AnOptimal Analytical Solution forProcessor Speed Control withThermal Constraints

Assemiconductor manufacturing technology scales tosmaller device sizes, thepowerconsumption ofclocked digital ICsbegins to increase. Dynamic voltage andfrequency scaling (DVFS) isawellknowntechnique forconserving energy. Recently, ithasalso been usedtocontrol theCPUtemperature aspart ofDynamic Thermal Management (DTM)techniques. Mostworksinthese areas assumethat theoptimum speed profile (for either minimizing energy ormaximizing performance) isaconstant profile. However, inthe presence ofthermal constraints, weshowthat theoptimal profile isingeneral, atime-varying function. Weformulate theproblem ofmaximizing theaverage throughput ofaprocessor overagiven timeperiod, subject tothermal andspeed constraints, asaproblem inthecalculus ofvariations. Thevariational approach provides a powerful framework forprecisely specifying andsolving thespeed control problem, andallows ustoobtain anexactanalytical solution. Thesolution methodology isverygeneral, andworks for anyconvex powermodel, andsimple lumped RCthermal models. Theresulting speed profiles werefound toconsist ofuptothree segments, ofwhich oneofthemisadecreasing function oftime, andtheothers areconstant. Weanialyze theeffect ofdifferent parameters like theinitial temperature, thermal capacitance andthe maximumrated speed onthenature andthecostoftheoptimum solution. Wealso propose atwo-speed solution that approximates the optimal speed curve. Thissolution wasfound toachieve aperformanceclose tothat oftheoptimum, andisalso easier toimplement inreal processors.