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batch Poisson arrivals; queue-size distribution; renewal theory; single vacation; transient state
A queueing system with batch Poisson arrivals and single vacations with the exhaustive service discipline is investigated. As the main result the representation for the Laplace transform of the transient queue-size distribution in the system which is empty before the opening is obtained. The approach consists of few stages. Firstly, some results for a ``usual'' system without vacations corresponding to the original one are derived. Next, applying the formula of total probability, the analysis of the original system on a single vacation cycle is brought to the study of the ``usual'' system. Finally, the renewal theory is used to derive the general result. Moreover, a numerical approach to analytical results is discussed and some illustrative numerical examples are given.
[1] Abate, J., Choudhury, G. L., Whitt, W.: An introduction to numerical transform inversion and its application to probability models. In: Computational Probability (W. Grassmann, ed.), Kluwer, Boston 2000, pp. 257-323. Zbl 0945.65008
[2] Bischof, W.: Analysis of $M/G/1$-queues with setup times and vacations under six different service disciplines. Queueing Syst. 39 (2001), 4, 265-301. DOI 10.1023/A:1013992708103 | MR 1885740 | Zbl 0994.60088
[3] Borovkov, A. A.: Stochastic Processes in Queueing Theory. Springer-Verlag 1976. MR 0391297 | Zbl 0319.60057
[4] Bratiichuk, M. S., Kempa, W. M.: Application of the superposition of renewal processes to the study of batch arrival queues. Queueing Syst. 44 (2003), 51-67. MR 1989866
[5] Bratiichuk, M. S., Kempa, W. M.: Explicit formulae for the queue length distribution of batch arrival systems. Stoch. Models 20 (2004), 4, 457-472. DOI 10.1081/STM-200033115 | MR 2094048
[6] Choudhury, G.: A batch arrival queue with a vacation time under single vacation policy. Comput. Oper. Res. 29 (2002), 14, 1941-1955. DOI 10.1016/S0305-0548(01)00059-4 | MR 1920586 | Zbl 1010.90010
[7] Hur, S., Ahn, S.: Batch arrival queues with vacations and server setup. Appl. Math. Model. 29 (2005), 12, 1164-1181. DOI 10.1016/j.apm.2005.03.002 | Zbl 1163.90425
[8] Kempa, W. M.: $GI/G/1/\infty$ batch arrival queueing system with a single exponential vacation. Math. Methods Oper. Res. 69 (2009), 1, 81-97. DOI 10.1007/s00186-008-0212-2 | MR 2476049 | Zbl 1170.60032
[9] Kempa, W. M.: Some new results for departure process in the $M^{X}/G/1$ queueing system with a single vacation and exhaustive service. Stoch. Anal. Appl. 28 (2010), 1, 26-43. DOI 10.1080/07362990903417920 | MR 2597978 | Zbl 1189.60168
[10] Kempa, W. M.: On departure process in the batch arrival queue with single vacation and setup time. Ann. UMCS, AI 10 (2010), 1, 93-102. MR 3116951 | Zbl 1284.60162
[11] Kempa, W. M.: Characteristics of vacation cycle in the batch arrival queueing system with single vacations and exhaustive service. Internat. J. Appl. Math. 23 (2010), 4, 747-758. MR 2731457 | Zbl 1208.60096
[12] Kempa, W. M.: On main characteristics of the $M/M/1/N$ queue with single and batch arrivals and the queue size controlled by AQM algorithms. Kybernetika 47 (2011), 6, 930-943. MR 2907852 | Zbl 1241.90035
[13] Kempa, W. M.: The virtual waiting time in a finite-buffer queue with a single vacation policy. Lecture Notes Comp. Sci. 7314 (2012), 47-60. DOI 10.1007/978-3-642-30782-9_4
[14] Prabhu, N. U.: Stochastic Storage Processes. Springer 1998. MR 1492990 | Zbl 0888.60073
[15] Takagi, H.: Queueing Analysis. A Foundation of Performance Evaluation. Volume 1: Vacation and Priority Systems. Part 1. North-Holland, Amsterdam 1991. MR 1149382 | Zbl 0744.60114
[16] Tang, Y., Tang, X.: The queue-length distribution for $M^{x}/G/1$ queue with single server vacation. Acta Math. Sci. (Eng. Ed.) 20 (2000), 3, 397-408. MR 1793213 | Zbl 0984.60097
[17] Tian, N., Zhang, Z. G.: Vacation Queueing Models. Theory and Applications. Springer, New York 2006. MR 2248264 | Zbl 1104.60004
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