Let {Φn} be a Markov chain on the state space [0,∞) 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 {Φn} in total variation; that is, proving the existence of a limiting probability measure π and a number r>1 such that limn→∞rn supA∈ℬ[0,∞) | Px[Φn ∈A]−π(A) | = 0 for every deterministic initial state Φ0≡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 is included.
