Abstract
We consider the stationary distribution of the M/GI/1 type queue when background states are countable. We are interested in its tail behavior. To this end, we derive a Markov renewal equation for characterizing the stationary distribution using a Markov additive process that describes the number of customers in system when the system is not empty. Variants of this Markov renewal equation are also derived. It is shown that the transition kernels of these renewal equations can be expressed by the ladder height and the associated background state of a dual Markov additive process. Usually, matrix analysis is extensively used for studying the M/G/1 type queue. However, this may not be convenient when the background states are countable. We here rely on stochastic arguments, which not only make computations possible but also reveal new features. Those results are applied to study the tail decay rates of the stationary distributions. This includes refinements of the existence results with extensions.
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References
G. Alsmeyer, On the Markov renewal theorem, Stochastic Process. Appl. 50 (1994) 37–56.
G. Alsmeyer, The ladder variable of a Markov random walk, Probab. Math. Statist. 20 (2000) 151–168.
E. Arjas and T.P. Speed, Symmetric Wiener-Hopf factorizations in Markov additive processes, Z. Wahrcheinlich. verw. Geb. 26 (1973) 105–118.
S. Asmussen, Applied Probability and Queues (Wiley, Chichester, 1987).
K.B. Athreya and P. Ney, A renewal approach to the Perron-Frobenius theory of non-negative kernels on general state spaces, Math. Zeitschrift 179 (1982) 507–529.
E. Çinlar, Markov additive processes I, II, Z. Wahrscheinlich. verw. Geb. 24 (1972) 85–93, 95–121.
E. Çinlar, Introduction to Stochastic Processes (Prentice-Hall, Englewood Cliffs, NJ, 1975).
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II (Wiley, New York, 1971).
M. Miyazawa, A Markov renewal approach to the asymptotic decay of the tail probabilities in risk and queueing processes, Probab. Engrg. Inform. Sci. 16 (2002) 139–150.
M. Miyazawa, A paradigm of Markov additive processes for queues and their networks, in: Proc. of 4th Internat. Conf. on Matrix Analytic Methods in Stochastic Models, Adelaide, Australia, 2002, pp. 265–289.
M. Miyazawa, Hitting probabilities in a Markov additive process with linear movements and upward jumps: Their applications to risk and queueing processes, Ann. Appl. Probab. (2004) to appear.
M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models: Algorithmic Approach (Johns Hopkins Univ. Press, Baltimore, MD, 1981).
M.F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications (Marcel Dekker, New York, 1989).
V. Ramaswami, A stable recursion for the steady state vector in Markov chains on M/G/1 type, Stochastic Models 4(1) (1988) 183–188.
E. Seneta, Non-negative Matrices and Markov Chains, 2nd ed. (Springer, New York, 1981).
V.M. Shurenkov, On the theory of Markov renewal, Theory Probab. Appl. 29 (1984) 247–265.
Y. Takahashi, K. Fujimoto and N. Makimoto, Geometric decay of the steady-state probabilities in a quasi-birth-and-death process with a countable number of phases, Stochastic Models 17(1) (2001) 1–24.
T. Takine, Geometric and subexponential asymptotics of Markov chains of M/G/1 type, Math. Oper. Res. (2004) to appear.
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Miyazawa, M. A Markov Renewal Approach to M/G/1 Type Queues with Countably Many Background States. Queueing Systems 46, 177–196 (2004). https://doi.org/10.1023/B:QUES.0000021148.33178.0f
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DOI: https://doi.org/10.1023/B:QUES.0000021148.33178.0f