Computation of the distribution of the completion time when the work requirement is a ph random variable

In this paper we derive the distribution of the completion time of a job with a PH-distributed work requirement, on a server modeled by a homogeneous Markov reward process. The interactions between the job in progress and the server are allowed to be either of resume or of repeat type in a probabilistic manner. The paper shows that the distribution of the completion time of a job with PH distributed requirement is PH under any mixture of preemptive policies. This result is stated as a closure property of the PH distribution class.