672 141.92091 141.79904 141.67083 141.60427 141.53604 141.46612-V =0.25 140.84969 140.65834 140.45391 140.23558 140.12098 140.00265 139.88055-V =0.30 140.81341 140.57729 140.32135 140.04405 139.89698 139.74409 139.58527-V =0.35 141.19841 140.93938 140.65476 140.34217 140.17475 139.99960 139.81654-V =0.40 142.26971 141.45563 141.15865 140.82833 140.64979 140.46190 140.26440-V =0.45 142.03672 142.00535 141.70753 141.37235 141.18963 140.99629 140.79198

Table 3

Fðp;-ST Þ p = 0.1 p = 0.3 p = 0.5 p = 0.7 p = 0.8 p = 0.9 p = 1

-ST =0.15 147.02492 146.97394 146.91453 146.84536 146.80673 146.76520 146.72063-ST =0.25 141.14325 141.02052 140.87762 140.71150 140.61882 140.51926 140.41250-ST =0.35 141.62826 141.45474 141.25285 141.01831 140.88754 140.74713 140.59665-ST =0.45 143.00201 142.79182 142.54735 142.26349 142.10528 141.93548 141.75354-ST =0.55 144.37432 144.13666 143.86033 143.53958 143.36087 143.16909 142.96366-ST =0.65 145.58134 145.32243 145.02146 144.67219 144.47763 144.26888 144.04529

Table 4

Fðp;aÞ p = 0.1 p = 0.3 p = 0.5 p = 0.7 p = 0.8 p = 0.9 p = 1

a=0.01 141.46280 141.23331 140.96643 140.65661 140.48396 140.29867 140.10017a=0.04 144.94583 144.72337 144.46467 144.16434 143.99699 143.81738 143.62496a=0.07 149.28706 149.07209 148.82210 148.53188 148.37016 148.19659 148.01065a=0.10 154.93001 154.72303 154.48233 154.20289 154.04717 153.88006 153.70102a=0.13 162.70901 162.51057 162.27979 162.01188 161.86259 161.70237 161.53072a=0.16 174.40808 174.21878 173.99864 173.74308 173.60067 173.44783 173.28409

Table 5

Fðp;-RÞ p = 0.1 p = 0.3 p = 0.5 p = 0.7 p = 0.8 p = 0.9 p = 1

-R=0.25 192.98512 192.77819 192.53754 192.25818 192.10250 191.93543 191.75643-R=0.35 162.98342 162.76939 162.52051 162.23157 162.07056 161.89776 161.71264-R=0.45 153.29046 153.07249 152.81903 152.52477 152.36080 152.18482 151.99630-R=0.55 148.88948 148.66901 148.41263 148.11499 147.94913 147.77113 147.58044-R=0.65 146.49299 146.27078 146.01239 145.71241 145.54524 145.36584 145.17365-R=0.75 145.03162 144.80814 144.54826 144.24657 144.07845 143.89802 143.70472

Table 6

Fðp;-C Þ p = 0.1 p = 0.3 p = 0.5 p = 0.7 p = 0.8 p = 0.9 p = 1

-C =0.1 147.66926 147.64743 147.62187 147.59196 147.57518 147.55708 147.53758-C =0.3 118.00731 117.93245 117.84510 117.74330 117.68638 117.62514 117.55937-C =0.5 143.70490 143.48004 143.21856 142.91501 142.74585 142.56431 142.36982-C =0.7 166.89628 166.49310 166.02537 165.48390 165.18287 164.86036 164.51549-C =0.8 176.09976 175.61407 175.05113 174.40010 174.03849 173.65131 173.23758-C =0.9 183.89340 183.33269 182.68328 181.93291 181.51640 181.07070 180.59470

Table 8

FðJ;-ST Þ J = 2 J = 3 J = 4 J = 5 J = 6 J = 7 J = 8

-ST =0.15 146.91698 146.88780 146.88124 146.87977 146.87944 146.87936 146.87935-ST =0.25 140.88352 140.81340 140.79765 140.79411 140.79331 140.79313 140.79309-ST =0.35 141.26117 141.16214 141.13991 141.13491 141.13379 141.13354 141.13348-ST =0.45 142.55742 142.43756 142.41065 142.40460 142.40324 142.40293 142.40286-ST =0.55 143.87172 143.73626 143.70585 143.69902 143.69748 143.69713 143.69705-ST =0.65 145.03386 144.88634 144.85323 144.84579 144.84411 144.84374 144.84365

Table 9

FðJ;aÞ J = 2 J = 3 J = 4 J = 5 J = 6 J = 7 J = 8

a=0.01 140.97743 140.84659 140.81722 140.81722 140.80913 140.80879 140.80872a=0.04 144.47533 144.34850 144.32003 144.32003 144.31219 144.31187 144.31180a=0.07 148.83240 148.70984 148.68233 148.68233 148.67475 148.67444 148.67437a=0.10 154.49225 154.37424 154.34775 154.34775 154.34045 154.34015 154.34008a=0.13 162.28930 162.17616 162.15077 162.15077 162.14377 162.14348 162.14342a=0.16 174.00771 173.89979 173.87556 173.87556 173.86889 173.86862 173.86855

Table 7

FðJ;-V Þ J = 2 J = 3 J = 4 J = 5 J = 6 J = 7 J = 8

-V =0.20 141.77023 141.73887 141.73574 141.73543 141.73540 141.73540 141.73540-V =0.25 140.41533 140.35388 140.34653 140.34565 140.34554 140.34553 140.34553-V =0.30 140.28403 140.19742 140.18546 140.18381 140.18358 140.18355 140.18354-V =0.35 140.62429 140.51850 140.50209 140.49954 140.49915 140.49908 140.49908-V =0.40 141.13731 141.01819 140.99784 140.99435 140.99375 140.99365 140.99363-V =0.45 141.69568 141.56846 141.54484 141.54045 141.53963 141.53947 141.53945

Table 10

FðJ;-RÞ J = 2 J = 3 J = 4 J = 5 J = 6 J = 7 J = 8

-R=0.25 192.54746 192.42948 192.40300 192.39705 192.39571 192.39541 192.39534-R=0.35 162.53077 162.40874 162.38136 162.37520 162.37381 162.37350 162.37343-R=0.45 152.82947 152.70521 152.67732 152.67104 152.66963 152.66932 152.66924-R=0.55 148.42319 148.29750 148.26929 148.26294 148.26152 148.26119 148.26112-R=0.65 146.02304 145.89635 145.86792 145.86152 145.86008 145.85976 145.85969-R=0.75 144.55897 144.43156 144.40297 144.39653 144.39509 144.39476 144.39469

Table 11

FðJ;-CÞ J = 2 J = 3 J = 4 J = 5 J = 6 J = 7 J = 8

-C =0.1 147.62293 147.61033 147.60750 147.60686 147.60672 147.60668 147.60668-C =0.3 117.84871 117.80577 117.79612 117.79395 117.79347 117.79336 117.79333-C =0.5 143.22934 143.10114 143.07237 143.06590 143.06444 143.06412 143.06404-C =0.7 166.04463 165.81573 165.76440 165.75286 165.75026 165.74968 165.74955-C =0.8 175.07429 174.79899 174.73726 174.72338 174.72026 174.71956 174.71940-C =0.9 182.70999 182.39258 182.32143 182.30543 182.30184 182.30103 182.30084


From Table 1, we observe that F(p, J) decreases as p or J increases. It is reasonable to infer that the optimal F(p, J) occurs atp = 1 and J approaching infinity. In other words, the vacation policy taking multiple vacation policy can minimize F(p, J)under these values of setting cost and system parameters. Tables 2–6 indicate that (i) the total cost decreases as p increases;and (ii) the change of the cost is tiny while p changes. From Table 2 we see that the cost function decreases in some intervalof -V while p is fixed and increases in the adjacent interval. The cost function has its minimum value at some value of -V.


The similar phenomena also hold for Table 3 and Table 6. Table 4 displays that the cost increases as a increases and Table 5displays that the cost decreases as -R increases. The result obtained for a and -R is as expected.

From Tables 7–11, we see that (i) the cost decreases as J increases; and (ii) the change of the cost is very small while Jchanges. Table 7 also shows that the cost function decreases in some interval of -V while J is fixed and increases in the adja-cent interval. Hence the cost function has its minimum value at some value of -V. Similarly, these properties can be seen inTable 8 and Table 11. Table 9 shows that the cost increases as a increases and Table 10 shows that the cost decreases as -R

increases.

7. Conclusions

This paper presented a complete analysis of an unreliable Geo/G/1 queue with startup and closedown times under ran-domized vacations and at most J vacations. The proposed model has the potential to be used in the VoIP system. The distri-butions of system queue length of various server states were derived. Some important performance characteristics such asthe mean lengths of turned-off, startup, busy, breakdown and closedown periods were also obtained. When a customerenters a service system, one of the most critical concerns is his spending time in the system. The analytical solutions forthe distributions and the mean waiting time in system (queue) were derived and the result confirmed Little’s formula. Fromthe view of the cost, the numerical investigation of the presented model can provide the manager to make a smart decision.

Acknowledgement

The author thanks the supports from Ministry of Science and Technology of Taiwan, under Contract No. NSC 100-2221-E-025-009-, and also thanks the referees’ comments leading to improve the paper considerably.

Appendix A

Proof of Theorem 1.Define umðzÞ ¼

P1k¼0zkpm;k;1; J ¼ 1;2; . . . ; J; J þ 5,

umðzÞ ¼X1k¼1

zkpm;k;1; m ¼ J þ 2; J þ 3;

uJþ4ðx; zÞ ¼X1k¼1

X1i¼1

zkxipJþ4;k;i;1:

Multiplying Eqs. (1), (2), (4), (5) and (7) by zk and summing over i after multiplying by xi and adding i yields

x� sx

G1ðx; zÞ ¼ �su1ðzÞ þ �kVðxÞpJþ5;0;1; ðA-1Þ

x� sx

Gmðx; zÞ ¼ �kVðxÞppm�1;0;1 � sumðzÞ; 2 6 m 6 J ðA-2Þ

x� sx

GJþ2ðx; zÞ ¼ SðxÞ kzpJþ1 þ sXJ

m¼1

umðzÞ � �kXJ

m¼1

pm;0;1

!� suJþ2; ðA-3Þ

x� �asx

GJþ3ðx; zÞ ¼ �asðBðxÞ � zÞ

zuJþ3ðzÞ þ suJþ4ðx; zÞ þ BðxÞvbðzÞ

� ; ðA-4Þ

x� sx

GJþ5ðx; zÞ ¼ �kCðxÞpJþ3;1;1 � suJþ5 ðA-5Þ

where s ¼ �kþ kz and vbðzÞ ¼ sðuJþ2ðzÞ þuJþ5ðzÞÞ � �kðpJþ3;1;1 þ pJþ5;0;1Þ.We multiply Eq. (6) by xi and sum over i. Then using the result, we multiply by yj and sum over j. Applying the same

process, multiplying by zk, and summing over k, finally yields

y� sy

GJþ4ðx; y; zÞ ¼ asRðyÞ GJþ3ðx; zÞx

þ ðBðxÞ � zÞz

uJþ3ðzÞ�

þ sðaRðyÞ � 1ÞuJþ4ðx; zÞ þ aBðxÞRðyÞvbðzÞ: ðA-6Þ

Setting x = s in (A-1)–(A-3) and (A-5), and y = s in (A-7), we get

u1ðzÞ ¼�kVðsÞ

spJþ5;0;1; ðA-7Þ


umðzÞ ¼p�kVðsÞ

spm�1;0;1; 2 6 m 6 J; ðA-8Þ

uJþ2ðzÞ ¼SðsÞ kzpJþ1 þ s

PJm¼1umðzÞ � �k

PJm¼1pm;0;1

� �s

; ðA-9Þ

uJþ5ðzÞ ¼�kCðsÞ

s pJþ3;1;1; ðA-10Þ

uJþ4ðx; zÞ ¼aRðsÞ

ð1� aRðsÞÞGJþ3ðx; zÞ

xþ ðBðxÞ � zÞ

zuJþ3ðzÞ þ

BðxÞvbðzÞs

� : ðA-11Þ

Substituting (A-9) into (A-3) and (A-10) into (A-5) and yields

GJþ2ðx; zÞ ¼xðSðxÞ � SðsÞÞvsðzÞ

x� s; ðA-12Þ

GJþ5ðx; zÞ ¼x�kðCðxÞ � CðsÞÞ

x� spJþ3;1;1; ðA-13Þ

where

vsðzÞ ¼ kzpJþ1 þ sXJ

m¼1

umðzÞ � �kXJ

m¼1

pm;0;1

!: ðA-14Þ

Substituting (A-11) into (A-4) and (A-6), respectively, yields

1��as

xð1� aRðsÞÞ

� GJþ3ðx; zÞ ¼

�að1� aRðsÞÞ

sðBðxÞ � zÞz

uJþ3ðzÞ þ BðxÞvbðzÞ�

; ðA-15Þ

GJþ4ðx; y; zÞ ¼yaðRðyÞ � RðsÞÞðy� sÞð1� aRðsÞÞ s GJþ3ðx; zÞ

xþ ðBðxÞ � zÞ

zuJþ3ðzÞ

� þ BðxÞvbðzÞ

� : ðA-16Þ

Let x ¼ r ¼ �asð1�aRðsÞÞ in (A-15), we have

uJþ3ðzÞ ¼zBðrÞvbðzÞsðz� BðrÞÞ ; ðA-17Þ

and

GJþ3ðx; zÞ ¼xz�aðBðxÞ � BðrÞÞvbðzÞ

ðz� BðrÞÞ½xð1� aRðzÞÞ � �as� : ðA-18Þ

Applying (A-17) and (A-18) in (A-16), we obtain

GJþ4ðx; y; zÞ ¼xyzaðBðxÞ � BðrÞÞðRðyÞ � RðsÞÞvbðzÞðy� sÞðz� BðrÞÞ½xð1� aRðsÞÞ � �as� : ðA-19Þ

Making z = 0 in (A-7), (A-8) and (A-10), we obtain

pm;0;1 ¼jm

ppJþ5;0;1; 1 6 m 6 J; ðA-20Þ

umðzÞ ¼�kVðsÞ

sjm�1pJþ5;0;1; 1 6 m 6 J; ðA-21Þ

pJþ3;1;1 ¼1

Cð�kÞpJþ5;0;1: ðA-22Þ

From Eq. (3) and (A-20) yields

pJþ1 ¼�kð�pjþ pjJ � jJþ1Þ

pkð1� jÞ pJþ5;0;1: ðA-23Þ

Applying (A-20)–(A-22) and (A-23), we can re-express vs(z) and vb(z) in term of pJ+5,0,1:

vsðzÞ ¼ �k#ðzÞpJþ5;0;1; ðA-24Þ


vbðzÞ ¼ �k #ðzÞSðsÞ þ CðsÞ � 1Cð�kÞ

� 1 �

pJþ5;0;1; ðA-25Þ

where #ðzÞ ¼ ðVðsÞ�ð1�zÞVð�kÞÞð1�jJÞ�zðj�jJ Þð1�jÞ .

Substituting (A-20) and (A-21) into (A-1) and (A-2) yields

Gmðx; zÞ ¼x�kjm�1ðVðxÞ � VðsÞÞ

x� spJþ5;0;1; 1 6 m 6 J: ðA-26Þ

Substituting (A-24) into (A-12), (A-25) into (A-18) and (A-19), and (A-22) into (A-14) yields

GJþ2ðx; zÞ ¼x�kðSðxÞ � SðsÞÞ#ðzÞ

ðx� sÞ pJþ5;0;1; ðA-27Þ

GJþ3ðx; zÞ ¼xz�k�aðBðxÞ � BðrÞÞ #ðzÞSðsÞ þ CðsÞ�1

Cð�kÞ � 1� �

ðz� BðrÞÞ½xð1� aRðsÞÞ � �as� pJþ5;0;1; ðA-28Þ

GJþ4ðx; y; zÞ ¼xyz�kaðBðxÞ � BðrÞÞðRðyÞ � RðsÞÞ #ðzÞSðsÞ þ CðsÞ�1

Cð�kÞ � 1� �

ðy� sÞðz� BðrÞÞ½xð1� aRðsÞÞ � �as� pJþ5;0;1; ðA-29Þ

GJþ5ðx; zÞ ¼x�kðCðxÞ � CðsÞÞðx� sÞCð�kÞ

pJþ5;0;1: ðA-30Þ

Appendix B

Proof of Theorem 2.Define the following PGFs and consider the five cases.Wjðzj jth vacationÞ � the PGF of the waiting time in the queue of a test customer conditioning that the server state is on

the jth vacation (j = 1, 2, . . ., J);WI(z|idle) � the PGF of the waiting time in the queue of a test customer conditioning that the server state is idle.WST(z|startup) � the PGF of the waiting time in the queue of a test customer conditioning that the server state is startup.WBDðzjbusy or breakdownÞ � the PGF of the waiting time in the queue of a test customer conditioning that the server state

is busy or broken-down.WQ(z) � the PGF of waiting time in the queue of a test customer.

Case 1: The test customer arrives during the jth vacation and find k customers (k P 0) in the system, the test customermust wait: (a) the remaining vacation; (b) a startup time; (c) the generalized service time of the preceding kcustomers.

Wjðzjjth vacationÞ � 1PVj

X1k¼0

X1i¼1

pj;k;izi�1SðzÞ½HðzÞ�k ¼ SðzÞGjðz;HðzÞÞzPVj

¼�kjj�1ðVðzÞ � Vð�kþ kHðzÞÞÞSðzÞ

PVj½z� ð�kþ kHðzÞÞ�

pJþ5;0;1;

1 6 j 6 J: ðB-1Þ

Case 2: The test customer that arrives while the server is idle and immediately is served. In this case, his waiting time inthe queue is a startup time,

WIðzjIdleÞ ¼ SðzÞ: ðB-2Þ

Case 3: The test customer arrives during startup period and find k customers (k P 1) in the system. The customer’s wait-ing time in the queue is remaining startup time plus generalized service time of the k customers in the queue.

WSTðzjstartupÞ ¼P1

k¼1

P1i¼1pJþ2;k;izi�1½HðzÞ�k

PST¼ GJþ2ðz;HðzÞÞ

zPST¼ ½SðzÞ � Sð�kþ kHðzÞÞ�#ðHðzÞÞ

zPST

�kpJþ5;0;1: ðB-3Þ

Case 4: The test customer that arrives during the generalized busy period and finds k customers in the system. Thereforethis customer’s waiting time in the queue consists of: (i) the remaining generalized service time of the customerbeing served; and (ii) the generalized service time of the k-1 customers in the queue.

WHðzjbusy or breakdownÞ ¼

X1

k¼1p_

H;k;izi�1½HðzÞ�k

PB þ PD¼ GHðz;HðzÞÞðPB þ PDÞzHðzÞ ¼

#ðHðzÞÞSð�kþ kHðzÞÞ þ Cð�kþkHðzÞÞ�1Cð�kÞ

h iðPB þ PDÞ½z� ð�kþ kHðzÞÞ�

�kpJþ5;0;1:

ðB-4Þ


where p_

H;k;i represents the probability that there are k customers in the system and the remaining generalized service time ofa customer being served is i slots when the state of the server is in the generalized busy period.

Remark. Setting a = 0 in GJ+3(x, z) yields the PGF GB(x, z) for joint distribution of the number of customers in the system andthe remaining service time immediately after an arbitrary boundary during the busy period for a reliable Geo/G/1 queue withat most J vacations and startup/closedown times. Replacing the service time with the generalized service time in GB(x, z)yields the PGF GH(x, z) for joint distribution of the number of customers in the system and the remaining generalized servicetime immediately after an arbitrary boundary during the generalized busy period as

GHðx; zÞ ¼xzðBðxÞ � BðsÞÞ #ðzÞSðsÞ þ CðsÞ�1

Cð�kÞ � 1� �

ðz� BðsÞÞ½x� s��kpJþ5;0;1:

Case 1: The test customer arrives while the server is closedown and finds k (k P 0) customers in the system. In this case,the waiting time in the queue of the customer consists of the remaining closedown time and the generalized ser-vice time of the k customers in the queue.

WCðzjclosedownÞ ¼ 1PC

X1k¼0

X1i¼1

pJþ5;k;izi�1½HðzÞ�k ¼ GJþ5ðz;HðzÞÞzPC

¼ CðzÞ � Cð�kþ kHðzÞÞPCCð�kÞ½z� ð�kþ kHðzÞÞ�

�kpJþ5;0;1: ðB-5Þ

Therefore, from (B-1) to (B-5), we have

WQ ðzÞ ¼XJ

j¼1

PVjWjðzjjth vacationÞ þ PIWIðzjIdleÞ þ PST WSTðzjstartupÞ þ ðPB þ PDÞWHðzjbusy or breakdownÞ

þ PCWCðzjclosedownÞ:

The waiting time of a customer in the system is easy to obtain WS(z) = WQ(z)H(z) .

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An unreliable Geo/G/1 queue with startup and closedown times under randomized finite vacations