The stationary tail asymptotics in the GI/G/1-type queue with countably many background states

Abstract

We consider the asymptotic behaviour of the stationary tail probabilities in the discrete-time GI/G/1-type queue with countable background state space. These probabilities are presented in matrix form with respect to the background state space, and shown to be the solution of a Markov renewal equation. Using this fact, we consider their decay rates. Applying the Markov renewal theorem, it is shown that certain reasonable conditions lead to the geometric decay of the tail probabilities as the level goes to infinity. We exemplify this result using a discrete-time priority queue with a single server and two types of customer.

Wiener-Hopf Factorizations for a Multidimensional Markov Additive Process and their Applications to Reflected Processes

“…For the one-dimensional case, we extend existing results of Miyazawa and Zhao [24] from the case of countably many background states to real-valued background states. This allows us to consider precise asymptotics of a system with autoregressive input.…”

“…We defer the proof of this lemma to Appendix E since it is similar to Lemma 4.2 of [24]. We are now ready to present a main result of this subsection.…”

“…One may think that such a multi-dimensionality can be reduced to the one dimensional case by incorporating auxiliary information into the background state. This is possible, but it complicates the background process, which may make analysis intractable (see, e.g., [24]). We keep the multidimensional additive component as long as possible.…”

“…We assume further conditions which enable us to extend the approach of [24] that considers an integer-valued Markov additive process with a countable background state space.…”

DUAL FORM OF MARKOV RENEWAL EQUATIONS AND AN APPLICATION TO ASYMPTOTIC ANALYSIS OF A SINGLE-SERVER QUEUE(<Special Issue>the 50th Anniversary of the Operations Research Society of Japan)

“…On the other hand, Miyazawa [6] As other related works, Miyazawa [7] and Miyazawa and Zhao [8] finite Ii], Takine [13] (12) A(e))ldOl,=, < 1.…”