Abstract

We study the problem of optimally controlling the use of sleep states in an energy-aware M/G/1 queue. In our model, we consider a family of policies where the server upon becoming idle can wait for a random period before entering, potentially randomly, any of a finite number of possible sleep states to save energy. The server becomes busy again after a possibly random number of jobs have arrived. However, jobs are served only after a random setup time. This kind of an energy-aware queuing system has been analyzed in recent papers under specific assumptions regarding the cost metrics and the distributions of the random variables. In this paper, we consider an essentially more general model. Notably we show that the optimal control of the idle time and sleep states is deterministic and does not benefit from randomization: either the system only uses the idle state and no sleep states, or the idle state is not used at all and the server immediately goes to some fixed sleep state and waits until a fixed number of jobs have arrived before starting the setup. We prove this result for two popular cost metrics, namely weighted sum of energy and response time (ERWS) and their product ERP.

Highlights

  • An increasing demand for green ICT has inspired the queueing community to consider energy-aware queueing systems

  • We prove that the optimal idling time is, either I = 0 or I = ∞ for the ERWS and ERP cost metrics even if we allow general distributions for service times and setup delays, multiple sleep states, and randomized rules to choose the sleep state i and the corresponding threshold k

  • ΠiE[DiR], i=0 k=1 where πi denotes the probability that the current cycle is related to state sleepi and that the arriving job is one of the jobs arriving during the corresponding setup delay Di, and DiR refers to the remaining part of the setup delay upon the arrival

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Summary

INTRODUCTION

An increasing demand for green ICT has inspired the queueing community to consider energy-aware queueing systems. Maccio and Down [14] considered this model but they restricted themselves to a single sleep state and deterministic rules to choose the corresponding threshold k They showed, explicitly only for the ERWS metric and assuming exponential service times and setup delays, that the optimal idling time is either I = 0 or I = ∞. We prove that the optimal idling time is, either I = 0 or I = ∞ for the ERWS and ERP cost metrics even if we allow general distributions for service times and setup delays, multiple sleep states, and randomized rules to choose the sleep state i and the corresponding threshold k.

ANALYSIS
OPTIMIZATION
Optimal idling time distribution
Optimal choice of sleep state and switchon threshold
E pik a
RESETTING THE IDLE TIME TIMER
NUMERICAL RESULTS
CONCLUSIONS
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