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# Machine interference problem involving unsuccessful switchover for a group of repairable servers with vacations

• * Corresponding author: Jau-Chuan Ke
• The purpose of this paper is to explore the multiple-server machine interference problem with standby unsuccessful switchover and Bernoulli vacation schedule. Failure times of operating and standby machines are assumed to have exponential distributions and repair times of the failed machines and vacation times of servers are also assumed to have exponential distributions. After the completion of service, the server either goes for a vacation or may continue serving for the next machine. The vacation policy we considered is a single vacation policy. In practice, the switchover may experience a significant failure. The matrix analytical method and recursive method are employed to obtain the steady-state probability vectors, and closed-form expressions of some important system characteristics are obtained. The problem of cost optimization dealt with a number of numerical examples is provided by the Quasi-Newton method, the pattern search method, and the Nelder-Mead simplex direct search method. Expressions of various system characteristics are derived. Sensitivity analysis is performed numerically for system parameters. This paper presents the first time that machine interference problem with unsuccessful switchover for a group of repairable servers with vacations has been obtained, which is quite useful for the decision makers.

Mathematics Subject Classification: Primary: 90B22; Secondary: 60K25.

 Citation: • • Figure 1.  Plot of the average cost function versus the mean service rate and mean vacation rate

Table 1.  The average cost function for given $(W, S)$ with $M = 6$, $\lambda = 0.5$, $\alpha = 0.2\lambda$, $\theta = 0.02$, $\beta = 0.02$

 $(S, W)$ (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (1, 7) (1, 8) $TAC$ 99.72 98.43 99.20 100.74 102.52 104.32 106.04 107.65 $(S, W)$ (1, 9) (1, 10) (1, 11) (1, 12) (2, 1) (2, 2) (2, 3) (2, 4) $TAC$ 109.13 110.49 111.74 112.88 92.32 $\mathbf{90.90}$ 92.45 94.40 $(S, W)$ (2, 5) (2, 6) (2, 7) (2, 8) (2, 9) (2, 10) (2, 11) (2, 12) $TAC$ 96.70 98.92 100.97 102.83 104.51 106.02 107.39 108.62 $(S, W)$ (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (3, 7) (3, 8) $TAC$ 97.43 94.99 96.04 97.96 100.03 102.00 103.83 105.47 $(S, W)$ (3, 9) (3, 10) (3, 11) (3, 12) (4, 1) (4, 2) (4, 3) (4, 4) $TAC$ 106.95 108.28 109.46 110.53 98.44 96.39 97.75 99.94 $(S, W)$ (4, 5) (4, 6) (4, 7) (4, 8) (4, 9) (4, 10) (4, 11) (4, 12) $TAC$ 102.17 104.22 106.04 107.65 109.07 110.33 111.45 112.44 $(S, W)$ (5, 9) (5, 10) (5, 11) (5, 12) (6, 1) (6, 2) (6, 3) (6, 4) $TAC$ 109.33 110.60 111.72 112.72 95.48 94.17 96.15 98.83 $(S, W)$ (6, 5) (6, 6) (6, 7) (6, 8) (6, 9) (6, 10) (6, 11) (6, 12) $TAC$ 101.39 103.66 105.63 107.35 108.84 110.16 111.33 112.37 $(S, W)$ (7, 2) (7, 3) (7, 4) (7, 5) (7, 6) (7, 7) (7, 8) (7, 9) $TAC$ 91.92 94.27 97.24 100.01 102.43 104.53 106.35 107.93 $(S, W)$ (7, 10) (7, 11) (7, 12) (8, 3) (8, 4) (8, 5) (8, 6) (8, 7) $TAC$ 109.32 110.54 111.64 91.97 95.24 98.25 100.85 103.10 $(S, W)$ (8, 8) (8, 9) (8, 10) (8, 11) (8, 12) (9, 4) (9, 5) (9, 6) $TAC$ 105.03 106.71 108.19 109.49 110.65 92.94 96.20 99.01 $(S, W)$ (9, 7) (9, 8) (9, 9) (9, 10) (9, 11) (9, 12) (10, 5) (10, 6) $TAC$ 101.41 103.48 105.28 106.85 108.24 109.48 93.93 96.95 $(S, W)$ (10, 7) (10, 8) (10, 9) (10, 10) (10, 11) (10, 12) (11, 6) (11, 7) $TAC$ 99.53 101.75 103.67 105.36 106.84 108.16 94.73 97.49 $(S, W)$ (11, 8) (11, 9) (11, 10) (11, 11) (11, 12) (12, 7) (12, 8) (12, 9) $TAC$ 99.87 101.92 103.72 105.31 106.71 95.32 97.86 100.06 $(S, W)$ (12, 10) (12, 11) (12, 12) $TAC$ 101.98 103.67 105.17

Table 2.  The minimum average cost function for varying values of $\lambda$ with $M = 6$, $\alpha = 0.2\lambda$, $\theta = 0.02$, $\beta = 0.02$

 QN method $\lambda$ 0.2 0.4 0.6 $TAC$ 68.34 85.47 95.95 $(W^*, S^*)$ (1, 1) (2, 2) (2, 2) $(\mu^*, \delta^*)$ (2.37, 5.43) (2.38, 2.14) (3.09, 4.09) Iterations 1204 1427 1583 CPU Time 6.26 5.62 5.64 MN method $TAC$ 68.34 85.47 95.95 $(W^*, S^*)$ (1, 1) (2, 2) (2, 2) $(\mu^*, \delta^*)$ (2.37, 5.43) (2.38, 2.14) (3.09, 4.09) Iterations 6741 6074 6176 CPU Time 6.39 5.77 5.62 PS method $TAC$ 68.34 85.47 95.95 $(W^*, S^*)$ (1, 1) (2, 2) (2, 2) $(\mu^*, \delta^*)$ (2.37, 5.43) (2.38, 2.14) (3.09, 4.09) Iterations 11679 13179 13294 CPU Time 22.44 23.57 25.80
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