The M/M/k with Deterministic Setup Times

Capacity management, whether it involves servers in a data center, or human staff in a call center, or doctors in a hospital, is largely about balancing a resource-delay tradeoff. On the one hand, one would like to turn off servers when not in use (or send home staff that are idle) to save on resources. On the other hand, one wants to avoid the considerable setup time required to turn an ''off'' server back ''on.'' This paper aims to understand the delay component of this tradeoff, namely, what is the effect of setup time on average delay in a multi-server system? Surprisingly little is known about the effect of setup times on delay. While there has been some work on studying the M/M/k with Exponentially-distributed setup times, these works provide only iterative methods for computing mean delay, giving little insight as to how delay is affected by k, by load, and by the setup time. Furthermore, setup time in practice is much better modeled by a Deterministic random variable, and, as this paper shows, the scaling effect of a Deterministic setup time is nothing like that of an Exponentially-distributed setup time. This paper provides the first analysis of the M/M/k with Deterministic setup times. We prove a lower bound on the effect of setup on delay, where our bound is highly accurate for the common case where the setup time is much higher than the job service time. Our result is a relatively simple algebraic formula which provides insights on how delay scales with the input parameters. Our proof uses a combination of renewal theory, martingale arguments and novel probabilistic arguments, providing strong intuition on the transient behavior of a system that turns servers on and off.