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On ergodicity conditions in a polling model with Markov modulated input and state-dependent routing

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Abstract

A polling system with switchover times and state-dependent server routing is studied. Input flows are modulated by a random external environment. Input flows are ordinary Poisson flows in each state of the environment, with intensities determined by the environment state. Service and switchover durations have exponential laws of probability distribution. A continuous-time Markov chain is introduced to describe the dynamics of the server, the sizes of the queues and the states of the environment. By means of the iterative-dominating method a sufficient condition for ergodicity of the system is obtained for the continuous-time Markov chain. This condition also ensures the existence of a stationary probability distribution of the embedded Markov chain at instants of jumps. The customers sojourn cost during the period of unloading the stable queueing system is chosen as a performance metric. Numerical study in case of two input flows and a class of priority and threshold routing algorithms is conducted. It is demonstrated that in case of light inputs a priority routing rule doesn’t seem to be quasi-optimal.

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Acknowledgments

This work was fulfilled as a part of State Budget Research and Development program No. 01201252499 “Mathematical modeling and construction of new methods for analysis of evolutionary systems and systems of optimization” of N.I. Lobachevsky State University of Nizhni Novgorod

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  1. N.I. Lobachevsky State University of Nizhni Novgorod, 23, Gagarina ave, Nizhni Novgorod , 603950, Russia

    Andrei V. Zorine

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  1. Andrei V. Zorine
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Correspondence to Andrei V. Zorine.

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Zorine, A.V. On ergodicity conditions in a polling model with Markov modulated input and state-dependent routing. Queueing Syst 76, 223–241 (2014). https://doi.org/10.1007/s11134-013-9385-3

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