Sojourn time distributions and time scale separation results for multiclass state-dependent processor sharing systems with a finite service pool and exponential service requirements

The paper addresses multiclass processor sharing systems with general state-dependent service rates, exponential service requirements and a finite service pool. By considering the amount of service received by a permanent customer and associating this service with the evolution of a Markov Reward process, the sojourn time distribution is formulated in terms of a matrix exponential expression. When the service rates are balanced, this expression can be diagonalized. Tail asymptotics are also discussed. The matrix exponential expression is subsequently exploited towards the study of time scale separation regimes. Unlike the standard practice of assuming a distinct time scale per class, the paper groups more realistically all customer classes in two time scales. Provably tight approximations, of a known, small degree of error, are developed for the sojourn time distribution of a given class (with either fast or slow dynamics), in terms of reduced models containing only the customer classes operating in the same time scale. The approximation for the fast classes gives rise to further characterization of the tail behavior. Additionally, the paper studies another, more specialized variant of the time scale separation regime, in which the service rates take a special form that leads to even simpler approximations. Finally, it is shown that the essence of the main results applies also to the more general setting of service requirement distributions with Markovian phase-type form.