M/M/3/3 and M/M/4/4 retrial queues

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A Markovian queue with two heterogeneous servers and multiple vacations

July 2009, 5(3): 431-451. doi: 10.3934/jimo.2009.5.431

M/M/3/3 and M/M/4/4 retrial queues

Tuan Phung-Duc ^1, , Hiroyuki Masuyama ^1, , Shoji Kasahara ^1, and Yutaka Takahashi ^2,

1.	Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan, Japan, Japan
2.	Dept. of Systems Science, Graduate School of Informatics, Kyoto University, Kyoto 606-8501

Received September 2008 Revised December 2008 Published June 2009

Abstract
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This paper studies M/M/$c$/$c$ retrial queues, where $c$ servers are all identical. In the retrial queues, an arriving customer is served immediately if it finds an idle server upon arrival, otherwise the customer tries to enter the system after an exponentially distributed time independently of other customers. As is well known, it is a challenging problem to obtain an analytical solution for the stationary joint distribution of the numbers of retrial customers and busy servers in the M/M/$c$/$c$ retrial queue especially for $c \ge 3$. Under some technical assumptions, a few analytical solutions have been presented for $c \ge 3$. This paper derives analytical solutions for M/M/3/3 and M/M/4/4 retrial queues without such technical assumptions. Through many numerical examples, we show that the derived analytical solutions can be computed by a numerically stable algorithm.

Keywords: analytical solution, minimal solution, Multiserver retrial queue, continued fractions., three-term recurrence relations.

Mathematics Subject Classification: Primary: 68M20, 90B22; Secondary: 60K2.

Citation: Tuan Phung-Duc, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. M/M/3/3 and M/M/4/4 retrial queues. Journal of Industrial & Management Optimization, 2009, 5 (3) : 431-451. doi: 10.3934/jimo.2009.5.431

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