Performability measure of acyclic Markovian models

Continuous-time Markov processes with a finite-state space are generally considered to model degradable fault-tolerant computer systems. The finite space is partitioned as ∪^mi=1Bi, where Bi stands for the set of states which corresponds to the configuration where the system has a performance level (or reward rate) equal to ri. The performability Yt is defined as the accumulated reward over a mission time [0,t]. In this paper, a renewal equation is established for the performability measure and solved for both ‘standard’ and uniform acyclic models. Two closed form expressions for the performability measure are derived for the two types of models. Furthermore, an algorithm with a low polynomial computational complexity is presented and applied to a degradable computer system.