Delay-Minimizing Capacity Allocation in an Infinite Server-Queueing System

We consider a service system with an infinite number of exponential servers sharing a finite service capacity. The servers are ordered by 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 ana-lyzed. 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 to ap-proximate the optimal allocation of service rates by constructing an appropriate dynamic program.