Two-Parameter Heavy-Traffic Limits for Infinite-Server Queues

TL;DR

This paper develops two-parameter heavy-traffic limit theorems for infinite-server queues with general service and arrival processes, extending previous models to more complex, real-world scenarios.

Contribution

It introduces new two-parameter stochastic process limits for infinite-server queues with general service and arrival distributions, broadening the scope of heavy-traffic approximations.

Findings

Established heavy-traffic limits for two-parameter processes

Extended previous models to non-exponential service times

Captured variability with Kiefer process

Abstract

In order to obtain Markov heavy-traffic approximations for infinite-server queues with general non-exponential service-time distributions and general arrival processes, possibly with time-varying arrival rates, we establish heavy-traffic limits for two-parameter stochastic processes. We consider the random variables $Q^e(t,y)$ and $Q^r(t,y)$ representing the number of customers in the system at time $t$ that have elapsed service times less than or equal to time $y$, or residual service times strictly greater than $y$. We also consider $W^r(t,y)$ representing the total amount of work in service time remaining to be done at time $t+y$ for customers in the system at time $t$. The two-parameter stochastic-process limits in the space $D([0,\infty),D)$ of $D$-valued functions in $D$ draw on, and extend, previous heavy-traffic limits by Glynn and Whitt (1991), where the case of discrete service-time distributions was treated, and Krichagina and Puhalskii (1997), where it was shown that the variability of service times is captured by the Kiefer process with second argument set equal to the service-time c.d.f.