{"title":"On Four Models of a Three Server Queue with Optional Server Vacations","authors":"Kailash C. Madan","volume":116,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":400,"pagesEnd":405,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/10005293","abstract":"We study four models of a three server queueing system with Bernoulli schedule optional server vacations. Customers arriving at the system one by one in a Poisson process are provided identical exponential service by three parallel servers according to a first-come, first served queue discipline. In model A, all three servers may be allowed a vacation at one time, in Model B at the most two of the three servers may be allowed a vacation at one time, in model C at the most one server is allowed a vacation, and in model D no server is allowed a vacation. We study steady the state behavior of the four models and obtain steady state probability generating functions for the queue size at a random point of time for all states of the system. In model D, a known result for a three server queueing system without server vacations is derived.","references":"[1] Borthakur, A. and Choudhury, G., On a Batch Arrival Poisson Queue with Generalized Vacation, Sankhya Ser. B, 59(1997), 369-383.\r\n[2] Ke Jau-Chuan et al, A Note on Multi-Server Queue with Vacations of Multiple Groups of Servers , Quality Technology and Quantitative Management, 10, 4(2013), 513-525\r\n[3] Krishnamoorthy A. and Srinivasan, C., An M\/M\/2 Queueing System with Heterogeneous Servers Including one Working Vacation, International Journal of Stochastic Analysis, Article ID 145867(2012), 16 pages.\r\n[4] Madan, Kailash C. et al , Steady State Analysis of an M\/D\/2 Queue with Bernoulli Schedule Server Vacations, Journal of Modern Applied Statistical Methods , 2, 1(2003), 202-209.\r\n[5] Madan, Kailash C. et al, A Two-Server Queue with Bernoulli Schedules and a Single Vacation Policy, Applied Math and Computation, 145, 1(2003), 59-71.\r\n[6] Madan, Kailash C., 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 Exhaustive Service of the Priority Units, Applied Mathematics, 3, 1(2011), 03-08.\r\n[7] Madan, Kailash C., On a M(X)\/(G1, G2)\/1 Queue with Third Stage Optional Service and Deterministic Server Vacations, Journal of Mathematical and Computational Science, 5, 2(2015), 195-206. [8] Xu Xiuli and Zhang, Z. George , Analysis of Multi-Server Queue with a Single Vacation (e, d)-policy, Performance Evaluation, 63(2006), 825-838.\r\n[9] Shanthikumar, J.G., On Stochastic Decomposition in the M\/G\/1 Type Queues with Generalized Vacations, Operations Research, 36(1988), 566-569.\r\n[10] Takagi, H., Time dependent Process of M\/G\/1 Vacation Models with Exhaustive Service, J. Appl. Prob., 29(1992), 418-429.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 116, 2016"}