Abstract
We revisit a single-server retrial queue with two independent Poisson streams (corresponding to two types of customers) and two orbits. The size of each orbit is infinite. The exponential server (with a rate independent of the type of customers) can hold at most one customer at a time and there is no waiting room. Upon arrival, if a type i customer \((i=1,2)\) finds a busy server, it will join the type i orbit. After an exponential time with a constant (retrial) rate \(\mu _i\), a type i customer attempts to get service. This model has been recently studied by Avrachenkov et al. (Queueing Syst 77(1):1–31, 2014) by solving a Riemann–Hilbert boundary value problem. One may notice that, this model is not a random walk in the quarter plane. Instead, it can be viewed as a random walk in the quarter plane modulated by a two-state Markov chain, or a two-dimensional quasi-birth-and-death process. The special structure of this chain allows us to deal with the fundamental form corresponding to one state of the chain at a time, and therefore it can be studied through a boundary value problem. Inspired by this fact, in this paper, we focus on the tail asymptotic behaviour of the stationary joint probability distribution of the two orbits with either an idle or a busy server by using the kernel method, a different one that does not require a full determination of the unknown generating function. To take advantage of existing literature results on the kernel method, we identify a censored random walk, which is an usual walk in the quarter plane. This technique can also be used for other random walks modulated by a finite-state Markov chain with a similar structure property.
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Acknowledgments
This work was done during the visit of the first two authors to the School of Mathematics and Statistics, Carleton University (Ottawa, Canada), who acknowledge the support provided by the School, and the visit of the third author to the School of Mathematics and Statistics, Nanjing University of Information Science and Technology (Nanjing, China), who acknowledges the support provided by the University. The first author also thanks the China Scholarship Council for supporting her visit to Carleton University through a scholarship. In addition, this work was supported in partial by the National Natural Science Foundation of China (11271373), and by the Natural Sciences and Engineering Research Council of Canada (NSERC). All authors thank the comments/suggestions made by two anonymous reviewers, which significantly improved the quality of the paper.
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In memory of Dr. Jesus R. Artalejo.
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Song, Y., Liu, Z. & Zhao, Y.Q. Exact tail asymptotics: revisit of a retrial queue with two input streams and two orbits. Ann Oper Res 247, 97–120 (2016). https://doi.org/10.1007/s10479-015-1945-y
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DOI: https://doi.org/10.1007/s10479-015-1945-y
Keywords
- Retrial queue
- Random walks in the quarter plane
- Random walks in the quarter plane modulated by a finite-state Markov chain
- Censored Markov chain
- Stationary distribution
- Generating function
- Kernel method
- Exact tail asymptotics