On optional deterministic server vacations in a batch arrival queueing system with a single server providing first essential service followed by one of the two types of additional optional service

Kailash C. Madan

doi:10.26637/mjm401/004

Abstract

We analyze a batch arrival queue with a single server providing first essential service (FES) followed by one of the two types of additional optional service (AOS). After completion of the FES, a customer has the option to leave the system or to choose one of the two types of AOS and as soon as a customer leaves (either after the FES or after completing one of the chosen AOS, the server may take a vacation or may continue staying in the system. The vacation times are assumed to be deterministic and the server vacations are based on Bernoulli schedules under a single vacation policy. We obtain explicit queue size distribution at a random epoch under the steady state. In addition, some important performance measures such as the steady state expected queue size and the expected waiting time of a customer at a random epoch are obtained. Further, some interesting particular cases are also discussed.

Keywords:

Batch arrivals, compound Poisson process, first essential service (FES), additional optional service (AOS), deterministic server vacation, queue size distribution

Mathematics Subject Classification:

60K25, 60J15, 90B10

Authors Imprint References Funding Related Articles Downloads

Kailash C. Madan Ahlia University, PO Box 10878, GOSI Complex, Kingdom of Bahrain.

Pages: 25-36
Date Published: 01-01-2016
Vol. 4 No. 01 (2016): Malaya Journal of Matematik (MJM)

Borthakur, A. and Chaudhury; G. (1997): On a batch arrival Poisson queue with generalized racation. Sankhya Ser. B, 59, 369-383.

Choudhury, $G$. (2000): An $M^F / G / 1$ queueing system with a setup period and a vacation period. Queueing Systems, $36,23-38$. DOI: https://doi.org/10.1023/A:1019170817355

Choudhury, $G$. (2002): A batch arrival queue with a vacation time under single vacation policy. Computers and Operations Research, 29, 1941-1955. DOI: https://doi.org/10.1016/S0305-0548(01)00059-4

Choudhury G. and Madan, Kailash C. (2006) A batch arrival queue with Bemoulli vacation schedule under multiple vacation policy: International Joumal of Management Sciences, Volume 12, Number 2, pp 1-18.

Gaver, D.P. (1962): A Wiiting Line with Interrupted Service Including Priorities. Journal of Royal Statistical Society, See, - B, $24(19,2), 73-90$. DOI: https://doi.org/10.1111/j.2517-6161.1962.tb00438.x

J. Keilson, J- and Servi, L.D. {1986}: Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules. Journal of Applied Probability, 23, 790-802. DOI: https://doi.org/10.1017/S0021900200111933

Krishnamoothy A. and Sreeniwasaan, C. (2012): An $M / M / 2$ queueing system with heterogeneous servers including one working vacation. Intemational Joumal of Stochastic Analysis, Article ID 145867 , 16 pages. DOI: https://doi.org/10.1155/2012/145867

Lee, HS. and Srinivasan, M.M. (1989). Control policies for the $M^x / mathrm{G} / 1$ queueing systems. Management Sciences, $35,706-721$ DOI: https://doi.org/10.1287/mnsc.35.6.708

Madan, Kailash C. (2011) : 'A non-preemptive priority queueing system with a single server serving two queues $M / G / 1$ and $M / D / 1$ with Optional server vacations based on exhanstive service of the priority units. Applied Mathematics, Vol. 3, No, 1, pp 03-08. DOI: https://doi.org/10.4236/am.2011.26106

Madan, Kailash $C$ (1999): An $M / G / 1$ Queue with optional deterministic server vacations. Metron, LVII No, 3-4 (1999) $83-95$

Madan, Kailash C. (2000): An M/G/1 queue with second optional service. Queueing Systems 34, 37 -46. DOI: https://doi.org/10.1023/A:1019144716929

Madan, Kailash C. and Abu-Rub, Adel (2004): On a single server queue with optional phase type server vacations based on exhaustive deterministic service with a single vacation policy, Applied Math. And Computation, $149,723-734$ DOI: https://doi.org/10.1016/S0096-3003(03)00174-7

Madan, Kailash $C$. and Abu-Rub, Adel (2004) Transient and steady state solution of a single server queue with modified Bemoulli server vacation based on exhaustive service and a single vacation policy: Investigacton Operacional, 25, 158-165.

Madan, Kailash C. and Abu-Rub, Adel (2005): On a Single Server Queue with Phase Type Service and Phase Type Server Vacations Based on Exhaustive Deterministic Service Under a Single Vacation Policy, Journal of Applied Statistical Science, 15(1), 49-66.

Madan, Kailash $C$. and Choudhury, $G$. (2004): $A n M(X) / G / 1$ queue with a Bernoulli vacation schedule under restricted admisibility policy: Sankhya, Vol. 66, Part 1, PP 175-193.

Madan, Kailash C. (2001): On a single server queue with two-stage heterogeneous service and deterministic server vacations. International J- of Systems Science 32, 7, 837-844. DOI: https://doi.org/10.1080/00207720121488

Madan, Kailash $C$ (2015) On a $M(X) /(G 1, G 2) / 1$ queue with third stage optional service and deterministic server vacations. foumal of Mathematical and Computational Science, VoL. 5, Nor. 2, pp. 195-206.

Rosenberg, E. and Yechiali, U. (1993): The quetie with single and multiple vacations ander LIFO service regime. Open. Res. Lett., 14,171-179. DOI: https://doi.org/10.1016/0167-6377(93)90029-G

Shanthikumar, J,G. (1988): On stochastic decomposition in The $M / G / 1$ type queues with generalized vacations, Operations Research, $36,566-569$. DOI: https://doi.org/10.1287/opre.36.4.566

Takagi, H (1992): Time dependent process of $M / G / 1$ vacation models with exhaustive service. J- Appl. Prob. $29,418-429$ DOI: https://doi.org/10.1017/S0021900200043163

Tao, Li, Zhang Lyuan and Gao, Shan. (2014): $M / M / 1$ retrial queue with working vacation interruption and feedback under N policy. Joumal of Applied mathematics, pages 1-11, Article Id 414739. DOI: https://doi.org/10.1155/2014/414739

Tegham, L.J. (1990): On a decomposition result for a class of vacation queueing systems. Journal of Applied Probability 27, 227-231. DOI: https://doi.org/10.2307/3214611