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 subsystems with a finite number of servers. Several stability measures are suggested and analyzed. 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 approximate the optimal allocation of service rates by constructing an appropriate dynamic program.