*(English)*Zbl 0847.60053

Summary: Let $\left\{{{\Phi}}_{n}\right\}$ 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 $\left\{{{\Phi}}_{n}\right\}$ in total variation; that is, proving the existence of a limiting probability measure $\pi $ and a number $r>1$ such that

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.