Queues with heterogeneous servers and uninformed customers: who works the most?

TL;DR

This paper analyzes queues with multiple heterogeneous servers and uninformed customers, revealing that faster servers are busier but also more productive than slower ones, providing insights into server utilization.

Contribution

It demonstrates that in such queueing systems, faster servers tend to be busier yet more effective, a novel insight into server dynamics with uninformed customers.

Findings

Probability fastest server is busy < probability slowest server is busy

Fastest server has higher effective service rate than slowest server

Heterogeneous servers exhibit distinct busy and productivity patterns

Abstract

In this paper, we consider systems that can be modelled by $M \mid M \mid n$ queues with heterogeneous servers and non informed customers. Considering any two servers: we show that the probability that the fastest server is busy is smaller than the probability that the slowest server is busy. Moreover, we show that the effective rate of service done by the fastest server is larger than effective rate of service done by the slowest server.