Markov chains on $${{\mathbb {Z}}^+}$$ Z +: analysis of stationary measure via harmonic functions approach

Denis Denisov (), Dmitry Korshunov () and Vitali Wachtel ()
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Denis Denisov: University of Manchester
Dmitry Korshunov: Lancaster University
Vitali Wachtel: University of Augsburg

Queueing Systems: Theory and Applications, 2019, vol. 91, issue 3, No 4, 265-295

Abstract: Abstract We suggest a method for constructing a positive harmonic function for a wide class of transition kernels on $${{\mathbb {Z}}^+}$$ Z + . We also find natural conditions under which this harmonic function has a positive finite limit at infinity. Further, we apply our results on harmonic functions to asymptotically homogeneous Markov chains on $${{\mathbb {Z}}^+}$$ Z + with asymptotically negative drift which arise in various queueing models. More precisely, assuming that the Markov chain satisfies Cramér’s condition, we study the tail asymptotics of its stationary distribution. In particular, we clarify the impact of the rate of convergence of chain jumps towards the limiting distribution.

Keywords: Transition kernel; Harmonic function; Markov chain; Stationary distribution; Renewal function; Exponential change of measure; Queues; 60J10; 60J45; 60K25; 60F10; 31C05 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s11134-019-09602-5

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