Analysis of a Multiserver Queue with Setup Times
Queueing Systems: Theory and Applications
Pergamum: replacing tape with energy efficient, reliable, disk-based archival storage
FAST'08 Proceedings of the 6th USENIX Conference on File and Storage Technologies
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In this paper, we consider an M/M/k queueing system with setup costs. Servers are turned off when there is no work to do, but turning on an off server incurs a setup cost. The setup cost takes the form of a time delay and a power penalty. Setup costs are common in manufacturing systems, data centers and disk farms, where idle servers are turned off to save on operating costs. Since servers in setup mode consume a lot of power, the number of servers that can be in setup at any time is often limited. In the staggered setup model, at most one server can be in setup at any time. While recent literature has analyzed an M/M/k system with staggered setup and exponentially distributed setup times, no closed-form solutions were obtained. We provide the first analytical closed-form expressions for the limiting distribution of the system states, the distribution of response times, and the mean power consumption for the above system. In particular, we prove the following decomposition property: the response time for an M/M/k system with staggered setup is equal, in distribution, to the sum of response time for an M/M/k system without setup, and the setup time.