Technical note: The MAP_t/Ph_t/∞ queueing system and multiclass [MAP_t/Ph_t/∞]~K queueing network

摘要

In this paper we demonstrate how a key adjustment to known numerically exact methods for evaluating time-dependent moments of the number of entities in the Ph_t/Ph_t/∞ queueing system and [Ph_t/Ph_t/∞]~K queueing network may be implemented to capture the effect of autocorrelation that may be present in arrivals to the more general MAP_t/Ph_t/∞ queueing system and multiclass [MAP_t/Ph_t/∞]~K queueing network. The MAP_t is more general than the Ph_t arrival process in that it allows for stationary nonrenewal point processes, as well as the time-dependent generalization of nonrenewal point processes. Modeling real-world systems with bursty arrival processes such as those in telecommunications and transportation, for example, necessitate the use of nonrenewal processes. Finally, we show that the covariance of the number of entities at different nodes and times may be described by a single closed differential equation.