The Number of Overlapping Customers in Erlang-A Queues: An Asymptotic Approach

TL;DR

This paper introduces an asymptotic approach using fluid and diffusion limits to approximate the mean and variance of overlapping customers in Erlang-A queues, enhancing understanding of queue dynamics.

Contribution

It develops new approximation methods for overlapping customers and waiting time metrics in Erlang-A queues using fluid and diffusion limit analysis.

Findings

Derived accurate mean and variance approximations for overlapping customers.

Provided new formulas for waiting time analysis in Erlang-A queues.

Enhanced understanding of queue overlap risks and system performance.

Abstract

In this paper, we investigate the number of customers that overlap or coincide with a virtual customer in an Erlang-A queue. Our study provides a novel approach that exploits fluid and diffusion limits for the queue to approximate the mean and variance of the number of overlapping customers. We conduct a detailed analysis of the fluid and diffusion limit differential equations to derive these approximations. We also construct new accurate approximations for the mean and variance of the waiting time in the Erlang-A queue by combining fluid limits with the polygamma function. Our findings have important implications for queueing theory and evaluating the overlap risk of more complicated service systems.