Multi-server machine repair problems under a (V,R) synchronous single vacation policy

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Multi-server machine repair problems under a (V, R)

synchronous single vacation policy

Chia-Huang Wu

a

, Jau-Chuan Ke

b,⇑ a

Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, ROC

b

Department of Applied Statistics, National Taichung University of Science and Technology, Taichung, Taiwan, ROC

a r t i c l e

i n f o

Article history:

Received 29 September 2010 Received in revised form 2 July 2013 Accepted 8 October 2013

Available online 30 October 2013 Keywords:

Machine repair problem Synchronous single vacation Standby

a b s t r a c t

This paper considers a machine repair problem with M operating machines and S standbys, in which R repairmen are responsible for supervising these machines and operate a (V, R) vacation policy. With such policy, if the number of the failed machines is reduced to R V (R > V) (there exists V idle repairmen) at a service completion, these V idle servers will together take a synchronous vacation (or leave for other secondary job). Upon returning from the vacation, they do not take a vacation again and remain idle until the ﬁrst arriving failed machine arrives. The steady-state probabilities are solved in terms of matrix forms and the system performance measures are obtained. Algorithmic procedures are provided to deal with the optimization problem of discrete/continuous decision variables while maintaining a minimum speciﬁed level of system availability.

1. Introduction

We consider a machine repair problem where a group of M operating machines is under the supervision of one or more repairmen (servers) in the repair facility. These operating units are assumed unreliable and may fail at any time. This inci-dence may lead to loss of production because the failed machine must stay in the repair facility for some time. To avoid any loss of production, the plant always keeps some standby machines, say S (S < M), so that a standby machine can immediately act as a substitute when an operating machine fails. When a machine fails, it is immediately sent to the repair facility for repair and backed up by a standby, if available. Meanwhile, the repairmen may together leave for a synchronous single vaca-tion of random length whenever there exists V (V < R) idle repairmen. The so-called single vacavaca-tion means that at the end of the vacation the repairmen remain idle until the ﬁrst arriving failed machine arrives. A real-world example of this vacation model can be realized in the manufacturing/production-assembly system where the servers in their idle time may be assigned to perform some extra operations such as additional work, preventive maintenance. On the other hand, machine repair related problems represent a group of very important problems used to analyze timesharing computer systems, mul-ti-programmed computer systems and multi-access communication channels (see[1]).

In a classical machine repair model, it is assumed that the servers remain idle until the failed machines present (i.e., each server is always available for the waiting failed ones). Such type of machine repair problems also received considerable attention in the literature, as shown by the literature surveys of Ke and Wang[2], Wang et al.[3,4]and Haque and Armstrong

[5]. So far, however, only a few works is taken into consideration the server vacations in machine repair problems (see[6–8]). Gupta[6]ﬁrst analyzed a machine repair problem with warm standbys and server vacations, in which the single server

http://dx.doi.org/10.1016/j.apm.2013.10.045

⇑ Corresponding author. Address: Department of Applied Statistics, National Taichung University of Science and Technology, No. 129, Sec. 3, Sanmin Rd., Taichung 404, Taiwan, ROC. Tel.: +886 4 22196077; fax: +886 4 22196331.

E-mail addresses:[email protected],[email protected](J.-C. Ke).

Contents lists available atScienceDirect

Applied Mathematical Modelling

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leaves a vacation when the repair facility is empty. Gupta’s work[6]gave an algorithm to compute the steady-state prob-ability distribution of the number of failed machines in the system. Gupta’s models were extended to server-breakdown case by Ke[7], who derived system performance measures and performed a cost sensitivity analysis. Ke and Lin[8]dealt with the reliability measures of a multi-server machine repair model with standby and multiple vacations. Ke and Wang[9]examined the steady-state results for a machine repair problem with two types of standby and multi-server vacations under two vaca-tion policies. They performed a sensitivity analysis to investigate the effect on the joint optimum number of standbys and servers if the system parameters take on other speciﬁc values. Later, Jain et al.[10]studied the machine repair problem with mixed standbys (warm and cold) where the failed unit may balk or renege in case of heavy load of failed units. Ke and Lin

[11]performed a sensitivity analysis of two machine repair models including various repair rates in each phase and two-phase with differing numbers of technicians. Ke et al.[12]considered three vacation policies of machine repair problem in production systems with spares and server vacations. Furthermore, Garg et al. [13] investigated the availability of crank-case manufacturing system in an automobile industry. They showed that the availability of the system can be im-proved using proper maintenance planning and scheduling. More recently, Yue et al.[14]considered a machine repair prob-lem with warm standbys and two heterogeneous repairmen. They investigated the probprob-lem from the viewpoints of both queueing and reliability. Wang et al.[15]provided an optimization analysis of the machine repair problem with balking and variable number of servers Recently, Ke et al.[16]proposed a multi-repairmen problem with warm standbys, pressure coefﬁcient, imperfect coverage and server breakdown. They performed a comparative analysis among two optimal ap-proaches for searching discrete and continuous parameters.

Existing research works on machine repair problems with multi-server vacations, including those above, mainly focused on server individual vacation at system empty (i.e., at each repair completion instant, the server individually takes a vacation each time system empty). From practical viewpoint, however, some servers may together take vacations when the number of failed machines reduced a predetermined threshold (see[17–19]). Ke and Wu[20]considered a machine repair problem operating a (R, V, K) synchronous vacation policy, where the vacation policy is a multiple (inﬁnite) vacation. It should be noted that multiple vacation policy is different from single vacation policy, which the former cannot be reduced to the latter (referred to[21,9]). Comparable work on machine repair problems with synchronous single vacation policy is rarely found in the literature. Thus, we develop a multi-server machine repair problem with standby where the servers apply a (V, R) syn-chronous single vacation policy when the number of failed machines is reduced to R V. Besides the lack of research work on this problem, our study is also motivated by some practical systems as follows.

Consider a firm has many departments including the research and development (R&D) department, the equipment department, the productive department, the quality control department, etc. The employees in the equipment department are responsible for and improving the machine reliability, evaluating the availability and performance of the machines, man-aging the components for the repair of machines, and creating and updating the relative log file. The main tasks of the employees are maintaining and keeping the operations of equipment in productive department. To provide enough produc-tion capacity to satisfy the orders placed by the customers. The number of operating machines should be greater than or equal to a threshold value called as M. It is assumed there are S standby machines as spares for the operating machines. In the equipment department, there are R employees who provide maintenance service for these machines. Suppose these employees are configurable. When the workload in the equipment department become lower, partial manpower, V of R employees, will be dispatched to organize the relative log file and loop up the maintain records to monitor the statues of each machines. That is, the employees may leave the system a random time. It can be regarded as the synchronous vacation of employees. For the machine maintain service in the equipment department, it is a multi-server queueing system with syn-chronous vacation policy where partial server will take a synsyn-chronous vacation together. We can investigate this queueing system to evaluate the employee’s performance.

Our model can be implemented to another practical problem based on the work of Chelst et al.[22]. Consider a coal trans-portation system with (M + S) identical trains which are response for transiting the coal from the mines to the unloader sys-tem to dump the coal. The unloader syssys-tem involves R employees to unload the train. Partial (V) employees may be assigned to execute some secondary tasks such as maintenance or clean when they are idle. Once the secondary tasks are completed, the employees will return to the unloader system. The employees and the trains correspond to servers and machines, respec-tively. The unloaded times and the times of executing secondary tasks can be regarded as service times and vacation times, respectively. In this study, we wish to develop a computational model that helps managers for the following important ques-tions: (1) Under a certain cost, what is the optimum spare machines and optimal (V, R) policy that minimize the expected cost of this system; That is how many standbys are needed and how many repairmen utilize their idle time during the oper-ation. (2) After standbys and (V, R) policy are decided, how to adjust the service rate and vacation rate such that the cost is possibly reduced.

The paper is organized as follows; In Section2, the system assumptions are described. In Section3, the steady-state equa-tions are obtained and the computable forms of the steady-state probabilities are derived using the matrix-analytic method. Some system performance measures are derived in Section4. In Section5, a cost model is developed to determine the opti-mal values of servers, standbys, vacation servers, service rate and vacation rate in order to minimize the total expected cost per unit time while maintaining a speciﬁed level of system availability. Some numerical examples and sensitivity analyses are provided. Section6concludes.

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2. System description

In this research, a machine repairable system with M identical and independent machines operating simultaneously in parallel, S standby machines, and R repairman who are responsible for maintaining these machines is considered. Our anal-ysis is based on the following assumptions.

2.1. Assumptions

1. M operating machines are required for the functioning of the system. In other words, the system is short if only if S + 1 or more machines fail.

2. Operating machines are subject to breakdowns according to an independent Poisson distribution with rate k. When an operating machine breaks down, it will immediately be backed up by an available standby.

3. Each of the standby machines fails independently of the others with Poisson rate

a

, where (0 6

a

6_{k). When a standby} machine moves into an operating state, its characteristics will be that of an operating machine.

4. Every failure machines are repaired by R repairmen in the order of failures, that is, the FCFS discipline. The repair time is assumed to be independent and identically exponentially distribution with parameter

l

.

5. When a failed standby machine is repaired, it is as good as a new one and goes into standby state unless the system is short. At this time, the repaired machine will be sent back to an operating state immediately.

6. Each repairman can repair only one failed unit at a time. The failed unit that on arriving at the repair facility ﬁnds all repairmen busy or on a vacation must wait in the queue until a repairman is available.

7. When the failure machines queueing up for repair is less than (R–V), that is, the number of idle repairman is more than V, these V idle repairmen will take a single vacation together.

8. The system allows only V repairmen on vacation at any time. The vacation time is distributed as an exponential with rate h. The various stochastic processes involved in this system are independent of each other.

It should be noted that the repairmen adopt a synchronous single vacation policy and they wait idly for the ﬁrst failed machine to arrive as the vacation period terminated.

3. Steady-state results

For the M/M/R machine repair model with standbys under a (V, R) synchronous single vacation policy, we describe the state of the system by the pairs {(i, n):i = V, B, and n = 0, 1, ..., M + S}, where i = V(B) indicates that there are V repairmen on vacation state (or not), and n represents the number of failed machines in the system. The mean failure rate knand mean

repair rate

l

nfor this system are given by

kn¼ Mk þ ðS nÞ

a

; 0 6 n 6 S; ½M ðn SÞk; S 6 n 6 M þ S; 0; otherwise 8 > < > : and

l

n¼ n

l

; 1 6 n 6 R; 0; otherwise:

In steady-state, the following notations are used

PV,n probability that there are n failed machines in the system when there are V repairmen on vacation.

PB,n probability that there are n failed machines in the system when there are no repairmen on vacation,

where 0 6 n 6 M þ S.

3.1. Steady-state equations

Using birth and death process and referring to the steady-transition-rate diagram shown inFig. 1, the steady-state equa-tions for the M/M/R machine repair problem with standbys under a (V, R) synchronous single vacation policy are obtained as follows.

(1) i = V

ðk0þ hÞPV;0¼

l

1PV;1; ð1Þ

RVPV;iþ1; R V þ 1 6 i 6 M þ S 1; ð4Þ

ð

l

_RVþ hÞPV;MþS¼ kMþS1PV;MþS1: ð5Þ

(2) i = B

k0PB;0¼

l

1PB;1þ hPV;0; ð6Þ

RPB;MþS¼ kMþS1PB;MþS1þ hPV;MþS: ð11Þ

There is no way of solving(1)–(11)in a recursive manner to develop the explicit expressions for the steady-state prob-abilities. In the next section, we provide a matrix-analytic method to deal with this problem.

3.2. Matrix-analytic solutions

To analyze the resulting system of linear equations(1)–(11), a matrix-form property is used. Following the concepts by Neuts[23], one ﬁnds that the transition rate matrix Q of this Markov chains can be partitioned as the following form:

Q ¼ CðMþSþ1ÞðMþSþ1Þ PðMþSþ1ÞðMþSþ1Þ UðMþSþ1ÞðMþSþ1Þ XðMþSþ1ÞðMþSþ1Þ

: ð12Þ

The matrix Q is a square matrix of order 2(M + S + 1) and each entry of the matrix Q is listed in the following:

l

RV+1andU= hIM+S+1. IM+S+1denotes the identity matrix of order

i, 0 6 i 6 M þ S, are such that the sum of each column of Q is zero.

Let P denote steady-state probability vector of Q. By partitioning the vector P ¼ ½PVPBT with PV and PB are both

(M + S + 1) 1 vector, one ﬁnds that the steady-state equations QP = 0 are given by

CPVþ PPB¼ 0; UPVþ XPB¼ 0:

ð15Þ

Using the following normalizing equation:

X i2fV;Bg X MþS n¼0 Pi;n¼ eTP ¼ 1; ð16Þ

where e represents a column vector with suitable size and each component equal to one. Eq.(16)is substituted into the ﬁrst (redundant) row in Eq.(15)to yield

QP ¼ C _P U X P ¼ ½1; 0; . . . ; 0T; ð17Þ C⁄ andP⁄

are the matrices which are obtained by replacing each elements in the ﬁrst row ofCandPwith one (for Eq.

(16), the normalization condition). The solution of Eq.(17)provides the steady-state probabilities as

P ¼ ðQ_Þ1 ½1; 0; . . . ; 0T¼ Q 1 11:2 Q 1 11:2II X1 X1_UQ1 11:2 X1UQ 1 11:2PX1þ X1 " # ½1; 0; . . . ; 0T: ð18Þ

where Q112=C⁄P⁄X-1U. Finally, it is observed that the steady-state probabilities PVand PBare equal to the ﬁrst column

of matrix Q1

11:2and X1UQ111:2, respectively.

4. Performance analysis

In this section, we deal with the steady-state availability and the mean time to system failure analysis. Also, the explicit expressions of some performance measures for the machine repair problem are included.

4.1. Availability and reliability analysis

It is assumed that the system breaks down if and only if (S + 1) or more machines fail. The steady-state availability can be calculated as A:V: ¼ Pð0 6 n 6 sÞ ¼X S n¼0 PV;nþ XS n¼0 PB;n¼ Vsþ1ðPVþ PBÞ ¼

v

sþ1ðIMþSþ1 X1UÞQ111:2½1; 0; . . . ; 0 T ; ð19Þ

where

v

krepresents a row vector with suitable size which the ﬁrst k elements are equal to 1 and zero otherwise.

To calculate the MTTF, we reduce the original transition rate matrix and delete the rows and columns for the absorbing state(s). The new matrix is called B as

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B ¼ Cr Pr Ur Xr

T

; ð20Þ

whereCr,Pr,UrandXrdenote the square sub-matrix ofC,P,UandX, respectively. The subscribe ‘‘r’’ means reducing the

matrix by deleting the (S + 2)th (M + S + 1)th rows and (S + 2)th (M + S + 1)th columns. Then, the expected time to reach an absorbing state is calculated from

E½TPð0Þ!PðabsorbingÞ ¼ Pð0Þ

TZ 1 0

eBt_{dt ¼ Pð0Þ}T

ðB1Þe: ð21Þ

where P(0) = [1, 0, ..., 0]T_{denotes the initial conditions for this problem.}

4.2. Other system performance measures

Our analysis is based on the following system performance measures. Let

E½F the expected number of failed machines in the system, E½Fq the expected number of failed machines in the queue,

E½O the expected number of operating machines in the system, E½S the expected number of acting standby machines in the system, E½B the expected number of busy repairmen in the system, E½V the expected number of vacation repairmen in the system, E½I the expected number of idle repairmen in the system,

M:A: machine availability (the fraction of the total time that the machines are working), O:U: operative utilization (the fraction of busy servers).

The expressions for E[F], E[Fq], E[O], E[S], E[B], E[V] and E[I] are obtained as follows:

E½F ¼X MþS n¼0 nðPV;nþ PB;nÞ ¼ ½0; 1; 2; . . . ; M þ SðPVþ PBÞ ¼ ½0; 1; 2; . . . ; M þ SðIMþSþ1 X1UÞQ111:2½1; 0; . . . ; 0 T ; ð22Þ E½Fq ¼ X MþS n¼0 maxf0; n ðR VÞgPV;nþ X MþS n¼0 maxf0; n RgPB;n¼ XMþS n¼RVþ1 ½n ðR VÞPV;nþ X MþS n¼Rþ1 ðn RÞPB;n; ð23Þ E½O ¼X MþS n¼0 minfM; M þ S ngðPV;nþ PB;nÞ; ð24Þ E½S ¼X MþS n¼0 maxf0; S ngðPV;nþ PB;nÞ ¼ XS1 n¼0 ðS nÞðPV;nþ PB;nÞ; ð25Þ E½B ¼X MþS n¼0 minfn; R VgPV;nþ X MþS n¼0 minfn; RgPB;n; ð26Þ E½V ¼X MþS n¼0 VPV;n¼ VeTPV¼ VeTQ1112½1; 0; . . . ; 0 T ; ð27Þ

E½I ¼ R E½B E½V: ð28Þ

Following Benson and Cox[24], the machine availability and the operative utilization are deﬁned by

M:A: ¼ 1 E½F

M þ S and O:U: ¼ E½B

R : ð29Þ

Furthermore, using the Little’s formula we obtain the expected waiting time in the system, E[W], and in the queue E[Wq]

as

E½W ¼ E½F=ke and E½Wq ¼ E½Fq=ke; ð30Þ

where ke¼Pi2fV;Bg

PMþS

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5. Cost analysis

In this section, we construct a total expected cost function per unit time based on the system performance measures, and impose a constraint on the system availability in which S, R and V are discrete decision variables.

First let

Ch cost per unit time when one failed machine joins the system,

Ce cost per unit time of a failed machine after all spares are exhaused (downtime cost),

Cs cost per unit time when one machine is functioning as a spare (inventory cost),

Cb cost per unit time when one repairmen is busy,

Ci cost per unit time when one repairmen is idle,

Cf cost per unit time of each repairmen,

c

reward per unit time when one repair is on vacation.

5.1. Direct search method for optimal S, R and V

Using the deﬁnitions of these cost elements listed above, the total expected cost function per unit time is given by

TcostðS; R; VÞ ¼ ChE½F þ CsE½S þ CeðM E½OÞ þ CbE½B þ CiE½I þ RCf

c

E½V: ð31Þ

Some production system always requires minimum of M machines in operation or a certain level of system availability. Our object is to determine the optimum number of standbys S, say S⁄_{, the optimum number of repairmen R, say R}⁄_{, and the}

optimum vacation policy level V, say V⁄_{, simultaneous which minimize the cost function T}

cost(S, R, V) and the system

avail-ability is maintained at a certain level.

Following the concept of Hilliard[25], the cost minimization problem can be illustrated mathematically as

Minimize S;R;V TcostðS; R; VÞ: ð32Þ Subject to A:V: ¼X S n¼0 ðPV;nþ PB;nÞ P A; ð33Þ

where A.V. is the steady-state probability that at least M machines are in operation and function properly (system availabil-ity) and A is the availability level required.

A direct search method may be used to obtain potentially useful results. The optimization algorithm is a direct search approach over a grid whose boundaries for decision variables are selected in order to guarantee that the global optimum is obtained in the interior region (see[25]). The direct search algorithm is applied in the set {M P S and M P R P V; S, R, V are positive integers}.

The speciﬁc steps in the direct search algorithm for obtaining the optimal value (S⁄_{, R}⁄_{, V}⁄_{) are as follows:}

Step 1. Find the optimal number of repairmen, and the optimal vacation policy level, for S standbys, i.e.,

Min

R;V TcostðS; R; VÞ ¼ TcostðS; R _;_V_Þ

subject to the availability constraint is satisﬁed.

Step 2. Step 2. Find the set of all minimum cost solutions for S = 1, 2, ..., M, i.e.,H= {Tcost(S, R⁄, V⁄):S = 1, 2, ..., M}.

Step 3. Find the minimum cost solutions in this set, i.e.,

min

S

H

¼ TcostðS _;_R_;_V

Þ:

We provide an example to illustrate the direct search algorithm.

Example. Consider M = 15 and the system parameters k ¼ 0:6,

l

= 2.5, h = 0.2,

a

= 0.3, the cost elements and availability as follows

Ch= 10, Ce= 125, Cs= 50, Cb= 75, Ci= 40, Cf= 80,

c

= 60, and A = 0.9.

Step 1. Find R⁄_{and V}⁄_{for S standbys necessary to satisfy the required availability, where S = 1, 2, ..., 15. (see}_{Table 1}_).

Step 2. FromTable 1,H={$1209.55,$1108.82,$1048.50,$1050.06,$1085.63,. . .}.

⁄, h⁄) = (1

0.48925) ⁄ 2000 = 1021.5).

Table 1

The expected cost Tcost(S ⁄

, R⁄

, V⁄

) and the system availability A.V. (k ¼ 0:6,l= 2.5, h = 0.2,a= 0.3).

S 6 7 8 9 10 (S⁄ , R⁄ , V⁄ ) (6, 8, 1) (7, 7, 1) (8, 7, 2) (9, 6, 1) (10, 6, 1) Tcost(S⁄, R⁄, V⁄) 1209.55 1108.82 1048.50 1050.06 1085.63 A.V. 0.91014 0.92727 0.90311 0.92069 0.93692 S 11 12 13 14 15 (S⁄ , R⁄ , V⁄ ) (11, 6, 1) (12, 6, 1) (13, 6, 1) (14, 6, 1) (15, 6, 1) Tcost(S⁄, R⁄, V⁄) 1122.34 1159.65 1197.19 1234.70 1271.99 A.V. 0.94896 0.95805 0.96502 0.97044 0.97472

Fig. 2. Cost and availability surfaces for (M, S⁄

, R⁄

, V⁄

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5.3. Sensitivity analysis

Considering M = 15, the same cost elements and system parameters listed in above example, we perform a sensitivity analysis for changes in the joint optimum value (S⁄_{, R}⁄_{, V}⁄_{) along with changes in speciﬁc values of the system parameters}

k,

l

, h and

a

. The minimum expected cost Tcost(S⁄, R⁄, V⁄) and values of various system performance measures A.V. E [F],

E[Fq], E[O], E[S], E[B], E[V], E[I], M.A., and O.U. at the optimum values (S⁄, R⁄, V⁄) are shown inTables 2 and 3for different

val-ues of (k,

a

) and (

l

, h).

FromTable 2, we observe that (i) Tcost(S⁄, R⁄, V⁄) increases as k or

a

increases; (ii) S⁄and R⁄increase as k increases; and (iii)

S⁄_{increases as}

_a

_{increases. One sees from}_{Table 3}_{that (i) T}

cost(S⁄, R⁄, V⁄) decreases as

l

increases; (ii) S⁄and R⁄decrease as

l

increases; and (iii) S⁄_{increases as h increases.}

Similarly, we are also interesting in the effect of (

l

⁄_{, h}⁄_{) by changing the values of other two continuous system}

param-eters k and

a

when M = 15 and (S⁄_{, R}⁄_{, V}⁄_{) = (8, 7, 2) are determined. In}_{Table 4}_{, the optimum value (}

_l

⁄_{, h}⁄_{) and several system} Table 2

System performance measures of the machine-repair problem with standbys under a (V, R) synchronous single vacation policy under optimal operating conditions (l= 2.5, h = 0.2). (k;aÞ (0.3, 0.3) (0.9, 0.3) (1.5, 0.3) (0.6, 0) (0.6, 0.3) (0.6, 0.6) (S⁄ , R⁄ , V⁄ ) (5, 4, 1) (11, 9, 2) (14, 13, 1) (8, 6, 1) (8, 7, 2) (10, 6, 1) Tcost 643.358 1416.81 2145.64 1022.11 1048.50 1073.32 A.V. 0.91941 0.90417 0.90509 0.92854 0.90311 0.90528 E[F] 2.61779 7.00162 9.93347 4.22717 4.84068 6.30274 E[Fq] 0.53192 1.17675 0.55327 0.66541 0.88737 1.81540 E[O] 14.8474 14.7711 14.7755 14.8407 14.7850 14.7891 E[S] 2.53484 4.22728 4.29107 3.93212 3.37432 3.90813 E[B] 2.08587 5.82487 9.38020 3.56177 3.95332 4.48734 E[V] 0.81645 1.77055 0.84624 0.81343 1.73861 0.88529 E[I] 1.09768 1.40459 2.77355 1.62481 1.30807 0.62737 M.A. 0.86911 0.73071 0.65747 0.81621 0.78954 0.74789 O.U 0.52147 0.64721 0.72155 0.59363 0.56476 0.74789 Table 3

System performance measures of the machine-repair problem with standbys under a (V, R) synchronous single vacation policy under optimal operating conditions (k ¼ 0:6,a= 0.3). (l, h) (1.2, 0.5) (2.4, 0.5) (3.6, 0.5) (2.5, 0.2) (2.5, 0.5) (2.5, 1.0) (S⁄ , R⁄ , V⁄ ) (14, 10, 10) (9, 6, 1) (6, 5, 1) (8, 7, 2) (8, 6, 1) (10, 6, 2) Tcost 1847.76 1065.20 822.23 1048.50 1030.20 1042.37 A.V. 0.90330 0.91424 0.93190 0.90311 0.91024 0.92091 E[F] 9.75034 5.30466 3.15025 4.84069 4.77319 5.76921 E[Fq] 1.24683 1.11666 0.42376 0.88737 0.80929 1.68337 E[O] 14.7644 14.8087 14.8681 14.7850 14.8057 14.8179 E[S] 4.48532 3.88670 2.98165 3.37432 3.42110 4.41291 E[B] 8.50350 4.18800 2.72649 3.95332 3.96390 4.08584 E[V] 0.01088 0.72677 0.70269 1.73861 0.71615 1.21740 E[I] 1.48562 1.08524 1.57082 1.30807 1.31995 0.69676 M.A. 6.63782 0.77897 0.84999 0.78954 0.79247 0.76923 O.U. 0.85035 0.69800 0.54530 0.56476 0.66065 0.68097 Table 4

System performance measures of the machine-repair problem with standbys under a (V, R) synchronous single vacation policy under optimal operating conditions (M = 15, S = 8, R = 7, V = 2). (k;aÞ) (0.3, 0.3) (0.6, 0.3) (0.9, 0.3) (0.6, 0) (0.6, 0.3) (0.6, 0.6) (l⁄ , h⁄ ) (1.5, 0.02) (2.9, 0.02) (4.3, 0.02) (2.7, 0.02) (2.9, 0.02) (3, 0.02) Tcost 1020.33 1021.50 1021.60 1026.46 1021.50 1017.43 A.V. 0.94548 0.95160 0.95404 0.95014 0.95160 0.94472 E[F] 4.36272 4.04968 3.92574 3.82927 4.04968 4.37077 E[Fq] 0.63529 0.54765 0.51498 0.51958 0.54765 0.64493 E[O] 14.8935 14.90267 14.9066 14.8936 14.90267 14.8917 E[S] 3.74381 4.04764 4.16762 4.27711 4.04764 3.73750 E[B] 3.72746 3.50203 3.41076 3.30969 3.50203 3.72585 E[V] 1.94268 1.96119 1.97082 1.94829 1.96119 1.97089 E[I] 1.32986 1.53677 1.61842 1.74202 1.53677 1.30326 M.A. 0.81032 0.82393 0.82932 0.83351 0.82393 0.80997 O.U. 0.53249 0.50029 0.48725 0.47281 0.50029 0.53226

(10)

performance measures for speciﬁc values of k and

⁄

=

l

Uthen adjusts (increases) the vacation rate h next.

6. Conclusions

The systematic methodology provided in this paper works efficiently for a machine repair model with standbys under a synchronous single vacation policy. The stationary probability vectors were obtained in terms of matrix forms using the technique of matrix partition. Firstly, we developed the steady-state solutions in matrix forms for the machine repair model by using the Markov process. These solutions were used to obtain the various system performance measures, such as the steady-state availability, MTTF, the expected number of failed machines in the queue / system, the expected number of idle, busy and vacation servers, machine availability, operative utilization, etc. Next, we developed a cost model for the machine repair model to determine the joint optimum number of standbys, servers and vacation servers in order to minimize the steady-state expected cost per unit time, while maintaining a specified level of system availability. After the determination of the three discrete decision variables, the optimal adjustments of service rate and vacation rate were also considered. Two procedures were provided to handle this optimization problem. Finally, a sensitivity analysis was performed to investigate the effect on the joint optimum values if the system parameters take on other specific values. We extended the traditional vacation policy to more generalization one. The investigated model can be used to evaluate the performance of some prac-tical queueing systems similar to mentioned earlier in Introduction. Specifically, it may be employed to fit the system with configurable servers which have multiple tasks.

Acknowledgments

The authors are grateful to Editor, Professor Cross, whose constructive comments have led to a substantial improvement in the presentation of the paper.

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