Fluid Model of A Many-Server Queueing Network with Abandonment and Markovian Routing

Abstract

This paper studies a fluid model for a non-Markovian many-server queueing network with abandonment, where externally arrived and internally routed customers are served under the non-idling global First-Come-First-Serve (FCFS) discipline at each station of many parallel servers. The routing follows a Markovian mechanism. Externally arrived and internally routed customers in each queue may have different service time distributions, as well as different patience time distributions, and all these distributions may depend on the station. The fluid model dynamics is described by the fluid contents of externally arrived customers and internally routed customers in each queue (both waiting and receiving service) and a set of four measure-valued processes, tracking the amount of service time each externally arrived customer in service has received, the amount of service time each internally routed customer in service has received, the waiting times of externally arrived customers and the waiting times of internally routed customers in queue. Under mild conditions on the service and patience time distributions, we prove the existence and uniqueness of a solution to the fluid model equations. We then characterize the invariant states of this fluid model when the arrival rates are constant. We also establish the convergence of the properly scaled stochastic evolution dynamics to the fluid model.