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Many computer systems change into a sleep mode when idle. While this conserves energy it also increases the average latency. In this paper we quantify the effect of state change times on latency and power consumption for a single queuing system with a sleep mode. We analyze the trade-off between power consumption and latency for Poisson arrivals and do a competitive analysis for worst-case arrivals. To calculate the latency and power consumption for Poisson arrivals we use a Markov chain. For the competitive analysis we present arrival patterns which result in high latency and energy consumption and prove that these patterns are the worst case. The latency increases approximately linear with the state change times for both Poisson and worst case arrivals. Power consumption asymptotically approaches 100% when the state change times increase. While this is intuitively clear, we provide an analytic derivation. We conclude that a single system with a single sleep mode must have fast state change times to be able to keep the latency low when conserving energy.