JOURNAL OF INDUSTRIAL AND doi:10.3934/jimo.2010.6.197 MANAGEMENT OPTIMIZATION Volume 6, Number 1, February 2010 pp. 197–207
COST ANALYSIS OF THE M/M/R MACHINE REPAIR PROBLEM WITH SECOND OPTIONAL REPAIR: NEWTON-QUASI METHOD
Kuo-Hsiung Wang, Chuen-Wen Liao and Tseng-Chang Yen Department of Applied Mathematics, National Chung-Hsing University Taichung 402, Taiwan
(Communicated by Wuyi Yue)
Abstract. This paper investigates the M/M/R machine repair problem with second optional repair. Failure times of the operating machines are assumed to be exponentially distributed with parameter λ. Repair times of the first essential repair and the second optional repair are assumed to follow exponen- tial distributions. A failed machine may leave the system either after the first essential repair with probability (1 − θ), or select to repair for second optional repair with probability θ (0 ≤ θ ≤ 1) at the completion of the first essential repair. We obtain the steady-state solutions through matrix-analytic method. A cost model is derived to determine the optimal number of the repairmen, the optimal values of the first essential repair rate, and the second optional repair rate while maintaining the system availability at a specified level. We use the direct search method to deal with the number of repairmen problem and the Newton-Quasi method for the repair rate problem to minimize the system operating cost while all the constraints are satisfied.
1. Introduction. In this paper, we consider the M/M/R machine repair problem with second optional repair where a group of identical operating machines are main- tained by one or more repairmen in the repair facilities. The first essential repair is required for all failed machines. A failed machine may leave the system either after the first essential repair with probability (1 − θ), or select to repair for second optional repair with probability θ(0 ≤ θ ≤ 1) at the completion of the first essential repair. A cost model is developed in order to determine the optimal values of the number of repairmen, the first essential repair rate µ1, and the second optional repair rate µ2 while maintaining a minimum specified level of system availability. Steady-state solutions of the machine repair problem are obtained for (1) the M/M/R model by Benson and Cox [4]; (2) M/G/1 and M/D/1 models by Ashcroft [3]; (3) the M/Ek/1 model by Benson and Cox [4]; (4) the D/D/R model by Allen [2]; (5) the M/Ek/R model by Maritas and Xirokostas [8]; and (6) the G/M/R model by Bunday and Scraton[5] and Sztrik [12]. Cost models in the machine repair problem have been investigated by While, Schmidt and Benett [16], Hilliard [6], Sivazlian and Wang [11], and many others. Recently, Wang, et al. [14] used the direct search
2000 Mathematics Subject Classification. Primary: 90B22. Key words and phrases. cost, matrix-analytic method, optimization, second optional repair, Newton-Quasi method.
197 198 KUO-HSIUNG WANG, CHUEN-WEN LIAO AND TSENG-CHANG YEN method and the steepest descent method to find the global maximum value satisfy- ing the availability, balking and reneging constraints. Wang and Kuo [15] analyzed the M/Ek/1 machine repair problem with a non-reliable service station. Wang and Kuo [15] determined the optimum number of machines to be assigned to the service station in order to maximize the total expected profit per machine per unit time. We should note that steady-state solutions of the M/M/R machine repair problem with second optional repair have not been investigated. In many systems, all failed machine requires the main repair and only some may require the subsidiary repair provided by the channel. Madan [7] first introduced the concept of second optional service. Madan [7] studied the time-dependent and steady-state behavior of an M/G/1 queue with second optional service. Medhi [9] proposed an M/G/1 queue with second optional server, developed the mean queue length and the mean waiting time. Al-Jararha and Madan [1] investigates both first essential service and second optional service following general distributions. They used supplementary variable technique to develop the time-dependent proba- bility generating functions and the corresponding steady-state results. In addition, Wang [13] studied an M/G/1 queue with second optional service and server break- downs. The primary objective of this paper is to investigate the M/M/R machine repair problem with second optional repair, and: 1: present a matrix-analytic method to develop the steady-state solutions; 2: use the results of system performance measures to determine the optimum ∗ ∗ ∗ value (R , µ1, µ2) and maintain the system availability at a certain level; 3: provide the numerical results of various system performance measures under the optimal operating conditions.
2. Description of the system. It is assumed that each of the operating machines fails independently to the state of the others, and has an exponential time-to-failure distribution with parameter λ. The first essential repair is required to all failed machines. The repair times of the first essential repair and the second optional repair have exponential distributions with mean 1/µ1 and 1/µ2, respectively. As soon as the first essential repair of a machine is completed, a machine may either leave the system with probability (1 − θ) or may select to the optional repair with probability θ(0 ≤ θ ≤ 1), at the completion of which the machine departs from the system and the next machine, if any, from the queue is taken up for the first essential repair. Each repairman can provide only one essential repair or second optional repair at a time. Failed machines at the system form a single waiting line and receive the repair in the order of their breakdowns. There are R repairmen who provide the first essential repair and the second optional repair to a failed machine. Suppose that the switchover time from failure to repair is instantaneous. Each repairman can repair only one failed machine at a time. If all repairmen are busy, then a failed machine must wait in the queue until a repairman is available.
3. Steady-state results. We investigate the M/M/R machine repair problem with N identical operating machines and second optional repair under steady-state conditions. We describe the states of the system by the pairs {(i, j) | 0 ≤ j ≤ R, and 0 ≤ i ≤ N − j}, where i denotes the number of failed machines in the first essential repair channel, and j denotes the number of failed machines in the second optional repair channel. If (i + j) is less than or equal to R, the failed machines COST ANALYSIS OF THE M/M/R MACHINE REPAIR 199 which upon to the repair facility will get repair immediately. If (i + j) is greater than R, the new arriving failed machines must wait in the queue until the repairmen finish the second optional channel. In steady-state condition, the notation Pi,j is the probability that there are i failed machines in the first essential repair channel and there are j failed machines in the second optional repair channel, where0 ≤ j ≤ R, and 0 ≤ i ≤ N − j. State-transition-rate diagram of an M/M/3 machine repair problem withN =5 identical operating and second optional repair is shown in Figure 1.
5λ - 4λ - 3λ - 2λ - λ - 0,0 1,0 2,0 3,0 4,0 5,0 Pi (1−θ)µ1 Pi 2(1−θ)µ1 Pi 3(1−θ)µ1 Pi 3(1−θ)µ1 Pi 3(1−θ)µ1 PP PP PP PP PP P θµ1 P 2θµ1 P 3θµ1 P 3θµ1 P 3θµ1 µ2 P P? µ2 P P? µ2 P P? µ2 P P? µ2 P P? 4λ - 3λ - 2λ - λ - 0,1 1,1 2,1 3,1 4,1 Pi (1−θ)µ1 Pi 2(1−θ)µ1 Pi 2(1−θ)µ1 Pi 2(1−θ)µ1 PP PP PP PP P θµ1 P 2θµ1 P 2θµ1 P 2θµ1 2µ2 P P? 2µ2 P P? 2µ2 P P? 2µ2 P P? 3λ - 2λ - λ - 0,2 1,2 2,2 3,2 Pi (1−θ)µ1 Pi (1−θ)µ1 Pi (1−θ)µ1 PP PP PP P θµ1 P θµ1 P θµ1 3µ2 P P? 3µ2 P P? 3µ2 P P? 2λ - λ - 0,3 1,3 2,3
Figure 1. State-transition-rate diagram for an M/M/R machine repair problem with N identical operating machines and second optional repair channel (R = 3 and N = 5).
3.1. Steady-state equations. The steady-state equations for an M/M/R machine repair problem with N identical operating machines and second optional repair are given by: (i) j =0
NλP0, 0 = µ2P0, 1 + (1 − θ)µ1P1, 0, (1)
[(N − i)λ + iµ1]Pi, 0 = (N − i + 1)λPi−1, 0 + µ2Pi, 1 + (i + 1)(1 − θ)µ1Pi+1, 0, 1 ≤ i ≤ R − 1 (2)
[(N − i)λ + Rµ1]Pi, 0 = (N − i + 1)λPi−1, 0 + µ2Pi, 1 + R(1 − θ)µ1Pi+1, 0, R ≤ i ≤ N − 1 (3)
Rµ1Pi, 0 = λPi−1, 0, i = N (4) 200 KUO-HSIUNG WANG, CHUEN-WEN LIAO AND TSENG-CHANG YEN
(ii) 1 ≤ j ≤ R − 1
[(N − j)λ + jµ2]P0, j = θµ1P1, j−1 + (1 − θ)µ1P1, j + (j + 1)µ2P0, j+1, (5)
[(N − i − j)λ + iµ1 + jµ2]Pi, j = (N − i − j + 1)λPi−1, j + (j + 1)µ2Pi, j+1
+(i + 1)θµ1Pi+1, j−1 + (i + 1)(1 − θ)µ1Pi+1, j , 1 ≤ i ≤ R − j − 1 (6)
[(N − i − j)λ + (R − j)µ1 + jµ2]Pi, j = (N − i − j + 1)λPi−1, j
+(j + 1)µ2Pi, j+1 + (R − j + 1)θµ1Pi+1, j−1 + (R − j)(1 − θ)µ1Pi+1, j , R − j ≤ i ≤ N − j − 1 (7)
[(R − j)µ1 + jµ2]Pi, j = λPi−1, j + (R − j + 1)θµ1Pi+1, j−1, i = N − j (8)
(iii) j = R
[(N − R)λ + Rµ2]P0, R = θµ1P1, R−1, (9)
[(N−i−R)λ+Rµ2]Pi, R = (N−i−R+1)λPi−1, R+θµ1Pi+1, R−1, 1 ≤ i ≤ N−R−1 (10)
Rµ2Pi, R = λPi−1, R + θµ1Pi+1, R−1, i = N − R. (11)
3.2. Matrix-analytic solutions. A matrix-geometric method was first introduced by Neuts [10] while studying the embedded Markov chains of many practical queue- ing systems. We provide a matrix-analytic method to develop the steady-state probabilities Pi, j ,0 ≤ j ≤ R and 0 ≤ i ≤ N − j. The corresponding transition rate matrix Q of this Markov chain has the following block-tridiagonal form:
A0 B0 C1 A1 B1 C A B 2 2 2 C3 A3 B3 Q = . . . ...... C − A − B − R 2 R 2 R 2 C − A − B − R 1 R 1 R 1 C A R R The rate matrix Q of this state process is similar to the quasi birth and death type, and this class of Markov process has been extensively investigated by Neuts [10]. R We should note that the matrix Q is a square matrix of order (R+1)×(N +1− 2 ). Each entry of the matrix Q is listed in the following: COST ANALYSIS OF THE M/M/R MACHINE REPAIR 201
x0, j z0, j 0 0 · · · 0 0 0 y1, j x1, j z1, j 0 · · · 0 0 0 0 y , j x , j z , j · · · 0 0 0 2 2 2 0 0 y3, j x3, j · · · 0 0 0 Aj = ...... , ...... 0 0 0 0 · · · x − − z − − 0 N 2 j, j N 2 j, j 0 0 0 0 · · · y − − x − − z − − N 1 j, j N 1 j, j N 1 j, j 0 0 0 0 · · · 0 y − x − N j, j N j, j 0 ≤ j ≤ R (12)
0 0 0 · · · 0 0 w1, j 0 0 · · · 0 0 0 w , j 0 · · · 0 0 2 0 0 w3, j · · · 0 0 Bj = , 0 ≤ j ≤ R − 1 (13) ...... ...... 0 0 0 · · · w − − 0 N 1 j, j 0 0 0 · · · 0 w − N j, j
−jµ2 0 · · · 0 00 0 −jµ2 · · · 0 00 ...... Cj = ...... , 1 ≤ j ≤ R (14) 0 0 ··· −jµ 0 0 2 0 0 · · · 0 −jµ 0 2 where Aj , Bj , and Cj are (N +1 − j) × (N +1 − j), (N +1 − j) × (N − j), (N +1 − j) × (N +2 − j) matrices, respectively, and
(N − i − j)λ + iµ1 + jµ2, 0 ≤ j ≤ R − 1, 0 ≤ i ≤ R − 1 − j xi, j = (N − i − j)λ + (R − j)µ1 + jµ2, 0 ≤ j ≤ R, R − j ≤ i ≤ N − 1 − j (R − j)µ1 + jµ2, 0 ≤ j ≤ R, i = N − j −i(1 − θ)µ1, 0 ≤ j ≤ R − 1, 0 ≤ i ≤ R − 1 − j yi, j = −(R − j)(1 − θ)µ1, 0 ≤ j ≤ R − 1, R − j ≤ i ≤ N − j 0, j = R, 0 ≤ i ≤ N − j zi, j = −(N − i − j)λ, 0 ≤ j ≤ R, 0 ≤ i ≤ N − 1 − j
0, 0 ≤ j ≤ R − 1, i =0 wi, j = −iθµ1, 0 ≤ j ≤ R − 1, 0
P0A0 + P1C1 =0, (15)
Pj−1Bj−1 + Pj Aj + Pj+1Cj+1 =0, 1 ≤ j ≤ R − 1 (16)
PR−1BR−1 + PRAR =0. (17) Thus we obtain after routine substitutions
−1 PR = −PR−1BR−1AR , (18)
Pj = Pj−1Xj , 1 ≤ j ≤ R − 1 (19)
P0(A0 + X1C1)=0, (20) where
−1 Xj = −Bj−1(Aj + Xj+1Cj+1) , 1 ≤ j ≤ R − 1 are (N +2 − j) × (N +1 − j) matrices, andAj ,Bj ,Cj are given in (12), (13) and −1 (14), respectively. Furthermore, we haveXR = −BR−1AR .
Equation (20) determines P0 up to a multiplicative constant. The other equations (18) and (19) determine PR, PR−1, PR−2,...,P1, up to the same constant, which is uniquely determined using the following normalizing equation
R Pj e =1, Xj=0 where e represents a column vector with each component equal to one.
An efficient computer program was developed to solve Pj and Pi, j for 0 ≤ j ≤ R and 0 ≤ i ≤ N − j.
4. System performance measures. Our analysis is based on the following sys- tem performance measures of an M/M/R machine repair problem with second op- tional repair. Let:
E[N1] ≡ average number of failed machines in the first essential repair, E[N2] ≡ average number of failed machines in the second optional repair, LS ≡ average number of failed machines in the system, E[O] ≡ average number of operating machines in the system, E[I] ≡ average number of idle repairmen in the system, E[B] ≡ average number of busy repairmen in the system, M.A. ≡ machine availability, O.U. ≡ operative utilization.
The expressions for E[N1], E[N2], LS, E[O], E[I], E[B], M.A. and O.U. are obtained as follows:
R N−j
E[N1] = i Pi, j , (21) Xj=0 Xi=0 COST ANALYSIS OF THE M/M/R MACHINE REPAIR 203
R N−j
E[N2] = j Pi, j , (22) Xj=0 Xi=0
LS = E[N1]+ E[N2], (23)
E[O] = N − LS, (24)
R−1 R−1 R−1−j E[I] = (R − i − j) Pi, j = (R − i − j) Pi, j , (25) i+Xj=0 Xj=0 Xi=0 E[B] = R − E[I], (26)
L M.A. = 1 − S , (27) N E[B] O.U. = . (28) R
There is no neater expression for Pi, j . Therefore, analytical results for various system performance measures are extremely difficult to obtain. An efficient Matlab computer program is used to calculate the measures of effectiveness for the M/M/R machine repair problem with second optional repair.
5. Cost analysis. We develop a steady-state expected cost function per unit time and impose a constraint on the system availability in which R, µ1 and µ2 are decision variables. The discrete variable R is required to be natural numbers, and the continuous variables µ1 and µ2 are positive numbers. The objective of this ∗ ∗ ∗ section is to determine the optimal value of (R, µ1, µ2), say (R , µ1, µ2) so as to minimize this function and maintain the system availability at a certain level. Let
AV ≡ the steady-state probability that at least one machine is in operation. A0 ≡ the minimum fraction of one machine is in operation.
We define the following cost elements:
C1 ≡ cost per unit time when one failed machine in the system, C2 ≡ cost per unit time when one repairman is busy, C3 ≡ cost per unit time when one repairman is idle, C4 ≡ fixed cost for every repair rate of the first essential repair, C5 ≡ fixed cost for every repair rate of the second optional repair. The cost minimization problem can be presented mathematically as
Minimize F (R, µ1, µ2) (29) R, µ1, µ2
Subject to: AV ≥ A0 where
F (R, µ1, µ2)= C1LS + C2E[B]+ C3E[I]+ C4µ1 + C5µ2. (30) 204 KUO-HSIUNG WANG, CHUEN-WEN LIAO AND TSENG-CHANG YEN
The cost parameters in (30) are assumed to be linear in the expected number of the indicated quantity. It is extremely difficult to obtain the useful analytical results for the optimum ∗ ∗ ∗ value (R , µ1, µ2), due to the highly non-linear and complex of the optimization ∗ ∗ ∗ problem. To find the optimal value (R , µ1, µ2), we first determine the major adjustment quantity R and then determine the minor adjustment quantities µ1 and ∗ µ2. Therefore, we first use the direct search method to find (R , µ1, µ2) when µ1 and µ2 are fixed. Next, based on this solution, we use Newton-Quasi method to ∗ ∗ ∗ ∗ find the optimal value (R , µ1, µ2) when R is fixed. 5.1. Direct search method. We fix the number of operating machines N = 10 and choose A0 =0.9. Due to the discrete property of R, we first use direct substi- tution of successive value of R into the cost function until the minimum value of ∗ F (R, µ1, µ2), say F (R , µ1, µ2) is achieved and constraint AV ≥ A0 is satisfied. Numerical results are provided by considering the following cost parameters:
C1 =$150/day, C2=$100/day, C3=$60/day, C4=$60/day, C5=$90/day. The cost minimization problem can be illustrated mathematically as
∗ F (R , µ1, µ2)= Minimize F (R, µ1, µ2) (31) R
Subject to: AV ≥ A0.
We fix (µ1, µ2) =(4.0, 2.0), N = 10, and choose different values of (λ, θ). The ∗ minimum expected cost F (R , µ1, µ2) and the values of various system perfor- mance measures LS, AV , E[O], E[I], E[B], M.A. and O.U., at the optimum value R∗ are shown in Table 1 for different values of (λ, θ). From Table 1, we observe ∗ that (i) F (R , µ1, µ2) increases as λ increases or θ increases; (ii) the optimum values for R∗ increases as λ increases or θ increases. Table 1. System performance measures of the M/M/R machine repair problem with second optional repair under optimal operating conditions (µ1 =4.0, µ2 =2.0, N=10)
(λ, θ) (0.8, 0.3) (1.0, 0.3) (1.5, 0.3) (0.8, 0.2) (0.8, 0.4) (0.8, 0.6) R∗ 3 4 4 3 4 4 ∗ F (R , µ1, µ2) 1129.53 1232.81 1444.72 1066.37 1185.16 1276.66 LS 2.926 3.080 4.323 2.553 2.810 3.328 AV 0.999 0.999 0.998 0.999 0.999 0.999 E[O] 7.073 6.919 5.676 7.446 7.189 6.671 E[I] 0.736 1.232 0.593 0.914 1.411 1.064 E[B] 2.263 2.767 3.406 2.085 2.588 2.935 M.A. 0.707 0.691 0.567 0.744 0.718 0.667 O.U. 0.754 0.691 0.851 0.695 0.647 0.733
We fix (λ, θ) =(0.6, 0.3), N = 10, and choose different values of (µ1, µ2). The ∗ minimum expected cost F (R , µ1, µ2) and the values of various system perfor- mance measures LS, AV , E[O], E[I], E[B], M.A. and O.U., at the optimum value ∗ R are shown in Table 2 for various values of (µ1, µ2). It appears from Table 2 ∗ that F (R , µ1, µ2) increases as µ1 decreases or µ2 increases. From the last three COST ANALYSIS OF THE M/M/R MACHINE REPAIR 205
∗ columns of Table 2, R is the same even though µ2 varies from 2.0 to 3.5. We may ∗ conclude that µ2 rarely affects R .
Table 2. System performance measures of the M/M/R machine repair problem with second optional repair under optimal operating conditions (λ =0.6, θ =0.3, N = 10)
(µ1, µ2) (3.5, 1.5) (3.0, 1.5) (2.5, 1.5) (2.5, 2.0) (2.5, 3.0) (2.5, 3.5) R∗ 3 3 4 3 3 3 ∗ F (R , µ1, µ2) 1010.46 1024.53 1050.02 1052.24 1094.19 1125.51 LS 2.666 2.926 2.809 2.998 2.711 2.630 AV 0.999 0.999 0.999 0.999 0.999 0.999 E[O] 7.333 7.073 7.190 7.001 7.288 7.369 E[I] 0.862 0.736 1.411 0.689 0.813 0.852 E[B] 2.137 2.263 2.588 2.310 2.186 2.147 M.A. 0.733 0.707 0.719 0.700 0.728 0.736 O.U. 0.712 0.754 0.647 0.770 0.728 0.715
5.2. Newton-Quasi method. Fixing R∗ and using Newton-Quasi method, we ∗ ∗ ∗ ∗ globally search (µ1, µ2) until the minimum value of F (R , µ1, µ2), say F (R , µ1, µ2) is obtained and constraint AV ≥ A0 is satisfied. The cost minimization problem can be illustrated mathematically as
∗ ∗ ∗ ∗ F (R , µ1, µ2)= Minimize F (R , µ1, µ2) (32) µ1, µ2
Subject to: AV ≥ A0. The steps of the Newton-Quasi method can be summarized as follows for any iteration. T 1. Let ~µn = [µ1, µ2] and set n = 0. 2. Set the initial trial solution for ~µn, and compute F (~µn). ~ T 3. Compute the cost gradient ∇F (~µn) = [∂F/∂µ1, ∂F/∂µ2] |~µ n and the cost Hessian matrix 2 2 2 ∂ F/∂µ1 ∂ F/∂µ1∂µ2 H(~µ)= 2 2 2 ∂ F/∂µ1∂µ2 ∂ F/∂µ2 −1 4. Find the new trial solution: ~µn+1 = ~µn − [H(~µ)] ∇~ F (~µn). 5. Set n = n+1 and repeat steps 2-4 until |∂F/∂µ1| < ε1 and |∂F/∂µ2| < ε2, where ε1 and ε2 are the tolerances. T ∗ ∗ 6. Find the global minimum value F (~µn )= F (µ1, µ2). We use the results shown in the fourth column of Table 1, that is, we select ∗ (λ, θ) = (0.8, 0.2), N = 10, and choose the initial trial solution (R , µ1, µ2) = ∗ (3, 4.0, 2.0) with initial value F (R , µ1, µ2) = 1066.366. We apply the Newton- Quasi method as mentioned above. It can be seen from Table 3 that after only four ∗ ∗ ∗ iterations, the minimum expected cost converges at this solution (R , µ1, µ2) = (3, 4.583876, 1.742317) with value 1058.173. Furthermore, we use the results shown in the first column of Table 2, that is, we select (λ, θ) = (0.6, 0.3), N = 10, and choose the initial trial solution ∗ ∗ (R , µ1, µ2) = (3, 3.5, 1.5) with initial value F (R , µ1, µ2) = 1010.46. We use 206 KUO-HSIUNG WANG, CHUEN-WEN LIAO AND TSENG-CHANG YEN
Table 3. Newton-Quasi method in searching the optimal solution (λ =0.8, θ =0.2, µ1 =4.0, µ2 =2.0, N = 10)
No. of iterations 0 1 2 3 4 ∗ F (R , µ1, µ2) 1066.366 1058.541 1058.174 1058.173 1058.173 AV 0.99989 0.99985 0.99988 0.99988 0.99988 R∗ 3 3 3 3 3 µ1 4.0 4.476601 4.580843 4.583874 4.583876 µ2 2.0 1.677610 1.739072 1.742308 1.742317 the Newton-Quasi method as mentioned above. It appears from Table 4 that af- ter only four iterations, the minimum expected cost also converges at this solution ∗ ∗ ∗ (R , µ1, µ2)=(3, 3.970005, 1.830623) with value 1000.347.
Table 4. Newton-Quasi method in searching the optimal solution (λ =0.6, θ =0.3, µ1 =3.5, µ2 =1.5, N = 10)
No. of iterations 0 1 2 3 4 ∗ F (R , µ1, µ2) 1010.460 1000.718 1000.348 1000.347 1000.347 AV 0.99984 0.99994 0.99995 0.99995 0.99995 R∗ 3 3 3 3 3 µ1 3.5 3.904177 3.968718 3.970005 3.970005 µ2 1.5 1.753625 1.825977 1.830605 1.830623
6. Conclusions. The matrix-analytic method presented in this paper works effi- ciently for the M/M/R machine repair problem with N identical operating machines and second optional repair. We have provided an efficient and effective Newton- Quasi method to determine the optimal number of the repairmen, the optimal values of the first essential repair rate, and the second optional repair rate while maintaining the system availability at a specified level. Various system performance measures are evaluated under optimal operating conditions.
REFERENCES
[1] J. Al-Jararha and K. C. Madan, An M/G/1 queue with second optional service with general service time distribution, Information and Management Sciences, 14 (2003), 47–56. [2] A. O. Allen, “Probability, Statistics and Queueing Theory with Computer Science Applica- tions,” Academic Press, New York, 1978. [3] H. Ashcroft, The productivity of several machines under the care of one operator, Journal of the Royal Statistical Society, B, 12 (1950), 145–151. [4] F. Benson and D. R. Cox, The productivity of machines requiring attention at random inter- vals, Journal of the Royal Statistical Society, B, 13 (1951), 65–82. [5] D. B. Bunday and R. E. Scraton, The G/M/r machine interference model, European Journal of Operational Research, 4 (1980), 399–402. [6] J. E. Hilliard, An approach to cost analysis of maintenance float systems, IIE Transactions, 8 (1976), 128–133. [7] K. C. Madan, An M/G/1 queue with second optional service, Queueing Systems, 34 (2000), 37–46. [8] D. G. Maritas and D. A. Xirokostas, The M/Ek/r machine interference model: steady state equations and numerical solutions, European Journal of Operational Research, 1 (1977), 112–123. COST ANALYSIS OF THE M/M/R MACHINE REPAIR 207
[9] J. Medhi, A single server Poisson input queue with a second optional server, Queueing Sys- tems, 42 (2002), 239–242. [10] M. F. Neuts, “Matrix Geometric Solutions in Stochastic Models,” The John Hopkins Univer- sity Press, Baltimore, 1981. [11] B. D. Sivazlian and K.-H. Wang, Economic analysis of the M/M/R machine repair problem with warm standbys, Microelectronics and Reliability, 29 (1989), 25–35. [12] J. Sztrik, On the finite-source G/M/r queue, European Journal of Operational Research, 20 (1985), 261–268. [13] J. Wang, An M/G/1 queue with second optional service and server breakdowns, Computers and Mathematics with Applications, 47 (2004), 1713–1723. [14] K.-H. Wang, J.-B. Ke and J.-C. Ke, Profit analysis of the M/M/R machine repair problem with balking, reneging, and standby switching failures, Computers and Operations Research, 34 (2007), 835–847. [15] K.-H. Wang and M.-Y. Kuo, Profit analysis of the M/Ek/1 machine repair problem with a non-reliable service station, Computers and Industrial Engineering, 32 (1997), 587–594. [16] J. A. White, J. W. Schmidt and G. K. Benett, “Analysis of Queueing System,” Academic Press, New York, 1975. Received January 2008; 1st revision August 2009; final revision October 2009. E-mail address: [email protected] E-mail address: spring [email protected] E-mail address: [email protected]