An Explicit Solution for a Series and Parallel Queue with Retrial, Losses, and Bernoulli Schedule
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An Explicit Solution for a Series and Parallel Queue with Retrial, Losses, and Bernoulli Schedule Shi-Zhong Zhou1 · Li-Wei Liu1 · Jian-Jun Li1
Received: 16 March 2015 / Revised: 28 July 2015 / Accepted: 23 September 2015 / Published online: 15 October 2015 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2015
Abstract This paper deals with the series and parallel queueing system in which there are two servers whose service time follow two exponential distributions. Each arriving customer either enters into the tandem service with probability or joins the service of the single server with complementary probability. We assume that the customers of arriving at the first server who find the first server is busy join an orbit and retry to enter the server after some time and of arriving at the second server who find the second server is busy are lost. For this model, we obtain the explicit expressions of the joint stationary distribution between the number of customers in the orbit and the states of the servers. Keywords Series and parallel queue · Retrial · Bernoulli schedule · Explicit solution · Hypergeometric function Mathematics Subject Classification
68M20 · 90B22 · 60K25
1 Introduction Tandem queues with retrial have been extensively studied in the queueing, communications, and manufacturing literature because of their pervasiveness and significance
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Shi-Zhong Zhou [email protected] Li-Wei Liu [email protected] Jian-Jun Li [email protected]
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Department of Statistics and Financial Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing 210094, China
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S.-Z. Zhou et al.
in real life. The pioneering studies of tandem queue with blocking presented the concept of tandem queue as an alternative to the classical network queue. The first result of the tandem queue with blocking was given by Hunt [1] who studied the blocking effects in a sequence of waiting lines. A number of survey papers have been published during the last three decades [2–6]. In 1980, Lantouche and Neuts [7] gave an algorithmic solution to two-stage exponential tandem queues with blocking and feedback. Kim etc. [8–10] studied the BMAP/G/1-./PH/1/M tandem queue with feedback and losses. Lian and Zhao [11] obtained the busy periods of a tandem network. On discrete-time tandem queue, Van Houdt and Alfa [12] introduced the discrete-time tandem queue with blocking, Markovian arrivals, and phase-type services in 2005. In that paper, they obtained main performance measures by constructing a Markov chain based on the age of the leading customer in the first queue. A number of papers studied the retrial tandem queue system [13–16]. Avrachenkov and Yechiali [17] in 2010 considered a tandem blocking queues with a common retrial queue by using mean value analysis and fixed point approach. They obtained an analytic result for two nods in tandem. In comparison with tandem queues, there is a lack of extensive research concer
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