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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. 

Multiserver-job systems, where jobs require concurrent service at many servers, occur widely in practice. Essentially all of the theoretical work on multiserver-job systems focuses on maximizing utilization, with almost nothing known about mean response time. In simpler settings, such as various known-size single-server-job settings, minimizing mean response time is merely a matter of prioritizing small jobs. However, for the multiserver-job system, prioritizing small jobs is not enough, because we must also ensure servers are not unnecessarily left idle. Thus, minimizing mean response time requires prioritizing small jobs while simultaneously maximizing throughput. Our question is how to achieve these joint objectives. We devise the ServerFilling-SRPT scheduling policy, which is the first policy to minimize mean response time in the multiserver-job model in the heavy traffic limit. In addition to proving this heavy-traffic result, we present empirical evidence that ServerFilling-SRPT outperforms all existing scheduling policies for all loads, with improvements by orders of magnitude at higher loads. Because ServerFilling-SRPT requires knowing job sizes, we also define the ServerFilling-Gittins policy, which is optimal when sizes are unknown or partially known. 

This document examines five performance questions which are repeatedly asked by practitioners in industry: (i) My system utilization is very low, so why are job delays so high? (ii) What should I do to lower job delays? (iii) How can I favor short jobs if I don't know which jobs are short? (iv) If some jobs are more important than others, how do I negotiate importance versus size? (v) How do answers change when dealing with a closed-loop system, rather than an open system? All these questions have simple answers through queueing theory. This short paper elaborates on the questions and their answers. To keep things readable, our tone is purposely informal throughout. For more formal statements of these questions and answers, please see [14].