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# Performance analysis of an infinite-buffer batch-size-dependent bulk service queue with server breakdown and multiple vacation

• In several manufacturing systems, server's breakdown is a common phenomenon for which renovation of the service station is required and has a definite effect on the system. The server's vacation is a significant tool to deal with the systems where a common server is shared by several users. The principal goal of this paper is to analyze a batch-service queueing model with server's breakdown, multiple vacations and batch-size-dependent service time. To be more precise, the center of attention is on the queue length together with server content distribution for which we obtain the bivariate probability generating function using supplementary variable technique. Server content distribution is very worthy for the manufacturing systems. The probabilities have been extracted and used to acquire the arbitrary epoch probabilities. In order to upgrade the qualitative aspects of the concerned model, diverse numerical results and graphical observations are illustrated.

Mathematics Subject Classification: Primary: 60K25, 68M20.

 Citation: • • Figure 1.  Total system cost versus service threshold value

Figure 2.  Total system cost versus arrival rate

Figure 3.  Average queue/system length versus breakdown probability

Figure 4.  Average waiting time in the queue/system versus breakdown probability

Figure 5.  Average queue/system length versus renovation rate

Figure 6.  Average waiting time in the queue/system versus renovation rate

Table 1.  Probability distributions at service completion epoch ($p_{n, r}^+, q_n^+, \gamma_n^+$) of an $M/G^{(a, b)}_n/1$ queue with service time $\sim D$, vacation time $\sim M$, renovation time $\sim HE_2$, $\sigma = 0.6$, $\lambda = 0.08$, $a = 5$, $b = 12$, $\mu_i = \frac{1.5}{i+1}$, $\rho = 0.627333$

 $n$ $p_{n, 5}^+$ $p_{n, 6}^+$ $p_{n, 7}^+$ $p_{n, 8}^+$ $p_{n, 9}^+$ $p_{n, 10}^+$ $p_{n, 11}^+$ $p_{n, 12}^+$ $p_n^+$ $q_n^+$ $\gamma_n^+$ 0 0.007077 0.003196 0.001409 0.000611 0.000263 0.000113 0.000048 0.000020 0.012741 0.002910 0.002665 1 0.018873 0.010228 0.005260 0.002610 0.001265 0.000603 0.000284 0.000148 0.039275 0.012533 0.009949 2 0.025164 0.016366 0.009819 0.005570 0.003037 0.001609 0.000835 0.000534 0.062936 0.029336 0.019634 3 0.022368 0.017457 0.012219 0.007921 0.004859 0.002861 0.001632 0.001299 0.070620 0.050256 0.027538 4 0.014912 0.013965 0.011405 0.008450 0.005831 0.003815 0.002394 0.002403 0.063178 0.071406 0.031126 5 0.007953 0.008938 0.008515 0.007210 0.005598 0.004069 0.002810 0.003618 0.048714 0.051931 0.030441 10 0.000035 0.000099 0.000204 0.000337 0.000471 0.000580 0.000645 0.004223 0.006597 0.010566 0.010236 15 0.000000 0.000000 0.000000 0.000001 0.000003 0.000007 0.000012 0.001178 0.001203 0.002149 0.002259 25 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000051 0.000051 0.000088 0.00010 35 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000002 0.000002 0.000003 0.000004 $\geq$40 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

Table 2.  Probability distributions at arbitrary epoch ($p_{n, r}, q_n, \gamma_n$) of an $M/G^{(a, b)}_n/1$ queue

 $n$ $p_{n, 5}$ $p_{n, 6}$ $p_{n, 7}$ $p_{n, 8}$ $p_{n, 9}$ $p_{n, 10}$ $p_{n, 11}$ $p_{n, 12}$ $p_n^{queue}$ $q_n$ $\gamma_n$ 0 0.031480 0.024983 0.019102 0.014285 0.010552 0.007748 0.005673 0.004151 0.122210 0.002578 0.001653 1 0.025211 0.021586 0.017355 0.013418 0.010132 0.007547 0.005579 0.007140 0.125249 0.011100 0.006175 2 0.016853 0.016150 0.014094 0.011568 0.009123 0.007013 0.005302 0.009185 0.127471 0.025983 0.012196 3 0.009424 0.010352 0.010035 0.008936 0.007509 0.006062 0.004759 0.010378 0.129095 0.044512 0.017123 4 0.004471 0.005714 0.006247 0.006130 0.005572 0.004795 0.003964 0.010767 0.130283 0.063244 0.019375 5 0.001829 0.002745 0.003419 0.003735 0.003713 0.003443 0.003030 0.010433 0.097320 0.045995 0.018972 10 0.000003 0.000012 0.000032 0.000066 0.000110 0.000162 0.000212 0.004268 0.020655 0.009358 0.006427 15 0.000000 0.000000 0.000000 0.000000 0.000000 0.000001 0.000002 0.000974 0.004310 0.001904 0.001427 25 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000041 0.000185 0.000078 0.000064 35 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000001 0.000007 0.000003 0.000003 $\geq$40 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 $L$=8.501607, $L_q$=4.115442, $L_{s}$=8.224059 $P_{busy}$=0.533333, $W$=10.627009, $W_q$=5.144302

Table 3.  Probability distributions at service completion epoch of an $M/G^{(a, b)}_n/1$ queue with service time$\sim$mixed, vacation time $\sim HE_2$, renovation time$\sim M$, $\lambda = 1.25$, $a = 4$, $b = 9$, $\mu_4 = 0.375000$, $\mu_5 = 0.300000$, $\mu_6 = 0.079365$, $\mu_7 = 0.068027$, $\mu_8 = 0.166667$, $\mu_9 = 0.150000$, $\rho = 0.127778$

 $n$ $p_{n, 4}^+$ $p_{n, 5}^+$ $p_{n, 6}^+$ $p_{n, 7}^+$ $p_{n, 8}^+$ $p_{n, 9}^+$ $p_n^+$ $q_n^+$ $\gamma_n^+$ 0 0.034180 0.007894 0.001682 0.000434 0.000176 0.000078 0.044446 0.116290 0.008889 1 0.019531 0.005263 0.001096 0.000297 0.000158 0.000072 0.026420 0.193773 0.008247 2 0.008370 0.002631 0.000714 0.000204 0.000071 0.000053 0.012047 0.233049 0.005158 3 0.003188 0.001169 0.000466 0.000140 0.000021 0.000037 0.005024 0.251047 0.002724 4 0.001138 0.000487 0.000304 0.000096 0.0000004 0.000025 0.002054 0.064234 0.001319 5 0.000390 0.000194 0.000199 0.000066 0.000000 0.000016 0.000865 0.016541 0.000613 10 0.000001 0.000001 0.000024 0.000010 0.000000 0.000002 0.000038 0.000020 0.000019 15 0.000000 0.000000 0.000003 0.000001 0.000000 0.000000 0.000004 0.000000 0.000002 $\geq$20 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

Table 4.  Probability distributions at arbitrary epoch of an $M/G^{(a, b)}_n/1$ queue

 $n$ $p_{n, 4}$ $p_{n, 5}$ $p_{n, 6}$ $p_{n, 7}$ $p_{n, 8}$ $p_{n, 9}$ $p_n^{queue}$ $q_n$ $\gamma_n$ 0 0.081646 0.024554 0.007868 0.002370 0.000640 0.000194 0.227867 0.099533 0.011059 1 0.033047 0.011458 0.005140 0.001629 0.000245 0.000181 0.228035 0.166072 0.010260 2 0.012219 0.004910 0.003361 0.001120 0.000067 0.000134 0.228122 0.199890 0.006417 3 0.004284 0.002000 0.002200 0.000771 0.000014 0.000092 0.228173 0.215419 0.003389 4 0.001450 0.000788 0.001442 0.000531 0.000002 0.000062 0.061510 0.055591 0.001641 5 0.000479 0.000303 0.000946 0.000366 0.000000 0.000041 0.017332 0.014432 0.000763 10 0.000001 0.000002 0.000116 0.000057 0.000000 0.000005 0.000226 0.000018 0.000024 15 0.000000 0.000000 0.000014 0.000009 0.000000 0.000000 0.000027 0.000000 0.000002 $\geq$20 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 $L$=2.722981, $L_q$=1.763711, $L_{s}$=4.574273 $P_{busy}$=0.209709, $W$=18.153210, $W_q$=11.758079

Table 5.  Performance measures for different probability of server's breakdown

 $\sigma$ $L_{q}$ $L$ $L_{s}$ $W_{q}$ $W$ 0.0 12.430488 18.920051 14.601525 6.215244 9.460026 0.05 12.541887 19.037950 14.616153 6.270943 9.518975 0.10 12.658263 19.160957 14.631074 6.329131 9.580479 0.15 12.779974 19.289436 14.646302 6.389987 9.644718 0.20 12.907417 19.423787 14.661850 6.453708 9.711894 0.25 13.041025 19.564453 14.677733 6.520513 9.782227 0.30 13.181282 19.711922 14.693964 6.590641 9.855961 0.35 13.328720 19.866734 14.710560 6.664360 9.933367 0.40 13.483936 20.029492 14.727539 6.741968 10.014746 0.45 13.647590 20.200867 14.744919 6.823795 10.100434 0.50 13.820427 20.381610 14.762720 6.910213 10.190805 0.55 14.003278 20.572562 14.780964 7.001639 10.286281 0.60 14.197083 20.774674 14.799673 7.098542 10.387337 0.65 14.402906 20.989018 14.818873 7.201453 10.494509 0.70 14.621952 21.216812 14.838590 7.310976 10.608406

Table 6.  Performance measures for different arrival rates

 $\lambda$ $L_{q}$ $L$ $L_{s}$ $W_{q}$ $W$ $P_{busy}$ $P_{idle}$ 1.75 11.310239 16.925046 14.438075 6.462994 9.671455 0.388889 0.611111 2.00 13.181282 19.711922 14.693964 6.590641 9.855961 0.444444 0.555556 2.25 15.621999 23.083005 14.922485 6.943111 10.259113 0.500000 0.500000 2.50 18.966311 27.371646 15.129603 7.586524 10.948658 0.555556 0.444444 2.75 23.855139 33.217037 15.319478 8.674596 12.078923 0.0.611111 0.388889 3.00 31.703060 42.033143 15.495129 10.567687 14.011048 0.666667 0.333333 3.25 46.372536 57.681670 15.658821 14.268472 17.748206 0.722222 0.277778 3.50 83.526050 95.824502 15.812296 23.864586 27.378429 0.777778 0.222222
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