Measure-valued stochastic recurrences and the stability of queues

TL;DR

This paper extends Loynes's Theorem to measure-valued stochastic recursions, providing stability criteria for complex queueing systems with infinitely many servers or single-server SRPT queues, ensuring the existence of stationary states.

Contribution

It introduces a stability criterion for measure-valued recursions, generalizing classical results to more complex queueing models with infinite measure spaces.

Findings

Established conditions for the existence of stationary measure-valued recursive sequences.

Provided a stability criterion applicable to queues with infinitely many servers.

Characterized the stationary states for the studied queueing systems.

Abstract

In this paper we present a stability criterion for finite measure-valued stochastic recursions, generalizing Loynes's Theorem to spaces of measures. This result provides conditions for the reach of a "total stationary state" for the queue with an infinity of servers and the single-server SRPT queue. Indeed, we give in both cases a condition of existence of a stationary measure-valued recursive sequence characterizing the queueing system exhaustively.