Convergence to equilibrium states for fluid models of many-server queues with abandonment

Abstract

Fluid models have become an important tool for the study of many-server queues with general service and patience time distributions. The equilibrium state of a fluid model has been revealed by Whitt (2006) and shown to yield reasonable approximations to the steady state of the original stochastic systems. However, it remains an open question whether the solution to a fluid model converges to the equilibrium state and under what condition. We show in this paper that the convergence holds under a mild condition. Our method builds on the framework of measure-valued processes developed in Zhang (2013), which keeps track of the remaining patience and service times.

Refined Models for Efficiency-Driven Queues with Applications to Delay Announcements and Staffing

“…The paper by Liu and Whitt (2012a) completed the story started in Whitt (2006) by bringing the model to the time-varying setting. Long and Zhang (2014) proved that the fluid model G/GI/n+GI converges to an equilibrium state, following the result for G/M/n+GI in Section 5 of Liu and Whitt (2011a). A sequence of works by Liu and Whitt (2011b, 2012b, 2014a comprehensively analyzed, from theory to algorithms, networks of many-server fluid queues in the time-varying setting.…”

Customer Service Chat Systems with General Service and Patience Times

“…The fluid limit was rigorously proven to serve as the fluid approximation in the many-server heavy traffic regime by Zhang (2013) using measure-valued processes. Long and Zhang (2014) proved that the fluid limit (which is a deterministic dynamic system) converges to the invariant state as time goes to infinite. It is clear in the literature that the invariant state of a fluid model provides an insightful approximation for the steady state of the original system.…”

“…In those models the abandonment probability (in the fluid limit) only depends on the mean service time and not on the distribution of service or abandonment times. However, the other performance measures (such as expected time in queue) may also depend on the entire patience time distribution for those systems, see Whitt (2006), Bassamboo and Randhawa (2016), Long and Zhang (2014) for more details.…”

A Note on Many-Server Fluid Models With Time-Varying Arrivals

“…In the second line of works tracking residual times, Zhang (2013) directly proves the existence and uniqueness of the many-server fluid model with a constant arrival rate only requiring continuity of the service time distribution and Lipschitz continuity of the patience time distribution. Moreover, it builds the foundation to prove the convergence to the equilibrium state in Long and Zhang (2014). However, the modeling approach in Zhang (2013) seems a bit inflexible as extending the analysis of the fluid model with a constant arrival rate to time-varying arrival rates is not that straightforward.…”