Delay-minimizing capacity allocation in an infinite server queueing system

TL;DR

This paper analyzes a queueing system with infinitely many servers sharing finite capacity, focusing on optimal service rate allocation to minimize delays using probabilistic analysis and dynamic programming.

Contribution

It introduces a novel approach to approximate optimal service rate allocation in infinite server queues by analyzing tail properties and stability measures.

Findings

Tail of optimal service rates decreases geometrically

Dynamic programming approximates optimal allocation

System stability measures are proposed and analyzed

Abstract

We consider a service system with an infinite number of exponential servers sharing a finite service capacity. The servers are ordered according to their speed, and arriving customers join the fastest idle server. A capacity allocation is an infinite sequence of service rates. We study the probabilistic properties of this system by considering overflows from sub-systems with a finite number of servers. Several stability measures are suggested and analysed. The tail of the series of service rates that minimizes the average expected delay (service time) is shown to be approximately geometrically decreasing. We use this property in order to approximate the optimal allocation of service rates by constructing an appropriate dynamic program.