Mean Field Queues with Delayed Information

TL;DR

This paper introduces a mean field queueing model with delayed information and customer abandonment, revealing a stability threshold independent of the number of queues, leading to periodic oscillations beyond this delay.

Contribution

It develops a novel mean field model with delay and abandonment, analyzing stability changes and oscillations, and computes the critical delay threshold.

Findings

System exhibits stability change at a critical delay threshold.

Periodic oscillations occur when delay exceeds the threshold.

Threshold is independent of the number of queues.

Abstract

In this paper, we consider a new queueing model where queues balance themselves according to a mean field interaction with a time delay. Unlike other work with delayed information our model considers multi-server queues with customer abandonment. In this setting, our queueing model corresponds to a system of mean field interacting delay differential equations with a point of non-differentiability introduced by the finite-server and abandonment terms. We show that this system of delay differential equations exhibits a change in stability when the delay in information crosses a critical threshold. In particular, the system exhibits periodic oscillations when the delay in information exceeds this critical threshold and we show that the threshold surprisingly does not depend on the number of queues. This is in stark contrast to other choice based queueing models with delayed information. We compute this critical threshold in each of the relevant parameter regions induced by the point of non-differentiability and show numerically how the critical threshold transitions through the point of non-differentiability.