2006 - A Geometric Process Model for MM1 Queueing System

European Journal of Operational Research 168 (2006) 100–121

www.elsevier.com/locate/ejor

Stochastics and Statistics

A geometric process model for M/M/1 queueing systemwith a repairable service station

Yeh Lam a,b,*, Yuan Lin Zhang c, Qun Liu a

a Northeastern University at Qinhuangdao, Qinhuangdao 066004, Chinab Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong

c Department of Mathematics, Southeast University, Nanjing 210018, China

Received 3 July 2003; accepted 19 November 2003

Available online 2 July 2004

Abstract

In this paper, we study a geometric process model for M/M/1 queueing system with a repairable service station. By

introducing a supplementary variable, some queueing characteristics of the system and reliability indices of the service

station are derived. Then a replacement policy N for the service station by which the service station will be replaced

following the Nth failure is applied. An optimal replacement policy N* for minimizing the long-run average cost per

unit time for the service station is then determined.

� 2004 Elsevier B.V. All rights reserved.

Keywords: M/M/1 Queueing system; Geometric process; Supplementary variable; The Laplace transform

1. Introduction

In classical queueing models, it is common to assume that the service station is not subject to failure.However, in many real cases, the service station may experience breakdowns. Therefore, a more realisticqueueing model is a model with a repairable service station. It has been developed tremendously sincethe pioneering work of Thiruvengadam (1963) and Avi-Itzhak and Naor (1963). The fundamental frameof a queueing model with a repairable service station can be stated as follows: assume that customers arriveaccording to an input process, the service times are independent and identically distributed (i.i.d.)random variables. A failed service station after repair is ‘‘as good as new’’. Wang (1995) proposed an

0377-2217/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2003.11.033

* Corresponding author. Address: Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road,

Hong Kong. Fax: +852 28589041.

E-mail addresses: [email protected] (Y. Lam), [email protected] (Y.L. Zhang), [email protected] (Q. Liu).

mailto:[email protected]



Nomenclature

a the ratio of GP {Xn,n = 1,2, . . .}Am (t) the availability of the service station at time tA�mðsÞ the Laplace transform of Am (t)�AmðtÞ the probability that the service station fails at time t�A�mðsÞ the Laplace transform of �AmðtÞb the ratio of GP {Yn,n = 1,2, . . .}Bi the ith busy period in GP M/M/1 model with distribution Bi (x)Bt a busy period in GP M/M/1 model starting with time t with conditional distribution Bt (x)~B a busy period in classicalM/M/1 model with distribution ~B ðxÞ, given that at the beginning the

number of customer in the system is 0 or 1~Bi the ith busy period in classical M/M/1 model with distribution ~Bi ðxÞ~BðnÞ ðxÞ the n-fold convolution of ~B ðxÞ with itself~B� ðsÞ the Laplace–Stieltjes transform of ~B ðxÞc the repair cost rate of the service stationC (N) the average cost under replacement policy Ncp the rate of part of the replacement cost that is proportional to ZF (x) the distribution of snF(n) (x) the n-fold convolution of F with itself, the density is f (n) (x)G (x) the distribution of vnI (t) the number of customers in the system at time tJ (t) 0, if the service station is operating at time t, 1 otherwiseK (t) the number of failures of the service station by time tmf (t) the rate of occurrence of failures at time t with the Laplace transform m�

f ðsÞMf (t) the expected number of failures of the service station by time tN the replacement policy by which the service station is replaced following the Nth failureN* an optimal replacement policy of Np a root of quadratic equation (3.31)pijk (t,m) the probability that the state at time t is (i, j,k) with the Laplace transform p�ijkðs;mÞq another root of quadratic equation (3.31)Qt the waiting time in the queue for a new arrival at time tQt (x) the distribution of Qtr the operating reward rate of the service stationR the basic replacement cost of the service stationSt the waiting time in the system for a new arrival at time tSt (x) the distribution of StTm the time to the first failure of the service stationTm (x) the distribution of Tm with the Laplace–Stieltjes transform T �

mðsÞVi the ith idle period in GP M/M/1 modelXn the operating time of the service station after the (n�1)th repairXn (t) the distribution of Xn

X ðnÞkþ1 ðxÞ the convolution of distributions Xk+1(x), . . .,Xk+ n (x)

Yn the repair time of the service station after the nth failureYn (t) the distribution of Yn

Y. Lam et al. / European Journal of Operational Research 168 (2006) 100–121 101

Y ðnÞkþ1 ðxÞ the convolution of distributions Yk+1(x), . . .,Yk+ n (x)

Z the replacement time of the service stationa the parameter in the distribution of X1b the parameter in the distribution of Y1k the parameter in the distribution of mnl the parameter in the distribution of vnmn the interarrival time between the (n�1)th and nth customerss the expected replacement time E (Z)vn the service time for the nth customer

102 Y. Lam et al. / European Journal of Operational Research 168 (2006) 100–121

M/M/1 queueing model with service station breakdowns. By using Markov renewal process, Cao andCheng (1982) investigated an M/G/1 queueing model with a repairable service station by assuming thatthe operating time of the service station has an exponential distribution and its repair time has a generaldistribution. Thereafter, Tang (1996) also studied the downtime of the service station in theM/G/1 queueingsystem with repairable service station. The study of a queueing model with repairable service station is animportant interdisciplinary topic in queueing theory and reliability theory, because it considers not only thequeueing characteristics for the queing system but also the reliability indices for the service station (see Neu-ts and Lucantoni (1979), Aissani and Artalejo (1998) for more references).However, in the study of a queuing model with a repairable service station, the assumption that a failed

service station after repair will be ‘‘as good as new’’, i.e., a repair is perfect is usually not realistic. In prac-tice, because of accumulated wearing and ageing effect, most of practical systems are deteriorating, so aremost of service stations. This means that the successive operating times of the service station after repairswill be decreasing, while the consecutive repair times after failures will be increasing. Therefore a more di-rect and more reasonable approach is to study a monotone process model for a repairable service station ina queueing system. To begin with, we state the definition of stochastic order here.

Definition 1. Given two random variables X and Y, X is said to be stochastically larger than Y or Y isstochastically less than X, if

P ðX > zÞP PðY > zÞ for all real z:

It is denoted by X P stY or Y 6 stX (see e.g., Ross (1996)). Furthermore, we say that a stochastic process{Xn, n = 1,2, . . .} is stochastically increasing (decreasing) if Xn+1 P st (6 st)Xn for all n = 1,2, . . ..As a simple monotone process, Lam (1988a,b) introduced the geometric process.

Definition 2. A sequence of random variables {Xn,n = 1,2, . . .} forms a geometric process (GP), if thereexists a real a > 0 such that {an� 1Xn,n = 1,2, . . .} forms a renewal process. The real a is called the ratioof the GP.Clearly, GP is a generalization of renewal process. It is also a simple monotone process. In fact, a GP is

stochastically increasing, if 0 < a 6 1, and stochastically decreasing if aP1. If a = 1, the GP will reduceto a renewal process.The GP has been satisfactorily applied to the maintenance problems for a one-component system and a

two-component series, parallel and cold standby systems (for reference, see Lam (2003), Zhang (1994b,1999), and Lam and Zhang (1996a,b). Furthermore, Lam (1992b), Lam and Chan (1998) and Lam et al.(2004) had applied the GP to analysis of data with a trend from a series of events. For more applications ofGP, see Lam et al. (2002), Zhang et al. (2001), Zhang (2002), Lam and Zhang (2003), Sheu (1999), Perez-Ocon and Torres-Castro (2002) for reference.


In this paper, we study a GP model for M/M/1 queueing system with a repairable service station. Themodel is formulated in Section 2, We assume that the successive operating times of the service station forma decreasing GP, while its consecutive repair times constitute an increasing GP. In Section 3, the busyperiod, the probability that the service station is idle, the probability mass function and the distribution ofthe waiting time are studied. Moreover, some reliability indices of the service station including the meantime to the first failure, the availability and the rate of occurrence of failures are considered in Section 4. InSection 5, a replacement policy N by which the service station will be replaced following the Nth failure isintroduced, then an optimal policy N* for minimizing the long-run average cost per unit time is determinedexplicitly.

2. A GP model for M/M/1 queueing system

A GP model for M/M/1 queueing system with a repairable service station is introduced by making thefollowing assumptions.

Assumption 1. At the beginning, a queueing system with a new service station and one repairman isinstalled, and there are m (P 0) customers in the system.

Assumption 2. Assume that the number of arrivals forms a Poisson process with rate k, so that the succes-sive interarrival times {mn,n = 1,2, . . .} are i.i.d. random variables each having an exponential distributionExp(k) with distribution

F ðxÞ ¼ Pðmn6 xÞ ¼ 1� e�kx; xP 0

and 0 otherwise. The customers will be served according to ‘‘first in first out’’ service discipline. Further-more, the consecutive service times {vn,n = 1,2, . . .} are also i.i.d. random variables each having an expo-nential distribution Exp(l) with distribution

GðxÞ ¼ P ðvn6 xÞ ¼ 1� e�lx; xP 0

and 0 otherwise. Assume that l > k.

Assumption 3. Whenever the service station breaks down, it is repaired immediately by the repairman.During the repair time, the system is closed so that no new arrival will be able to join the system. In otherwords, no more customers will arrive. However, the customers waiting in the system will remain in the sys-tem, while the service to the customer being served will be stopped. The system will be recovered after com-pletion of the repair, this means that the service station restarts its service to the customer whose service wasstopped due to failure of the service station with the same exponential distribution Exp(l), and new arrivalwill be able to join the system. If there is no customer in the system, the service station will remain in theoperating condition.

Assumption 4. Let Xn, n = 1,2, . . ., be the operating time of the service station after the (n�1)th repair, andlet Yn, n = 1,2, . . ., be the repair time of the service station after the nth failure. Then {Xn, n = 1,2, . . .}forms a decreasing GP with ratio aP1. Moreover, Xn has an exponential distribution Exp(a

n� 1a) witha > 0 and distribution

X ðxÞ ¼ P ðX 6 xÞ ¼ 1� expð�an�1axÞ xP 0
n n


and 0 otherwise. On the other hand, {Yn,n = 1,2, . . .} follows an increasing GP with ratio 0<b6 1, and Yn

has an exponential distribution Exp(bn� 1b) with b > 0 and distribution

Y nðxÞ ¼ P ðY n6 xÞ ¼ 1� expð�bn�1bxÞ; xP 0

and 0 otherwise.

Assumption 5. The sequences {mn,n = 1,2, . . .}, {vn,n = 1,2, . . .}, {Xn,n = 1,2, . . .}, and {Yn,n = 1,2, . . .}are all independent sequences of independent random variables.

2.1. Remarks

(1) Assumption l > kmakes theM/M/1 queueing system a real queue, otherwise the length of the queueingsystem might tend to infinity.

(2) Assumption 3 is reasonable. For example, consider a computer network system in which several work-stations are connected together to form a local area network as a ‘‘system’’, and a printer is the ‘‘servicestation’’. Whenever a workstation needs to submit a print job that will queue up as a ‘‘customer’’ in thesystem. If the printer breaks down due to cut supply of power or short of ink, it will be repaired, whilethe job being printed for an earlier part of printing time will be stopped, and the printer will close suchthat any new print job will be rejected. In other words, no more ‘‘customer’’ will arrive. However, theprint jobs already submitted will remain in the ‘‘system’’. After completion of the repair, the printer willresume the job, and the ‘‘system’’ will be recovered. Because the exponential distribution is memoryless,the later part of the printing time (the residual service time) for the job stopped when the printer breaksdown will have the same distribution Exp(l).

(3) Assumption 4 just means that the service station is deteriorating. This is a GP model introduced byLam (1988a,b).

Now, the system state (I (t),J (t)) at time t is defined in the following way: I (t) = i, if at time t, the numberof customers in the system is i; J (t) = 0, if at time t the service station is operating, i.e., is in an up state, andJ (t) = 1, if at time t the service station breaks down, i.e., is in a down state. Then, the state space isX = {(i, j), i = 0,1, . . .; j = 0,1}, the set of up states is U = (i,0), i = 0,1, . . ., and the set of down states isD = {(i,1), i = 1,2, . . .}. Clearly, the stochastic process {(I (t),J (t)), t P 0} is not a Markov process.Now, we shall introduce a supplementary variable K (t) such that if the number of failures of the servicestation by time t is k, then

KðtÞ ¼ k; k ¼ 0; 1; . . . :

Afterwards, the process (I (t),J (t),K (t)), t P 0 will be a three-dimensional continuous-time Markovprocess.Since the system state at time t = 0 is (I (0),J (0),K (0)) = (m,0,0), the probability mass function of

the Markov chain at time t is the transition probability from (I (0),J (0),K (0)) = (m,0,0) to (I (t),J (t),K (t)) = (i, j,k), it is given by

pijk ðt;mÞ ¼ PfðIðtÞ; JðtÞ;KðtÞÞ ¼ ði; j; kÞjðIð0Þ; Jð0Þ;Kð0ÞÞ ¼ ðm; 0; 0Þg

with

i ¼ 0; 1; . . . ; j ¼ 0; k ¼ 0; 1; . . .


or

i ¼ 1; 2; . . . ; j ¼ 1; k ¼ 1; 2; . . .
By applying a classical probability analysis, it is straightforward to derive the following differential equa-
tions:

d

dtp00k ðt;mÞ ¼ �kp00k ðt;mÞ þ lp10k ðt;mÞ; k ¼ 0; 1; . . . ; ð2:1Þ

d

dtpi00 ðt;mÞ ¼ �ðk þ l þ aÞpi00 ðt;mÞ þ kpi�100 ðt;mÞ þ lpiþ100 ðt;mÞ; i ¼ 1; 2; . . . ; ð2:2Þ

d

dtpi0k ðt;mÞ ¼ �ðk þ l þ akaÞpi0k ðt;mÞ þ kpi�10k ðt;mÞ þ lpiþ10k ðt;mÞ þ bk�1bpi1k ðt;mÞ;

i ¼ 1; 2; . . . ; k ¼ 1; 2; . . . ; ð2:3Þ

d

dtpi1k ðt;mÞ ¼ �bk�1bpi1k ðt;mÞ þ ak�1api0k�1 ðt;mÞ; i ¼ 1; 2; . . . ; k ¼ 1; 2; . . . : ð2:4Þ

The initial conditions are

pi00 ð0;mÞ ¼ dmi; ð2:5Þ

pijk ð0;mÞ ¼ 0; i ¼ 0; 1; . . . ; j ¼ 0; 1; k ¼ 1; 2; . . . ; ð2:6Þ

where dmi is the Kronecker d defined by

dmi ¼1; i ¼ m;

0; i 6¼ m:

�
Let
p�ijkðs;mÞ ¼Z 1

0

e�stpijk ðt;mÞdt ð2:7Þ

be the Laplace transform of pijk (t,m). Then by taking the Laplace transform on the both sides of (2.1)–(2.4)with the help of initial conditions (2.5) and (2.6), it follows that:

ðsþ kÞp�00kðs;mÞ ¼ lp�10kðs;mÞ þ d0kdm0; k ¼ 0; 1; . . . ; ð2:8Þ

ðsþ k þ l þ aÞp�i00ðs;mÞ ¼ kp�i�100ðs;mÞ þ lp�iþ100ðs;mÞ þ dmi; i ¼ 1; 2; . . . ; ð2:9Þ

ðsþ k þ l þ akaÞp�i0kðs;mÞ ¼ kp�i�1 0kðs;mÞ þ lp�iþ1 0kðs;mÞ þ bk�1bp�i1kðs;mÞ;i ¼ 1; 2; . . . ; k ¼ 1; 2; . . . ; ð2:10Þ

ðsþ bk�1bÞp�i1kðs;mÞ ¼ ak�1ap�i0k�1ðs;mÞ; i ¼ 1; 2; . . . ; k ¼ 1; 2; . . . ð2:11Þ

3. Some queueing characteristics of the system

3.1. The distribution of busy period

It is well known that a busy period is a time interval that starts when the number of customers in thesystem increases from 0 to nonzero, and ends at the time whenever the number of customers in the system


reduces to 0. In classical model forM/M/1 queueing system (or simply the classicalM/M/1 model) in whichthe service station is not subject to failure, assume that at the beginning there is no customer or just 1 cus-tomer in the system, let ~Bi be the ith busy period. Then it is well known that f~B1; ~B2; . . .g are i.i.d. randomvariables each having a common distribution ~B ðxÞ ¼ P ð~B26 x) with the Laplace–Stieltjes transform givenby

~B�ðsÞ ¼Z 1

0

e�std~B ðtÞ ¼sþ k þ l �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsþ k þ lÞ2 � 4kl

q2k

ð3:1Þ

and

Eð~BÞ ¼ � d~B�ðsÞds

js¼0 ¼1

l � k> 0 ð3:2Þ

(see e. g. Kleinrock (1975) for reference). In general, if at the beginning there are m customers in the system,then ~B1; ~B2; . . . are still independent but the distribution of ~B1 is given by

~BðmÞ ðxÞ ¼ ~B � ~B � � � � � ~B ðxÞ; ð3:3Þ
the m-fold convolution of ~B ðxÞ with itself, while the distribution of ~Bi; i ¼ 2; 3; . . . is still given by B (x).In the GP model forM/M/1 queueing system with a repairable service station (or simply the GPM/M/1
model), a busy period will include the total service time plus the total repair time of the service station. LetBi be the ith busy period in the GPM/M/1 model. Then, each Bi will consist of two parts, the first part is thetotal service time of the service station corresponding to busy period ~Bi in the classical M/M/1 model, thesecond part is the total repair time of the service station. Consequently, by summing up the total repair timeof the service station, the sum of the first n+1 busy periods given that the total number of repairs is k isgiven by

Xnþ1i¼1Bi ¼

Xnþ1i¼1

~Bi þXki¼1Y i: ð3:4Þ

Note that the distribution ofPnþ1

i¼1~Bi is given by ~Bðmþ nÞ ðxÞ, the (m+n)-fold convolution of ~B ðxÞ with

itself.Now, denote the convolution of distributions Xk+1(x), . . .,Xk+ n (x) by

X ðnÞkþ1 ðxÞ ¼ X kþ1 � X kþ2 � � � � � X kþn ðxÞ;

and the convolution of distributions Yk+1(x), . . .,Yk+ n (x) by

Y ðnÞkþ1 ðxÞ ¼ Y kþ1 � Y kþ2 � � � � � Y kþn ðxÞ:

Moreover, define

X ð0Þkþ1 ðxÞ 1; Y ð0Þ

kþ1 ðxÞ 1 for xP 0: ð3:5Þ

Then, we have the following result.

Theorem 1. The distribution of the first busy period B1 is given by

B1ðxÞ ¼X1k¼0

Z x

0

Y ðkÞ1 ðx� uÞ½X ðkÞ

1 ðuÞ � X ðkþ1Þ1 ðuÞ�d~Bðm_1Þ ðuÞ: ð3:6Þ

where m _ 1 ¼ maxfm; 1g.


Proof. First assume that m > 0. It follows from (3.4) that

B1ðxÞ ¼ PfB16 xg

¼X1k¼0

P ~B1 þXkj¼1Y j6 x;

Xkj¼1X j < ~B16

Xkþ1j¼1X j

( )ð3:7Þ

¼X1k¼0

Z x

0

PXkj¼1Y j6 x� u;

Xkj¼1X j < u6

Xkþ1j¼1X j

( )d~BðmÞ ðuÞ ð3:8Þ

¼X1k¼0

Z x

0

Y ðkÞ1 ðx� uÞ½X ðkÞ

1 ðuÞ � X ðkþ1Þ1 ðuÞ�d~BðmÞ ðuÞ: ð3:9Þ

Here (3.7) follows from (3.4). Furthermore, because at the beginning there are m customers in the system,then from (3.3) we have (3.8). On the other hand, (3.9) holds since {Xn,n = 1,2, . . .} and {Yn,n = 1,2, . . .}are independent by Assumption 5.Second, if m = 0, the first busy period B1 will start when the number of customer in the system increases

to 1. Then the distribution of B1 is the same as the case m = 1, and (3.6) follows. This completes the proof ofTheorem 1. h

The distribution of other busy period depends on the system state at the beginning of the busy period.Therefore, in general it is different from the distribution of B1. Assume that a busy period Bt starts at time twith state (I (t),J (t),K (t)) = (1,0,‘). Then the conditional distribution Bt (x) is given by the followingtheorem.

Theorem 2. Given that a busy period Bt starts with (I (t), J (t), K(t)) = (1,0,‘), the conditional distribution of Bt

is given by

BtðxÞ ¼X1k¼‘

Z x

0

Y ðk�‘Þ‘þ1 ðx� uÞ X ðk�‘Þ

‘þ1 ðuÞ � X ðk�‘þ1Þ‘þ1 ðuÞ

h id~B ðuÞ: ð3:10Þ

Proof. It follows from (3.4) that

BtðxÞ ¼ PfBt6 x j ðIðtÞ; JðtÞ;KðtÞÞ ¼ ð1; 0; ‘Þg

¼X1k¼‘

P ~BþXkj¼‘þ1

Y j6 x;XL‘þ1 þXkj¼‘þ2

X j < ~B6XL‘þ1 þXkþ1j¼‘þ2

X j

( )ð3:11Þ

¼X1k¼‘

Z x

0

PXkj¼‘þ1

Y j6 x� u;Xkj¼‘þ1

X j < u6Xkþ1j¼‘þ1

X j

( )d~B ðuÞ ð3:12Þ

¼X1k¼0

Z x

0

Y ðk�‘Þ‘þ1 ðx� uÞ X ðk�‘Þ

‘þ1 ðuÞ � X ðk�‘þ1Þ‘þ1 ðuÞ

h id~B ðuÞ: ð3:13Þ

Here the service station after the ‘th repair, an earlier part of operating time X‘+1 was spent in the last busyperiod, and the later part XL‘þ1 of operating time X‘+1 is used for giving service in the present busy period.Then (3.11) follows. Because of memoryless property of exponential distribution, XL‘þ1 will have the samedistribution as X‘+1 has. Thus (3.12) holds. Once again, (3.13) holds since {Xn,n = 1,2, . . .} and {Yn,n = 1,2, . . .} are independent. This completes the proof of Theorem 2. h


3.2. The probability of idle service station

Now, we shall derive the probability that the service station is idle. This is given by the following theo-rem.

Theorem 3. The probability that the service station is idle is given by

p00k ðt;mÞ ¼X1n¼0

Z t

0

F ðnÞ � Y ðkÞ1 ðt � uÞ � F ðnþ1Þ � Y ðkÞ

1 ðt � uÞh i

� X ðkÞ1 ðuÞ � X ðkþ1Þ

1 ðuÞh i

d~BðmþnÞ ðuÞ; k ¼ 1; 2; . . . ð3:14Þ

and

p000 ðt;mÞ ¼X1n¼0

Z t

0

F ðnÞ ðt � uÞ � F ðnþ1Þ ðt � uÞ� �

e�aud~BðmþnÞ ðuÞ; ð3:15Þ

where

F ðnÞ ðxÞ ¼ F � F � � � � � F ðxÞ;

~BðnÞ ðxÞ ¼ ~B � ~B � � � � � ~B ðxÞ;
are, respectively, the n-fold convolutions of F(x) and ~B ðxÞ with themselves, and F ðnÞ � Y ðkÞ
1 ðxÞ is the convolution

of F(n) (x) and Y ðkÞ1 ðxÞ.

Proof. First of all, assume that m > 0 and k > 0. Since the idle period Vi and busy period Bi in the queueingsystem will occur alternatively, we have

p00k ðt;mÞ ¼ PfðIðtÞ; JðtÞ;KðtÞÞ ¼ ð0; 0; kÞ j ðIð0Þ; Jð0Þ;Kð0ÞÞ ¼ ð0; 0;mÞg

¼X1n¼0

P B1 þXni¼1

ðV i þ Biþ1Þ < t6B1 þXni¼1

ðV i þ Biþ1Þ þ V nþ1;(

total number of repairs on the service station by time t is k

)

¼X1n¼0

PXni¼1V i þ

Xnþ1i¼1

~Bi þXki¼1Y i < t6

Xnþ1i¼1V i þ

Xnþ1i¼1

~Bi þXki¼1Y i;

(

Xki¼1X i<

Xnþ1i¼1

~Bi6Xkþ1i¼1X i

)ð3:16Þ

¼X1n¼0

Z t

0

PXni¼1V i þ

Xki¼1Y i < t � u6

Xnþ1i¼1V i

(þXki¼1Y i;Xki¼1X i < u6

Xkþ1i¼1X i

)d~BðmþnÞ ðuÞ

ð3:17Þ

¼X1n¼0

Z t

0

½F ðnÞ � Y ðkÞ1 ðt � uÞ � F ðnþ1Þ � Y ðkÞ

1 ðt � uÞ� X ðkÞ1 ðuÞ � X ðkþ1Þ

1 ðuÞh i

d~BðmþnÞ ðuÞ: ð3:18Þ

Here (3.16) follows from (3.4), while (3.17) is due to that the distribution ofPnþ1

i¼1~Bi is given by ~BðmþnÞðxÞ. On

the other hand, as {Vi, i = 1,2, . . .} depends on {mn,n = 1,2, . . .} only, then Assumption 5 implies that{Vi, i = 1,2, . . .} and {Yi, i = 1,2, . . .} are independent, hence (3.18) follows.


Now, assume that m > 0 but k = 0. A similar argument shows that

p000 ðt;mÞ ¼ PfðIðtÞ; JðtÞ;KðtÞÞ ¼ ð0; 0; 0Þ j ðIð0Þ; Jð0Þ;Kð0ÞÞ ¼ ðm; 0; 0Þg

¼X1n¼0

PXni¼1V i þ

Xnþ1i¼1

~Bi < t6Xnþ1i¼1V i þ

Xnþ1i¼1

~Bi;Xnþ1i¼1

~Bi6X 1

( )

¼X1n¼0

Z t

0

PXni¼1V i < t � u6

Xnþ1i¼1V i; u6X 1

( )d~BðmþnÞ ðuÞ

¼X1n¼0

Z t

0


ð1� X 1ðuÞÞd~BðmþnÞ ðuÞ

¼X1n¼0

Z t

0


e�aud~BðmþnÞ ðuÞ: ð3:19Þ

If m = 0, then we can think B1 = 0, the proof is similar. This completes the proof of Theorem 3. h

In particular, it follows from (3.15) that

p000 ðt; 0Þ ¼ e�kt þX1n¼1

Z t

0


e�aud~BðnÞ ðuÞ: ð3:20Þ

3.3. The probability mass function pijk(t,m)

Now we shall determine the Laplace transform of pijk (t,m) recursively. For this purpose, some wellknown results are reviewed here. Assume that X1, . . .,Xn are independent and Xi has exponential distribu-tion Exp(ki). Then the density function of

Pni¼1X i is given by

xðnÞ1 ðxÞ ¼ ð�1Þn�1k1k2 � � � knXni¼1

e�kixQnj¼1j 6¼i

ðki � kjÞfor x > 0 ð3:21Þ

and 0 otherwise (see Chiang (1980) for reference). In particular, if a > 1 and ki = ai� 1a, then (3.21) gives

xðnÞ1 ðxÞ ¼ ð�1Þn�1anðn�1Þ2 a

Xni¼1

e�ai�1ax

Qnj¼1j 6¼i

ðai�1 � aj�1Þfor x > 0 ð3:22Þ

and 0 otherwise. Consequently, the distribution function ofPn

i¼1X i is given by

X ðnÞ1 ðxÞ ¼ 1�

Xni¼1

Ynj¼1j 6¼i

aj�1

aj�1 � ai�1

0BB@

1CCAe�ai�1ax for xP 0 ð3:23Þ

and 0 otherwise. However, if a = 1, thenPn

i¼1X i will have a gamma distribution with, respectively, the fol-lowing density and distribution functions:

xðnÞ1 ðxÞ ¼ an

ðn� 1Þ! xn�1e�ax for x > 0 ð3:24Þ


and 0 otherwise, and

X ðnÞ1 ðxÞ ¼

X1i¼n

ðaxÞi

i!e�ax for xP 0 ð3:25Þ

and 0 otherwise.By using Theorem 3, an explicit expression for p�00kðs;mÞ is given by the following theorem.

Theorem 4

p�00kðs;mÞ ¼

Qkj¼1

bj�1bsþbj�1b

Pkþ1i¼1ai�k�1

Qkþ1r¼1r 6¼i

ar�1

ar�1�ai�1

0B@

1CA ½~B�ðsþai�1aÞ�m

sþk�k~B�ðsþai�1aÞ for a > 1;

ð�1Þk ak

k!

Qkj¼1

bj�1bsþbj�1b

( )dk

dak½~B�ðsþaÞ�m

sþk�k~B�ðsþaÞ

n ofor a ¼ 1:

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

ð3:26Þ

Proof. Assume first a > 1, then from (3.14) and (3.23), we have

p�00kðs;mÞ ¼Z 1

0

e�stp00k ðt;mÞdt

¼Z 1

0

e�stX1n¼0

Z t

0

½F ðnÞ � Y ðkÞ1 ðt � uÞ � F ðnþ1Þ � Y ðkÞ

1 ðt � uÞ�(

�½X ðkÞ1 ðuÞ � X ðkþ1Þ

1 ðuÞ�d~BðmþnÞ ðuÞ)dt

¼X1n¼0

Z 1

0

Z 1

0

e�sv½F ðnÞ � Y ðkÞ1 ðvÞ � F ðnþ1Þ � Y ðkÞ

1 ðvÞ�dv� �

� e�su½X ðkÞ1 ðuÞ � X ðkþ1Þ

1 ðuÞ�d~BðmþnÞ ðuÞ

¼X1n¼0

1

sk

sþ k

� �n� k

sþ k

� �nþ1" # Ykj¼1

bj�1bsþ bj�1b

!

�Z 1

0

e�suXkþ1i¼1

Ykþ1r¼1r 6¼i

ar�1

ar�1 � ai�1 e�ai�1au �

Xki¼1

Ykr¼1r 6¼i

ar�1

ar�1 � ai�1 e�ai�1au

8><>:

9>=>;d~BðmþnÞ ðuÞ

¼X1n¼0

kn

ðsþ kÞnþ1Ykj¼1

bj�1bsþ bj�1b

!Xkþ1i¼1ai�k�1

Ykþ1r¼1r 6¼i

ar�1

ar�1 � ai�1

0BB@

1CCAZ 1

0

e�ðsþai�1aÞud~BðmþnÞ ðuÞ

¼Ykj¼1

bj�1bsþ bj�1b

Xkþ1i¼1ai�k�1

Ykþ1r¼1r 6¼i

ar�1

ar�1 � ai�1

0BB@

1CCAX

1

n¼0

kn

ðsþ kÞnþ1½~B�ðsþ ai�1aÞ�mþn

¼Ykj¼1

bj�1bsþ bj�1b

Xkþ1i¼1ai�k�1

Ykþ1r¼1r 6¼i

ar�1

ar�1 � ai�1

0BB@

1CCA ½~B�ðsþ ai�1aÞ�m

sþ k � k~B�ðsþ ai�1aÞ:


For a = 1, by a similar way, it follows from (3.14) and (3.25) that

p�00kðs;mÞ ¼Z 1

0

e�stp00k ðt;mÞdt

¼X1n¼0

Z 1

0

Z 1

0

e�sv½F ðnÞ � Y ðkÞ1 ðvÞ � F ðnþ1Þ � Y ðkÞ

1 ðvÞ�dv� �

� e�su X ðkÞ1 ðuÞ � X ðkþ1Þ

1 ðuÞh i

d~BðmþnÞ ðuÞ

¼X1

n¼01

sk

sþ k

� �n� k

sþ k

� �nþ1" # Ykj¼1

bj�1bsþ bj�1b

!Z 1

0

e�suðauÞk

k!e�aud~BðmþnÞ ðuÞ

¼Ykj¼1

bj�1bsþ bj�1b

X1n¼0

kn

ðsþ kÞnþ1ð�1Þk ak

k!dk½~B�ðsþ aÞ�mþn

dak

¼ ð�1Þk ak

k!

Ykj¼1

bj�1bsþ bj�1b

( )dk

dak½~B�ðsþ aÞ�m

sþ k � k~B�ðsþ aÞ

� �:

This completes the proof of Theorem 4. h

Especially, for k = 0 and aP 1, from (3.26) we obtain

p�000ðs;mÞ ¼½~B�ðsþ aÞ�m

sþ k � k~B�ðsþ aÞ: ð3:27Þ

On the basis of Theorem 4, we can derive the Laplace transform p�100ðs;mÞ of p100 (t,m) directly. In fact, itfollows from (2.8) that

p�100ðs;mÞ ¼sþ k

lp�000ðs;mÞ �

dm0l

¼ ðsþ kÞ½~B�ðsþ aÞ�m

l½sþ k � k~B�ðsþ aÞ�� dm0

l: ð3:28Þ

Furthermore, we have the following theorem.

Theorem 5

p�i00ðs; 0Þ ¼kðpi � qiÞ~B�ðsþ aÞ þ lðpqi � piqÞ

lðp � qÞ½sþ k � k~B�ðsþ aÞ�; i ¼ 0; 1; . . . ; ð3:29Þ

p�i00ðs;mÞ ¼

½ðpi�qiÞðsþkÞþlðpqi�piqÞ�½~B�ðsþaÞ�mlðp�qÞ½sþk�k~B�ðsþaÞ� � pi�m�qi�m

lðp�qÞ ;

i ¼ 0; 1; . . . ; m ¼ 1; 2; . . . ; i� 1;½ðpi�qiÞðsþkÞþlðpqi�piqÞ�½~B�ðsþaÞ�m

lðp�qÞ½sþk�k~B�ðsþaÞ� ;

i ¼ 0; 1; . . . ; m ¼ i; iþ 1; . . .

8>>>><>>>>:

ð3:30Þ

where p and q are two roots of the quadratic equation

lt2 � ðsþ k þ l þ aÞt þ k ¼ 0: ð3:31Þ

Proof. Assume that m = 0, then from (2.9) we have

p�iþ100ðs; 0Þ ¼sþ k þ l þ a

lp�i00ðs; 0Þ �

klp�i�100ðs; 0Þ; i ¼ 1; 2; . . . ð3:32Þ


Because the discriminant of Eq. (3.31) is positive, p and q are two distinct real roots of (3.31). Then (3.32)becomes

p�iþ100ðs; 0Þ � pp�i00ðs; 0Þ ¼ qðp�i00ðs; 0Þ � pp�i�100ðs; 0ÞÞ ð3:33Þ

or

p�iþ100ðs; 0Þ � qp�i00ðs; 0Þ ¼ pðp�i00ðs; 0Þ � qp�i�100ðs; 0ÞÞ: ð3:34Þ

By iteration, it is straightforward that

p�i00ðs; 0Þ � pp�i�100ðs; 0Þ ¼ qi�1 ðp�100ðs; 0Þ � pp�000ðs; 0ÞÞ
and
p�i00ðs; 0Þ � qp�i�100ðs; 0Þ ¼ pi�1 ðp�100ðs; 0Þ � qp�000ðs; 0ÞÞ:
Consequently,
p�i00ðs; 0Þ ¼pi � qip � q p

�100ðs; 0Þ þ

pqi � piqp � q p�000ðs; 0Þ ¼

kðpi � qiÞ~B�ðsþ aÞ þ lðpqi � piqÞlðp � qÞðsþ k � k~B�ðsþ aÞÞ

; i ¼ 0; 1; . . .

Thus (3.29) follows. To prove (3.30), consider the case m< i. By a similar argument to (3.33), from (2.9) wehave

p�iþ100ðs;mÞ � pp�i00ðs;mÞ ¼ qi�m ðp�mþ100ðs;mÞ � pp�m00ðs;mÞÞ ¼ qi�m qðp�m00ðs;mÞ � pp�m�1 00ðs;mÞÞ �1

l

% &

¼ qi�mþ1 ðp�m00ðs;mÞ � pp�m�1 00ðs;mÞÞ �1

lqi�m:

Therefore, we have

p�i00ðs;mÞ � pp�i�100ðs;mÞ ¼ qi�1 ðp�100ðs;mÞ � pp�000ðs;mÞÞ �1

lqi�m�1: ð3:35Þ

Similarly

p�i00ðs;mÞ � qp�i�100ðs;mÞ ¼ pi�1 ðp�100ðs;mÞ � qp�000ðs;mÞÞ �1

lpi�m�1: ð3:36Þ

Then for the case m< i, (3.30) follows from (3.35) and (3.36) directly. On the other hand, for the casem P i, (3.30) is trivial. This completes the proof of Theorem 5. h

As a result, on the basis of Theorems 4 and 5, all the Laplace transforms p�ijkðs;mÞ can be determinedfrom Eqs. (2.8)–(2.11) recursively, so are the probability mass functions pijk (t,m).As an application, now we are able to determine the Laplace transform of the distribution of I (t), i.e., the

Laplace transform of the distribution of the number of customers in the system at time t. In fact

P ðIðtÞ ¼ i j ðIð0Þ; Jð0Þ;Kð0ÞÞ ¼ ðm; 0; 0ÞÞ ¼X1j¼0

X1k¼0

PfðIðtÞ; JðtÞ;KðtÞÞ ¼ ði; j; kÞ j ðIð0Þ; Jð0Þ;Kð0ÞÞ

¼ ðm; 0; 0Þg ¼X1j¼0

X1k¼0

pijk ðt;mÞ:

Therefore the Laplace transform of P ðIðtÞ ¼ i j ðIð0ÞJð0Þ;Kð0ÞÞ ¼ ðm; 0; 0ÞÞ can be determined accord-ingly.


3.4. The distribution of waiting time

Let Qt be the waiting time in the queue for a new arrival arriving at time t. Assume that the systemstate at time t is (I (t),J (t), K (t)) = (i,0,k). Then when the new arrival arrives, there are i customers in thequeue and the first one is being served. The new arrival must wait for the time until the services to these icustomers are completed. As the earlier part of the service time to the first customer was conducted be-fore the new arrival arrives, then the total service time to these i customers since the new arrival arrives isgiven by

Gi ¼ vL1 þXij¼2

vj;

Gi is the sum of the later part vL1 of the service time v1 for the first customer and the total service time forthe other i�1 customers. Because of the memoryless property of exponential distribution, vL1 and v1 willhave the same exponential distribution function G (x). Consequently, the distribution of Gi will be givenby

GðiÞ ðxÞ ¼ G � G � � � � � GðxÞ;
the i-fold convolution of G with itself. Furthermore, at time t, J (t) = 0 and K (t) = k, then the servicestation is in an up state and the service station has been repaired for k times. To complete the servicesto i customers in the queue, the services may be completed in time XLkþ1, the later part of operatingtime Xk+1 of the service station after the kth repair, since the earlier part of operating time X
Lkþ1

has been used for service before the new arrival arrives. As a result, the event Qt6 x is equivalent tothe event

Gi6 x; Gi6XLkþ1:

In general, the services may be completed after n-time more repairs on the service station, n = 1,2, . . ., thenthe real time for completion of services to these i customers is Gi þ

Pkþnj¼kþ1Y j. Consequently, the event

Qt6 x now is equivalent to

Gi þXkþnj¼kþ1

Y j6 x; XLkþ1 þXkþnj¼kþ2

X j < Gi6XLkþ1 þXkþnþ1j¼kþ2

X j:

Note that when the new arrival arrives, the service station has served for the earlier part of operating timeXk+1. The later part X

Lkþ1 of Xk+1 is the residual operating time of the service station. Again, due to the

memoryless property of exponential distribution, XLkþ1 and Xk+1 will have the same distribution functionXk+1(x). According to the above explanation, we can prove the following theorem.

Theorem 6. Let the distribution of Qt be Qt (x), then

QtðxÞ ¼ Qtð0Þ þX1i¼1

X1k¼0

pi0k ðt;mÞ(Z x

0

e�akaudGðiÞ ðuÞ

þX1n¼1

Z x

0

Y ðnÞkþ1 ðx� uÞ X

ðnÞkþ1 ðuÞ � X

ðnþ1Þkþ1 ðuÞ

h idGðiÞ ðuÞ

)for x > 0; ð3:37Þ

with Qtð0Þ ¼P1

k¼0p00k ðt;mÞ:


Proof. For x = 0, the result is trivial. Now consider the case x > 0, we have

QtðxÞ ¼ PðQt6 xÞ ¼ PðQt ¼ 0Þ þ Pð0 < Qt6 xÞ

¼ Qtð0Þ þX1i¼1

X1k¼0

pi0k ðt;mÞPfQt 6 xjðIðtÞ; JðtÞ;KðtÞÞ ¼ ði; 0; kÞg

¼ Qtð0Þ þX1i¼1

X1k¼0

pi0k ðt;mÞ

��PðGi6 x; Gi6XLkþ1Þ þ

X1n¼1

P�Gi þ

Xkþnj¼kþ1

Y j6 x; XLkþ1 þXkþnj¼kþ2

X j < Gi6XLkþ1 þXkþnþ1j¼kþ2

X j

��

¼ Qtð0Þ þX1i¼1

X1k¼0

pi0k ðt;mÞ(Z x

0

ð1� Xkþ1 ðuÞÞdGðiÞ ðuÞ

þX1n¼1

Z x

0

PXkþnj¼kþ1

Y j6 x� u;XLkþ1 þXkþnj¼kþ2

X j < u6XLkþ1 þXkþnþ1j¼kþ2

X j

!dGðiÞ ðuÞ

)

¼ Qtð0Þ þX1i¼1

X1k¼0

pi0k ðt;mÞ(Z x

0

e�akaudGðiÞ ðuÞ þ

X1n¼1

Z x

0



ðnþ1Þkþ1 ðuÞ

h idGðiÞ ðuÞ

):

This completes the proof of Theorem 6. h

Now, let St be the waiting time in the system for a new arrival arriving at time t. It is equal to the waitingtime in the queue plus the service time to the new arrival. Thus, by a similar argument, we can easily obtainthe following theorem.

Theorem 7. Let the distribution of St be St (x), then

StðxÞ ¼X1i¼0

X1k¼0

pi0k ðt;mÞ(Z x

0

e�akaudGðiþ1Þ ðuÞþ

X1n¼1

Z x

0



ðnþ1Þkþ1 ðuÞ

h idGðiþ1Þ ðuÞ

);

for x > 0

and St (x) = 0 otherwise.

4. Some reliability indices of the service station

4.1. Mean time to the first failure (MTTFF)

Given that there are m customers in the system at the beginning, let Tm be the time to the first failure ofthe service station, and let the distribution of Tm be

T mðxÞ ¼ PfTm6 x j ðIð0Þ; Jð0Þ;Kð0ÞÞ ¼ ðm; 0; 0Þg: ð4:1Þ

Then we have the following theorem.

Theorem 8. The Laplace–Stieltjes transform of Tm (x) is given by

T �mðsÞ ¼

asþ a

� as½~B�ðsþ aÞ�m

ðsþ aÞ½sþ k � k~B�ðsþ aÞ�: ð4:2Þ


Proof. Once again, let Vi be the ith idle period. Note that all the busy periods prior to the first failure arethe same as that in the classical M/M/1 model. Then it follows from (4.1) that

TmðxÞ ¼PfX 16 x;X 16 ~B1 j ðIð0Þ; Jð0Þ;Kð0ÞÞ ¼ ðm; 0; 0Þg

þX1n¼1

PXni¼1V i þ X 16 x;

Xni¼1

~Bi < X 16Xnþ1i¼1

~BijðIð0Þ; Jð0Þ;Kð0ÞÞ ¼ ðm; 0; 0Þ( )

ð4:3Þ

¼Z x

0

P ð~B1 P uÞdX 1ðuÞ þX1n¼1

Z x

0

PXni¼1V i6 x� u;

Xni¼1

~Bi < u6Xnþ1i¼1

~Bi

( )dX 1ðuÞ

¼Z x

0

ð1� ~BðmÞ ðuÞÞae�auduþX1n¼1

Z x

0

F ðnÞ ðx� uÞ½~Bðmþn�1Þ ðuÞ � ~BðmþnÞ ðuÞ�ae�audu; ð4:4Þ

where (4.3) is obtained by conditioning on the number of idle periods prior the first failure of the servicestation, while (4.4) is due to (3.3). h

Let the common density of mn be f (x), n = 1,2, . . ., and let

f ðnÞ ðxÞ ¼ dFðnÞ ðxÞdx

¼ f � f � � � � f ðxÞ

be the n-fold convolution of f with itself. Then by taking the Laplace–Stieltjes transform of Tm (x), we have

T �mðsÞ¼

Z 1

0

e�sxdTmðxÞ

¼Z 1

0

e�sxð1� ~BðmÞ ðxÞÞae�axdx

þX1n¼1

Z 1

0

e�sxZ x

0

f ðnÞ ðx�uÞ ~Bðmþn�1Þ ðuÞ� ~BðmþnÞ ðuÞ� �

ae�audu� �

dx

¼ asþa

�Z 1

0

a~BðmÞ ðxÞe�ðsþaÞxdxþX1n¼1

Z 1

0

e�svf ðnÞ ðvÞdvZ 1

0

ae�ðsþaÞu ~Bðmþn�1Þ ðuÞ� ~BðmþnÞ ðuÞ� �

du� �

¼ asþa

�a½~B�ðsþaÞ�m

sþaþX1n¼1

�ksþk

�n asþa

½~B�ðsþaÞ�mþn�1�½~B�ðsþaÞ�mþnn o

¼ asþa

�a½~B�ðsþaÞ�m

sþaþak½~B�ðsþaÞ�mð1� ~B�ðsþaÞÞ

ðsþaÞ½sþk�k~B�ðsþaÞ�

¼ asþa

� as½~B�ðsþaÞ�m

ðsþaÞ½sþk�k~B�ðsþaÞ�:

Consequently, with the help of (3.1), the expectation of Tm, i.e., the mean time to the first failure(MTTFF) of the service station, will be given by

EðT mÞ ¼Z 1

0

tdTmðtÞ ¼ � dT�mðsÞds

''''s¼0

¼ 1aþ ½~B�ðaÞ�m

k½1� ~B�ðaÞ�

¼ 1aþ

a þ k þ l �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða þ k þ lÞ2 � 4kl

q� �m2m�1kmf�a þ k � l þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða þ k þ lÞ2 � 4kl

qg: ð4:5Þ


4.2. Availability of the service station

Let the availability of the service station at time t be

AmðtÞ ¼ Pfthe service station operates at time tjðIð0Þ; Jð0Þ;Kð0ÞÞ ¼ ðm; 0; 0Þg:
Moreover, let the probability that the service station breaks down at time t be
�AmðtÞ ¼ Pfthe service station breaks down at time tjðIð0Þ; Jð0Þ;Kð0ÞÞ ¼ ðm; 0; 0Þg:
Now, denote the Laplace transforms of Am (t) and �AmðtÞ by A�
mðsÞ and �A�mðsÞ respectively. Then, the fol-

lowing theorem follows directly.

Theorem 9

A�mðsÞ ¼

X1i¼0

X1k¼0

p�i0kðs;mÞ:

Proof. It is clear that

AmðtÞ ¼X1i¼0

X1k¼0

PfðIðtÞ; JðtÞ;KðtÞÞ ¼ ði; 0; kÞjðIð0Þ; Jð0Þ;Kð0ÞÞ ¼ ðm; 0; 0Þg ¼X1i¼0

X1k¼0

pi0k ðt;mÞ:

Therefore

A�mðsÞ ¼

X1i¼0

X1k¼0

p�i0k ðs;mÞ:

Furthermore, due to the fact AmðtÞ þ �AmðtÞ ¼ 1, we have

A�mðsÞ þ �A

�mðsÞ ¼

1

s:

Thus

�A�mðsÞ ¼

1

s�X1i¼0

X1k¼0

p�i0kðs;mÞ: �

4.3. The rate of occurrence of failures (ROCOF)

The ROCOF is one of important indices in reliability theory. LetMf (t) be the expected number of failuresof the service station that have occurred by time t, then its derivativemf ðtÞ ¼ M 0

f ðtÞ is called the rate of occur-rence of failures (ROCOF). According to Lam (1997), the ROCOF can be evaluated in the following way:

mf ðtÞ ¼X1i¼1

X1k¼0

akapi0k ðt;mÞ:

Therefore, the Laplace transform of mf (t) is given by

m�f ðsÞ ¼

X1i¼1

X1k¼0

akap�i0kðs;mÞ:

Since p�i0kðs;mÞ has been determined in Section 3, we can then evaluate m�f ðsÞ and hence mf (t) accordingly.


5. Optimal replacement model for service station

5.1. Long-run average cost under policy N

For aM/M/1 queueing system with a repairable service station, Zhang (1994a) considered a replacementpolicy based on the number of customers� service completed. In this section, a replacement policy N bywhich the service station will be replaced by a new identical one following the Nth failure is studied. Weshall then make an additional assumption below.

Assumption 6. Assume that the repair cost rate of the service station is c, the operating reward rate of theservice station is r, and the replacement cost comprises two parts: one part is the basic replacement cost R,the other part is proportional to the replacement time Z at rate cp with E (Z) = s. Moreover, {Xn,n = 1, 2,. . .}, and {Yn,n = 1,2, . . .} and Z are independent.

Our objective is to determine an optimal policy N*, such that the long-run average cost per unit time isminimized.To do this, we say that a cycle is completed if a replacement of service station is completed. In other

words, a cycle is a time interval between the installation and the first replacement or two successivereplacements. Thus the successive cycles and the costs incurred in each cycle will form a renewal rewardprocess. Assume that a replacement policy N is applied, then by using the standard result in renewal re-ward process, the long-run average cost per unit time (or simply the average cost) of the service station isgiven by

CðNÞ ¼ the average cost incurred in a cyclethe average length of a cycle

¼E c

PN�1n¼1Y n þ Rþ cpZ � r

PNn¼1Xn

� �

EPNn¼1Xn þ

PN�1n¼1Y n þ Z

� �

¼cb

PN�1n¼1

1bn�1 þ Rþ cps � r

a

PNn¼1

1an�1

1a

PNn¼1

1an�1 þ 1

b

PN�1n¼1

1bn�1 þ s

ð5:1Þ

(see Ross (1996) for reference).Lam (2003) had derived the optimal policy N* for minimizing the average cost C (N). To do this, we shall

first define an auxiliary function

gðNÞ ¼ðcþ rÞ

PNk¼1ak �

PN�1k¼1bk þ asaN

� �½Rþ ðcp þ rÞs�ðbbN�1 þ aaN Þ : ð5:2Þ

Then it can be shown that

CðN þ 1Þ¼>

<CðNÞ () gðNÞ¼

>

<1

and g (N) is nondecreasing in N. Consequently, Lam (2003) showed that an optimal replacement policy N*

for the service station is given by

N � ¼ minfN jgðNÞP 1g: ð5:3Þ


Furthermore, if g (N*) > 1, then the optimal policy N* is unique.We can also determine an optimal replacement policy N* for minimizing average cost C (N) numerically.

In fact, it follows from (5.1) that for a > 1 and 0<b<1, we have

CðNÞ ¼c

ð1�bÞb ðb2�N � bÞ þ ðRþ cpsÞ � rða�1Þa ða� a1�N Þ

1ða�1Þa ða� a1�N Þ þ 1

ð1�bÞb ðb2�N � bÞ þ s: ð5:4Þ

Now let C (x) be a function defined by substituting x for N in C (N). Then we can differentiate C (x) anddenote the numerator of C 0 (x) by

DðxÞ ¼ � cb2�x ln b

ð1� bÞb � ra1�x ln a

ða� 1Þa

% &ðb2�x � bÞð1� bÞb þ ða� a1�xÞ

ða� 1Þa þ s

% &

� cðb2�x � bÞð1� bÞb þ ðRþ cpsÞ �

rða� a1�xÞða� 1Þa

% &� b2�x ln bð1� bÞb þ a1�x ln a

ða� 1Þa

% &: ð5:5Þ

Obviously, C 0 (x) and D (x) will have the same sign. Therefore, if there exists x1<x2 such that D (x1)<0 andD (x2) > 0. Then, an integer N* might be found such that x16N*6 x2 for minimizing C (N).

5.2. A numerical example

Now, we study a numerical example with the following parameter values: a = 1.05, b = 0.95, R = 4000,a = 0.04, b = 0.1, c = 5, r = 60, cp = 10 and s = 24. Two methods based on (5.3) and (5.5) respectively areused for finding an optimal policy N*.

Method 1

Step 1. Initially set N = 1, according to (5.2) evaluate g (1) = 0.1659<1. Then set N = 20, calculateg (20) = 2.0233 > 1. Thus from (5.3), 1<N*6 20.

Step 2. Reduce the range of N*. Finally find g (13) = 1.0321 > 1, and g (12) = 0.9108. Then N* = 13 is theunique optimal policy, and C (N*) = C (13) = �22.5516 is the minimum average cost.

Method 2

Step 1. Initially set N = 1, according to (5.5) evaluate D (1) = �162880<0. Then set N = 20, calculateD (20) = 191200 > 0. Thus, 16N*6 20.

Step 2. Reduce the range of N*. Finally find N* = 13 and C (N*) = C (13) = �22.5516 are respectively theoptimal policy and the minimum average cost.

The numerical results are presented in Fig. 1 and Table 1 respectively.To study the influence of the ratio of GP on the optimal solution, we tabulate the optimal replacement

policy N* and the minimum average cost C (N*) for different values of a > 1 and 0<b<1 in Tables 2 and 3,respectively.Note that the GP will become a RP when a = b = 1. In Table 2, when b and other parameters are fixed,

then N* is nonincreasing in a, but C (N*) is nondeareasing in a. In Table 3, when a and other parameters arefixed, then N* is nondecreasing in b, but C (N*) is nonincreasing in b. According to Tables 2 and 3, we cansee that the optimal replacement policy N* and minimum average cost C (N*) are sensitive to the change of aor b, when the other parameters are fixed. Therefore, introducing the GP M/M/1 model is necessary andappropriate.

Table 1Results obtained from formulas (5.1) and (5.2)

N C (N) g (N) N C (N) g (N) N C (N) g (N)

1 55.9184 0.1659 11 �22.2155 0.7970 21 �20.3455 2.17482 16.4405 0.1860 12 �22.4595 0.9108 22 �19.8693 2.32633 0.4613 0.2153 13 �22.5516 1.0321 23 �19.3698 2.47714 �7.9821 0.2540 14 �22.5228 1.1601 24 �18.8509 2.62655 �13.0588 0.3025 15 �22.3959 1.2941 25 �18.3162 2.77406 �16.3380 0.3609 16 �22.1887 1.4333 26 �17.7686 2.91897 �18.5428 0.4291 17 �21.9148 1.5766 27 �17.2109 3.06098 �20.0526 0.5071 18 �21.5853 1.7234 28 �16.6456 3.19979 �21.0859 0.5947 19 �21.2092 1.8725 29 �16.0748 3.334710 �21.7776 0.6915 20 �20.7939 2.0233 30 �15.5005 3.4660

Table 2Optimal N* and C (N*) obtained for different values of a

a b = 1 b = 0.95

N* C (N*) N* C (N*)

1 18 �29.03241.01 45 �35.0761 16 �27.53951.02 32 �32.2986 15 �26.10671.03 26 �30.1360 14 �24.87251.04 23 �28.2997 14 �23.68151.05 20 �26.6810 13 �22.55161.06 19 �25.2184 13 �21.48361.07 17 �23.8787 12 �20.46721.08 16 �22.6380 12 �19.51021.09 15 �21.4757 12 �18.57991.10 15 �20.3931 11 �17.7002

0 5 10 15 20 25 30 35 4030

20

10

0

10

20

30

40

50

60

N

C(N)

g(N)

Fig. 1. The plots of C(N) and g(N) against failure number.


Table 3Optimal N* and C (N*) obtained for different values of b

b a = 1 a = 1.05

N* C (N*) N* C (N*)

1 20 �26.6810

0.99 42 �35.6186 18 �25.71050.98 29 �33.3387 16 �24.82120.97 23 �31.6405 15 �24.01380.96 20 �30.2431 14 �23.25980.95 18 �29.0324 13 �22.55160.94 16 �27.9684 12 �21.87380.93 15 �26.9930 12 �21.26160.92 14 �26.1028 11 �20.65910.91 13 �25.2884 11 �20.09560.90 12 �24.5316 10 �19.5398


6. Concluding remarks

In this paper, we study the GP M/M/1 model for a queueing system. Assume that the service station issubject to failure and a failed service station after repair is not ‘‘as good as new’’ such that the successiveoperating times of the service station form a decreasing GP and the consecutive repair times of the servicestation constitute an increasing GP. Many queueing characteristics are derived. Moreover, some reliabilityindices of the service station are also studied. Furthermore, a maintenance model is introduced for the serv-ice station, an optimal replacement policy is determined explicitly. This work is an interdiscipline work be-tween queueing and reliability. It is a generalization of the existing work, as by putting a = b = 1, the GPM/M/1 model will reduce to a M/M/1 queueing model with a repairable service station in which repair isperfect. Therefore, our work should have not only theoretical interest but also some potential practicalapplication.Although GP has been wildly applied to the maintenance problem and data analysis from a series of

events with trend, this is the first work to apply GP to a queueing system with a repairable service station.It seems that GP is a powerful tool for dealing with a deteriorating or improving system. We hope thatmore work on the GP will appear in the future.Finally, we note that the work in Section 5 is still true even without the exponential distribution assump-

tion. In other words, the replacement model for the service station can be formulated by assuming that {Xn,n = 1,2, . . .} forms a decreasing GP, and {Yn,n = 1,2, . . .} constitutes an increasing GP without the expo-nential distribution assumption. Then an optimal replacement policy N* is also determined by (5.3). On theother hand, in Section 5, we only study the monotonicity of optimal policy N* on a or b while the otherparameters are fixed. In practice, it might be interesting to study the monotonicity of optimal policy N*

on the other parameters c, r, cp, R, a, b, s. For the study of these two problems, please see Lam (2003)for reference.

Acknowledgement

The authors are grateful to the referees and editor for their comments and suggestions which led to muchimprovement in the presentation.


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