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Geometric convergence rates for stochastically ordered Markov chains. (English) Zbl 0847.60053
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 $\lim_{n \to \infty} r^n \sup_{A \in {\mathcal B} [0, \infty)} \bigl |P_x [\Phi_n \in A] - \pi (A) \bigr |= 0$ 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:
 60J25 Continuous-time Markov processes on general state spaces 60K25 Queueing theory (aspects of probability theory)
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