We consider a Dynamic Voltage and Frequency Scaling (DVFS) processor executing jobs with
obsolescence deadlines. Forcing hard deadlines requires to consider the worst cases with only
a finite number of jobs. In this paper we use soft real-time constraints instead, i.e., jobs
may miss their deadlines, at some cost. In they do, they are immediately removed from the
system. The objective is to design a dynamic speed policy for the processor that minimizes its
average energy consumption plus an obsolescence cost per deadline miss.

Under Poisson arrivals and exponentially distributed deadlines and job sizes, we show that
this problem can be modeled as a continuous time Markov decision process (MDP) with un-
bounded state space and unbounded rates. Inspired by the scaling method introduced recently
by Blok and Spieksma ,we show the existence of an optimal speed profile that is increasing in
the number of jobs in the system and upper bounded by some constant, that does not depend
on the deadlines and arrival rates. In addition, it yields a simple approximation for the optimal
policy and several numerical tests show that such approximation is accurate in heavy-traffic
conditions.