Article Contents
Article Contents

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

• *Corresponding author: Sourav Pradhan
• 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
•  [1] L. Abolnikov and A. Dukhovny, Markov chains with transition delta-matrix: Ergodicity conditions, invariant probability measures and applications, J. Appl. Math. Stochastic Anal., 4 (1991), 333–-355. doi: 10.1155/S1048953391000254. [2] R. Arumuganathan and S. Jeyakumar, Analysis of a bulk queue with multiple vacations and closedown times, Internat. J. Inform. Management Sci., 15 (2004), 45-60. [3] R. Arumuganathan and S. Jeyakumar, Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy and closedown times, Applied Mathematical Modelling, 29 (2005), 972-986. [4] A. Banerjee and U. C. Gupta, Reducing congestion in bulk-service finite-buffer queueing system using batch-size-dependent service, Performance Evaluation, 69 (2012), 53-70. [5] A. Banerjee, U. C. Gupta and V. Goswami, Analysis of finite-buffer discrete-time batch-service queue with batch-size-dependent service, Computers and Industrial Engineering, 75 (2014), 121-128. [6] A. Banerjee, U. C. Gupta and K. Sikdar, Analysis of finite-buffer bulk-arrival bulk-service queue with variable service capacity and batch-size-dependent service: $M^X/G^Y_r/1/N$, Int. J. Math. Oper. Res., 5 (2013), 358-386.  doi: 10.1504/IJMOR.2013.053629. [7] A. D. Banik, U. C. Gupta and S. S. Pathak, Finite buffer vacation models under E-limited with limit variation service and Markovian arrival process, Oper. Res. Lett., 34 (2006), 539-547.  doi: 10.1016/j.orl.2005.08.006. [8] S. H. Chang and D. W. Choi, Performance analysis of a finite-buffer discrete-time queue with bulk arrival, bulk service and vacations, Computers and Operations Research, 32 (2005), 2213-2234. [9] B. D. Choi and D. H. Han, $GI/M^{(a, b)}/1$ queues with server vacations, J. Oper. Res. Soc. Japan, 37 (1994), 171-181.  doi: 10.15807/jorsj.37.171. [10] D. Claeys, B. Steyaert, J. Walraevens, K. Laevens and H. Bruneel, Analysis of a versatile batch-service queueing model with correlation in the arrival process, Performance Evaluation, 70 (2013), 300-316. [11] D. Claeys, B. Steyaert, J. Walraevens, K. Laevens and H. Bruneel, Tail probabilities of the delay in a batch-service queueing model with batch-size dependent service times and a timer mechanism, Comput. Oper. Res., 40 (2013), 1497-1505.  doi: 10.1016/j.cor.2012.10.009. [12] D. Claeys, J. Walraevens, K. Laevens and H. Bruneel, A queueing model for general group screening policies and dynamic item arrivals, European J. Oper. Res., 207 (2010), 827-835.  doi: 10.1016/j.ejor.2010.05.042. [13] D. Claeys, J. Walraevens, B. Steyaert and H. Bruneel, Applicability of a static model in a dynamic context in group-screening decision making, Comput. Oper. Res., 51 (2014), 313-322.  doi: 10.1016/j.cor.2014.06.017. [14] B. T. Doshi, Queueing systems with vacations - A survey, Queueing Systems Theory Appl., 1 (1986), 29-66.  doi: 10.1007/BF01149327. [15] V. Goswami and G. Panda, Synchronized abandonment in discrete-time renewal input queues with vacations, J. Ind. Manag. Optim., (2021). doi: 10.3934/jimo.2021163. [16] G. K. Gupta and A. Banerjee, On finite buffer bulk arrival bulk service queue with queue length and batch size dependent service, Int. J. Appl. Comput. Math., 5 (2019), Paper No. 32, 20 pp. doi: 10.1007/s40819-019-0617-z. [17] G. K. Gupta, A. Banerjee and U. C. Gupta, On finite-buffer batch-size-dependent bulk service queue with queue-length dependent vacation, Quality Technology and Quantitative Management, (2009), 1–27. [18] U. C. Gupta and S. Pradhan, Queue length and server content distribution in an infinite-buffer batch-service queue with batch-size-dependent service, Adv. Oper. Res., (2015), Art. ID 102824, 12 pp. doi: 10.1155/2015/102824. [19] U. C. Gupta, S. K. Samanta, R. K. Sharma and M. L. Chaudhry, Discrete-time single-server finite-buffer queues under discrete Markovian arrival process with vacations, Performance Evaluation, 64 (2007), 1-19. [20] U. C. Gupta and K. Sikdar, Computing queue length distributions in ${MAP/G/1/N}$ queue under single and multiple vacation, Appl. Math. Comput., 174 (2006), 1498-1525.  doi: 10.1016/j.amc.2005.07.001. [21] U. C. Gupta and K. Sikdar, On the finite buffer bulk service ${M/G/1}$ queue with multiple vacations, Journal of Probability and Statistical Science, 3 (2005), 175-189. [22] U. C. Gupta and K. Sikdar, The finite-buffer ${M/G/1}$ queue with general bulk-service rule and single vacation, Performance Evaluation, 57 (2004), 199-219. [23] S. Jeyakumar and R. Arumuganathan, A non-Markovian bulk queue with multiple vacations and control policy on request for re-service, Quality Technology and Quantitative Management, 8 (2011), 253-269. [24] S. Jeyakumar and B. Senthilnathan, A study on the behaviour of the server breakdown without interruption in a ${M^X/G^{(a, b)}/1}$ queueing system with multiple vacations and closedown time, Appl. Math. Comput., 219 (2012), 2618-2633.  doi: 10.1016/j.amc.2012.08.096. [25] J. C. Ke, Batch arrival queues under vacation policies with server breakdowns and startup/closedown times, Applied Mathematical Modelling, 31 (2007), 1282-1292. [26] H. W. Lee, S. S. Lee and K. C. Chae, A fixed-size batch service queue with vacations, J. Appl. Math. Stochastic Anal., 9 (1996), 205-219.  doi: 10.1155/S1048953396000196. [27] H. W. Lee, S. S. Lee, K. C. Chae and R. Nadarajan, On a batch service queue with single vacation, Applied Mathematical Modelling, 16 (1992), 36-42. [28] K. C. Madan, A. D. Walid and M. Gharaibeh, Steady state analysis of two queue models with random breakdowns, Internat. J. Inform. Management Sci., 14 (2003), 37-51. [29] S. Pradhan, A discrete-time batch transmission channel with random serving capacity under batch-size-dependent service: ${Geo^X/G^Y_n/1}$, Int. J. Comput. Math. Comput. Syst. Theory, 5 (2020), 175-197.  doi: 10.1080/23799927.2020.1792998. [30] S. Pradhan, On the distribution of an infinite-buffer queueing system with versatile bulk-service rule under batch-size-dependent service policy: ${M/G_n^{(a, Y)}/1}$, Int. J. Math. Oper. Res., 16 (2020), 407-434.  doi: 10.1504/IJMOR.2020.106908. [31] S. Pradhan and U. C. Gupta, Analysis of an infinite-buffer batch-size-dependent service queue with Markovian arrival process, Ann. Oper. Res., 277 (2019), 161-196.  doi: 10.1007/s10479-017-2476-5. [32] S. Pradhan and U. C. Gupta, Stationary queue and server content distribution of a batch-size-dependent service queue with batch Markovian arrival process: $BMAP/G^{(a, b)}_n/1$, Comm. Statist. Theory Methods, 51 (2022), 4330-4357.  doi: 10.1080/03610926.2020.1813304. [33] S. Pradhan and U. C. Gupta, Stationary distribution of an infinite-buffer batch-arrival and batch-service queue with random serving capacity and batch-size-dependent service, Int. J. Oper. Res., 40 (2021), 1-31.  doi: 10.1504/IJOR.2021.111951. [34] G. V. K. Reddy and R. Anitha, Markovian bulk service queue with delayed vacations, Comput. Oper. Res., 25 (1998), 1159-1166.  doi: 10.1016/S0305-0548(98)00003-3. [35] G. V. K. Reddy and R. Anitha, Non-Markovian bulk service queue with different vacation policies, Internat. J. Inform. Management Sci., 10 (1999), 59-67. [36] G. V. K. Reddy, R. Nadarajan and R. Arumuganathan, Analysis of a bulk queue with N-policy multiple vacations and setup times, Comput. Oper. Res., 25 (1998), 957-967.  doi: 10.1016/S0305-0548(97)00098-1. [37] S. K. Samanta, M. L. Chaudhry and U. C. Gupta, Discrete-time ${Geo^X/G^{(a, b)}/1/N}$ queues with single and multiple vacations, Math. Comput. Modelling, 45 (2007), 93-108.  doi: 10.1016/j.mcm.2006.04.008. [38] S. K. Samanta, U. C. Gupta and R. K. Sharma, Analysis of finite capacity discrete-time ${GI/Geo/1}$ queueing system with multiple vacations, Journal of the Operational Research Society, 58 (2007), 368-377. [39] S. K. Samanta, U. C. Gupta and R. K. Sharma, Analyzing discrete-time D-BMAP/G/1/N queue with single and multiple vacations, European J. Oper. Res., 182 (2007), 321-339.  doi: 10.1016/j.ejor.2006.09.031. [40] S. K. Samanta and R. Nandi, Analysis of ${GI^X/D-MSP/1/\infty}$ queue using $RG$-factorization, J. Ind. Manag. Optim., 17 (2021), 549-573.  doi: 10.3934/jimo.2019123. [41] K. Sikdar, Analysis of Finite/Infinite Buffer Bulk Service Queue with Poisson/Markovian Arrival Process and Server Vacations, PhD Thesis, IIT, Kharagpur, 2003. [42] K. Sikdar and U. C. Gupta, Analytic and numerical aspects of batch service queues with single vacation, Computers and Operations Research, 32 (2005), 943-966. [43] K. Sikdar and U. C. Gupta, The queue length distributions in the finite buffer bulk-service ${MAP/G/1}$ queue with multiple vacations, Top, 13 (2005), 75–103. doi: 10.1007/BF02578989. [44] A. S. Vadivu and R. Arumuganathan, Cost analysis of ${MAP/G^{(a, b)}/1/N}$ queue with multiple vacations and closedown times, Quality Technology & Quantitative Management, 12 (2015), 605-626.

Figures(6)

Tables(6)