Efficient and accurate Lyapunov function-based truncation technique for multi-dimensional Markov chains with applications to discriminatory processor sharing and priority queues

Abstract

Online service providers aim to satisfy the tail performance requirements of customers through Service-Level Objectives (SLOs). One approach to ensure tail performance requirements is to model the service as a Markov chain and obtain its steady-state probability distribution. However, obtaining the distribution can be challenging, if not impossible, for certain types of Markov chains, such as those with multi-dimensional or infinite state-space and state-dependent transitions. Examples include M/M/1 with Discriminatory Processor Sharing (DPS) and preemptive M/M/c with multiple priority classes and customer abandonment.

To address this fundamental problem, we propose a Lyapunov function-based state-space truncation technique that leverages moments or bounds on moments of the state variables. This technique allows us to obtain tight truncation bounds while ensuring arbitrary probability mass guarantees for the truncated chain. We highlight the efficacy of our technique for multi-dimensional DPS and M/M/c priority queue with abandonment, demonstrating a substantial reduction in state space (up to 74%) compared to existing approaches. Additionally, we present three practical use cases that highlight the applicability of our truncation technique by analyzing the performance of the DPS system.