Large-time limit of nonlinearly coupled measure-valued equations that model many-server queues with reneging

TL;DR

This paper analyzes the long-term behavior of measure-valued equations modeling many-server queues with reneging, proving convergence to a unique invariant state under broad conditions and using different proof techniques based on service distribution properties.

Contribution

It establishes the large-time convergence of solutions to a unique invariant state for a broad class of queue models with reneging, extending previous results to more general distributions.

Findings

Stationary distributions converge to the invariant state.

Solutions converge to the invariant state when traffic intensity is not one.

Different proof techniques are used depending on hazard rate properties.

Abstract

The large-time behavior of a nonlinearly coupled pair of measure-valued transport equations with discontinuous boundary conditions, parameterized by a positive real-valued parameter $\lambda$, is considered. These equations describe the hydrodynamic or fluid limit of many-server queues with reneging (with traffic intensity $\lambda$), which model phenomena in diverse disciplines, including biology and operations research. For a broad class of reneging distributions with finite mean and service distributions with finite mean and hazard rate function that is either decreasing or bounded away from zero and infinity, it is shown that if the fluid equations have a unique invariant state, then the Dirac measure at this state is the unique random fixed point of the fluid equations, which implies that the stationary distributions of scaled $N$-server systems converge to the unique invariant state of the corresponding fluid equations. Moreover, when $\lambda\ne 1$, it is shown that the solution to the fluid equation starting from any initial condition converges to this unique invariant state in the large time limit. The proof techniques are different under the two sets of assumptions on the service distribution. When the hazard rate function is decreasing, a reformulation of the dynamics in terms of a certain renewal equation is used, in conjunction with recursive asymptotic estimates. When the hazard rate function is bounded away from zero and infinity, the proof uses an extended relative entropy functional as a Lyapunov function. Analogous large-time convergence results are also established for a system of coupled measure-valued equations modeling a multiclass queue.