Specific bounds for a probabilistically interpretable solution of the Poisson equation for general state-space Markov chains with queueing applications

2019/09/17 by Hiroyuki Masuyama, Masuyama, Hiroyuki
Business, Management and Accounting · Computer Science · Engineering · #Advanced Queuing Theory Analysis #Age of Information Optimization #FOS: Mathematics #Probability (math.PR) #Reliability and Maintenance Optimization

paper · pdf · doi:10.48550/arxiv.1909.07567

openalex publication_date 2019/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper considers the Poisson equation for general state-space Markov chains in continuous time. The main purpose of this paper is to present specific bounds for the solutions of the Poisson equation for general state-space Markov chains. The solutions of the Poisson equation are unique in the sense that they are expressed in terms of a certain probabilistically interpretable solution (called the \it standard solution). Thus, we establish some specific bounds for the standard solution under the f-modulated drift condition (which is a kind of Foster-Lyapunov-type condition) and some moderate conditions. To demonstrate the applicability of our results, we consider the workload processes in two queues: MAP/GI/1 queue, and M/GI/1 queue with workload capacity limit.

Citations

• The stability of a queue with non-independent inter-arrival and service times

Related

• Computable Bounds on the Solution to Poisson's Equation for General Harris Chains
• Stochastic Approximation in a Markovian Framework Revisited: Lipschitz Continuity of the Poisson Equation
• A Numerical Truncation Approximation with A Posteriori Error Bounds for the Solution of Poisson's Equation
• New Look at Finite Single Server Queue with Poisson Input and Semi-Markov Service Times
• Solutions of Poisson's Equation for Stochastically Monotone Markov\n Chains