Feasibility of on-line speed policies in real-time systems

Overview & Synopsis

We consider a real-time system where a single processor with variable speed executes an infinite sequence of sporadic and independent jobs. We assume that job sizes and relative deadlines are bounded by $C$ and $∆$ respectively. Furthermore, $Smax$ denotes the maximal speed of the processor. In such a real-time system, a speed selection policy dynamically chooses (i.e.,on-line) the speed of the processor to execute the current, not yet finished, jobs. We say that an on-line speed policy is feasible if it is able to execute any sequence of jobs while meeting two constraints: the processor speed is always below $Smax$ and no job misses its deadline. In this paper, we compare the feasibility region of four on-line speed selection policies in single-processor real-time systems, namely Optimal Available(OA)[1], Average Rate(AVR)[1],(BKP)[2], and aMarkovian Policy based on dynamic programming(MP)[3]. We prove the following results: • (OA)is feasible if and only if $Smax≥C(h∆−1+ 1)$, where $hn$ is the $n-th$ harmonic number $(hn=∑ni=11/i≈logn)$. • (AVR) is feasible if and only if $Smax≥Ch∆$. • (BKP) is feasible if and only if $Smax≥eC(wheree= exp(1))$. • (MP) is feasible if and only if $Smax≥C$. This is an optimal feasibility condition because when $Smax< C$ no policy can be feasible. This reinforces the interest of (MP) that is not only optimal for energy consumption (on average) but is also optimal regarding feasibility.