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Zbl 0847.60053
Lund, Robert B.; Tweedie, Richard L.
Geometric convergence rates for stochastically ordered Markov chains. (English)
[J] Math. Oper. Res. 21, No.1, 182-194 (1996). ISSN 1526-5471; ISSN 0364-765X/e

Summary: Let $\{\Phi_n\}$ be a Markov chain on the state space $[0, \infty)$ that is stochastically ordered in its initial state; that is, a stochastically larger initial state produces a stochastically larger chain at all other times. Examples of such chains include random walks, the number of customers in various queueing systems, and a plethora of storage processes. A large body of recent literature concentrates on establishing geometric ergodicity of $\{\Phi_n \}$ in total variation; that is, proving the existence of a limiting probability measure $\pi$ and a number $r > 1$ such that $$\lim_{n \to \infty} r^n \sup_{A \in {\cal B} [0, \infty)} \bigl |P_x [\Phi_n \in A] - \pi (A) \bigr |= 0$$ for every deterministic initial state $\Phi_0 \equiv x$. We seek to identify the largest $r$ that satisfies this relationship. A dependent sample path coupling and a Foster-Lyapunov drift inequality are used to derive convergence rate bounds; we then show that the bounds obtained are frequently the best possible. Application of the methods to queues and random walks are included.

MSC 2000:: *60J25 Markov processes with continuous parameter
60K25 Queueing theory

Keywords: coupling; storage processes; queueing systems; geometric ergodicity; Foster-Lyapunov drift inequality

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