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On steady-state joint distribution of an infinite buffer batch service Poisson queue with single and multiple vacation. (English) Zbl 07319780

Summary: This article considers a single server, infinite buffer, bulk service Poisson queue with single and multiple vacation. The customers are served in batches following ‘general bulk service’ (GBS) rule. The customers are arriving according to the Poisson process, and the service time of the batches follows an exponential distribution. Using bivariate probability generating function (PGF) method the steady-state joint distributions of the queue content and server content (when server is busy), and joint distribution of the queue content and type of the vacation taken by the server (when server is in vacation) have been obtained. Here by the ‘type of the vacation’ we mean the queue length at vacation initiation epoch. The information about these joint distributions may help in increasing the system performance. Finally, several numerical examples are carried out using MAPLE software to verify the analytical results.

MSC:

90Bxx Operations research and management science

Software:

Maple
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References:

[1] Altman, E.; Nain, P., Optimality of a threshold policy in the \({M}/{M}/1\) queue with repeated vacations, Math. Methods Oper. Res., 44, 1, 75-96 (1996) · Zbl 0863.90065
[2] Baba, Y., Analysis of a \({GI}/{M}/1\) queue with multiple working vacations, Oper. Res. Lett., 33, 2, 201-209 (2005) · Zbl 1099.90013
[3] Banerjee, A.; Gupta, UC, Reducing congestion in bulk-service finite-buffer queueing system using batch-size-dependent service, Perform. Eval., 69, 1, 53-70 (2012)
[4] Banerjee, A.; Gupta, UC; Chakravarthy, SR, Analysis of a finite-buffer bulk-service queue under Markovian arrival process with batch-size-dependent service, Comput. Oper. Res., 60, 138-149 (2015) · Zbl 1348.90193
[5] Barbhuiya, F.P., Gupta, U.C.: A discrete-time \({GI}^X/{Geo}/1\) queue with multiple working vacations under late and early arrival system. Methodol. Comput. Appl. Probab. 1-26 (2019) · Zbl 1450.60053
[6] Chang, SH; Choi, DW, Performance analysis of a finite-buffer discrete-time queue with bulk arrival, bulk service and vacations, Comput. Oper. Res., 32, 9, 2213-2234 (2005) · Zbl 1067.60090
[7] Chaudhry, ML; Templeton, JGC, A First Course in Bulk Queues (1983), New York: Wiley, New York · Zbl 0559.60073
[8] Choi, BD; Han, DH, \({G}/{M}^{(a, b)}/1\) queues with server vacations, J. Oper. Res. Soc. Jpn., 37, 3, 171-181 (1994) · Zbl 0818.60089
[9] Cong, TD, Application of the method of collective marks to some \({M}/{G}/1\) vacation models with exhaustive service, Queueing Syst., 16, 1-2, 67-81 (1994) · Zbl 0801.90045
[10] Doshi, BT, Queueing systems with vacationsa survey, Queueing Syst., 1, 1, 29-66 (1986) · Zbl 0655.60089
[11] Frey, A.; Takahashi, Y., An explicit solution for an \({M}/{GI}/1/{N}\) queue with vacation time and exhaustive service discipline, J. Oper. Res. Soc. Jpn., 41, 3, 430-441 (1998) · Zbl 1002.60087
[12] Gupta, G.K., Banerjee, A., Gupta, U.C.: On finite-buffer batch-size-dependent bulk service queue with queue-length dependent vacation. Qual. Technol. Quant. Manag. 1-27 (2019) · Zbl 1411.90094
[13] Gupta, UC; Banik, AD; Pathak, SS, Complete analysis of \({M}{A}{P}/{G}/{1}/{N}\) queue with single (multiple) vacation (s) under limited service discipline, Int. J. Stoch. Anal., 2005, 3, 353-373 (2005) · Zbl 1107.60057
[14] Gupta, U.C., Pradhan, S.: Queue length and server content distribution in an infinite-buffer batch-service queue with batch-size-dependent service. Adv. Oper. Res. 2015 (2015) · Zbl 1387.90064
[15] Gupta, UC; Samanta, SK; Sharma, RK; Chaudhry, ML, Discrete-time single-server finite-buffer queues under discrete Markovian arrival process with vacations, Perform. Eval., 64, 1, 1-19 (2007)
[16] Gupta, UC; Sikdar, K., The finite-buffer \({M}/{G}/1\) queue with general bulk-service rule and single vacation, Perform. Eval., 57, 2, 199-219 (2004)
[17] Gupta, UC; Sikdar, K., Computing queue length distributions in \({MAP}/{G}/1/{N}\) queue under single and multiple vacation, Appl. Math. Comput., 174, 2, 1498-1525 (2006) · Zbl 1103.60077
[18] Haridass, M.; Arumuganathan, R., Analysis of a \({M}^X/{G (a, b)}/1\) queueing system with vacation interruption, RAIRO Oper. Res., 46, 4, 305-334 (2012) · Zbl 1268.60113
[19] Jain, M.; Singh, P., State dependent bulk service queue with delayed vacations, Eng. Sci., 16, 1, 3-15 (2005)
[20] Jeyakumar, S.; Senthilnathan, B., Modelling and analysis of a bulk service queueing model with multiple working vacations and server breakdown, RAIRO Oper. Res., 51, 2, 485-508 (2017) · Zbl 1367.60114
[21] Kalidass, K.; Gnanaraj, J.; Gopinath, S.; Kasturi, R., Transient analysis of an \({M}/{M}/1\) queue with a repairable server and multiple vacations, Int. J. Math. Oper. Res., 6, 2, 193-216 (2014) · Zbl 1390.90225
[22] Karaesmen, F.; Gupta, SM, The finite capacity \({GI}/{M}/1\) queue with server vacations, J. Oper. Res. Soc., 47, 6, 817-828 (1996) · Zbl 0855.90059
[23] Ke, JC; Wu, CH; Zhang, ZG, Recent developments in vacation queueing models: a short survey, Int. J. Oper. Res., 7, 4, 3-8 (2010)
[24] Kempa, WM, Transient workload distribution in the \({M}/{G}/1\) finite-buffer queue with single and multiple vacations, Ann. Oper. Res., 239, 2, 381-400 (2016) · Zbl 1338.90131
[25] Kim, S.J., Kim, N.K., Park, H.M., Chae, K.C., Lim, D.E.: On the discrete-time \({Geo}^X/{G}/1\) queues under \({N}\)-policy with single and multiple vacations. J. Appl. Math. 2013 (2013) · Zbl 1397.90118
[26] Laxmi, PV; Rajesh, P., Analysis of variant working vacations on batch arrival queues, OPSEARCH, 53, 2, 303-316 (2016) · Zbl 1360.90097
[27] Lee, HW; Lee, SS; Chae, KC, A fixed-size batch service queue with vacations, Int. J. Stoch. Anal., 9, 2, 205-219 (1996) · Zbl 0858.60085
[28] Lee, HW; Lee, SS; Chae, KC; Nadarajan, R., On a batch service queue with single vacation, Appl. Math. Model., 16, 1, 36-42 (1992) · Zbl 0752.60081
[29] Levy, Y.; Yechiali, U., Utilization of idle time in an \({M}/{G}/1\) queueing system, Manag. Sci., 22, 2, 202-211 (1975) · Zbl 0313.60067
[30] Li, H.; Zhu, Y., Analysis of \({M}/{G}/1\) queues with delayed vacations and exhaustive service discipline, Eur. J. Oper. Res., 92, 1, 125-134 (1996) · Zbl 0913.90110
[31] Mao, B.; Wang, F.; Tian, N., Fluid model driven by an \({M}/{M}/1\) queue with multiple vacations and \({N}\)-policy, J. Appl. Math. Comput., 38, 1-2, 119-131 (2012) · Zbl 1296.60248
[32] Medhi, J., Stochastic Models in Queueing Theory (2002), Cambridge: Academic Press, Cambridge · Zbl 0743.60100
[33] Nadarajan, R., Subramanian, A.: A general bulk service queue with server’s vacation. Oper. Res. Manag. Syst. 127-135 (1984)
[34] Neuts, MF, A general class of bulk queues with poisson input, Ann. Math. Stat., 38, 3, 759-770 (1967) · Zbl 0157.25204
[35] Niranjan, S.; Chandrasekaran, V.; Indhira, K., Performance characteristics of a batch service queueing system with functioning server failure and multiple vacations, J. Phys. Conf. Ser., 1000, 012112 (2018)
[36] Panda, G.; Banik, AD; Guha, D., Stationary analysis and optimal control under multiple working vacation policy in a \({GI}/{M}^{(a, b)}/1\) queue, J. Syst. Sci. Complex., 31, 4, 1003-1023 (2018) · Zbl 1406.90028
[37] Pradhan, S., Gupta, U.C.: Analysis of an infinite-buffer batch-size-dependent service queue with markovian arrival process. Ann. Oper. Res. 1-36 (2017)
[38] Pradhan, S.; Gupta, UC; Samanta, SK, Queue-length distribution of a batch service queue with random capacity and batch size dependent service: \({M}/{G^Y_r}/1\), OPSEARCH, 53, 329-343 (2016) · Zbl 1360.60168
[39] Samanta, SK; Chaudhry, ML; Gupta, UC, Discrete-time \({Geo}^X/{G}^{(a, b)}/1/{N}\) queues with single and multiple vacations, Math. Comput. Model., 45, 1-2, 93-108 (2007) · Zbl 1138.60338
[40] Scholl, M.; Kleinrock, L., On the \({M}/{G}/1\) queue with rest periods and certain service-independent queueing disciplines, Oper. Res., 31, 4, 705-719 (1983) · Zbl 0523.60088
[41] Servi, LD; Finn, SG, \({M}/{M}/1\) queues with working vacations \({M}/{M}/1/{WV}\), Perform. Eval., 50, 1, 41-52 (2002)
[42] Sikdar, K.; Gupta, UC, Analytic and numerical aspects of batch service queues with single vacation, Comput. Oper. Res., 32, 4, 943-966 (2005) · Zbl 1071.90016
[43] Sikdar, K.; Gupta, UC, On the batch arrival batch service queue with finite buffer under servers vacation: \({M}^X/{G}^Y/1/{N}\) queue, Comput. Math. Appl., 56, 11, 2861-2873 (2008) · Zbl 1165.90407
[44] Sikdar, K.; Samanta, SK, Analysis of a finite buffer variable batch service queue with batch Markovian arrival process and servers vacation, OPSEARCH, 53, 3, 553-583 (2016) · Zbl 1360.90101
[45] Takagi, H., Time-dependent analysis of \({M}/{G}/1\) vacation models with exhaustive service, Queueing Syst., 6, 1, 369-389 (1990) · Zbl 0696.60091
[46] Takagi, H., Queueing Analysis: A Foundation of Performance Evaluation (1991), New York: North-Holland, New York · Zbl 0744.60114
[47] Thangaraj, M.; Rajendran, P., Analysis of batch arrival queueing system with two types of service and two types of vacation, Int. J. Pure Appl. Math., 117, 11, 263-272 (2017)
[48] Tian, N.; Zhang, ZG, Vacation Queueing Models: Theory and Applications (2006), Berlin: Springer, Berlin · Zbl 1104.60004
[49] van der Duyn Schouten, FA, An \({M}/{G}/1\) queueing model with vacation times, Zeitschrift für Oper. Res., 22, 1, 95-105 (1978) · Zbl 0378.60089
[50] Wu, DA; Takagi, H., \({M}/{G}/1\) queue with multiple working vacations, Perform. Eval., 63, 7, 654-681 (2006)
[51] Wu, W.; Tang, Y.; Yu, M., Analysis of an \({M}/{G}/1\) queue with \({N}\)-policy, single vacation, unreliable service station and replaceable repair facility, OPSEARCH, 52, 4, 670-691 (2015) · Zbl 1365.90110
[52] Yang, D.; Ke, J., Cost optimization of a repairable \({M}/{G}/1\) queue with a randomized policy and single vacation, Appl. Math. Model., 38, 21-22, 5113-5125 (2014) · Zbl 1428.90047
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