Exponential Convergence Rates for Stochastically Ordered Markov\n Processes with Random Initial Conditions

2018/10/17 by Julia Gaudio, Saurabh Amin, Gaudio, Julia +3
Business, Management and Accounting · Mathematics · #Advanced Queuing Theory Analysis #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1810.07732

openalex publication_date 2018/10/17 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28

Abstract

In this brief paper we find computable exponential convergence rates for a\nlarge class of stochastically ordered Markov processes. We extend the result of\nLund, Meyn, and Tweedie (1996), who found exponential convergence rates for\nstochastically ordered Markov processes starting from a fixed initial state, by\nallowing for a random initial condition that is also stochastically ordered.\nOur bounds are formulated in terms of moment-generating functions of hitting\ntimes. To illustrate our result, we find an explicit exponential convergence\nrate for an M/M/1 queue beginning in equilibrium and then experiencing a change\nin its arrival or departure rates, a setting which has not been studied to our\nknowledge.\n

Citations

• Renewal theory and computable convergence rates for geometrically ergodic Markov chains

Related

• Convergence Rates in Uniform Ergodicity by Hitting Times and L²-exponential Convergence Rates
• Stationary Distributions and Convergence for M/M/1 Queues in Interactive Random Environment
• Bounds on the rate on convergence for Markovian queueing models with\n catastrophes
• On the Three Methods for Bounding the Rate of Convergence for some Continuous-time Markov Chains
• Quantitative Convergence Rates for Stochastically Monotone Markov Chains