A Universal Maximum Entropy Solution for Complex Queueing Systems and Networks

An analytic framework is presented for a unified exposition of entropy maximization and complex queueing systems and networks. In this context, a universal maximum entropy (ME) solution is characterized, subject to appropriate mean value constraints, for the joint state probability distribution of a complex single server queueing system with finite capacity, distinct either priority or nonpriority classes of jobs, general (G-type) class interarrival and service time processes and either complete (CBS) or partial (PBS) buffer sharing scheme. The ME solution leads to the establishment of closed-form expressions for the aggregate and marginal state probabilities and, moreover, it is stochastically implemented by making use of the generalized exponential (GE) distribution towards the least biased approximation of G-type continuous time distributions with known first two moments. Subsequently, explicit analytic formulae are presented for the estimation of the Lagrangian coefficients via asymptotic connections to the corresponding infinite capacity queue and GE-type formulae for the blocking probabilities per class. Furthermore, it is shown that the ME solution can be utilized, in conjunction with GE-type flow approximation formulae, as a cost effective building block towards the determination of an extended ME product-form approximation and a queue-by-queue decomposition algorithm for the performance analysis of complex open queueing network models (QNMs) with arbitrary configuration and repetitive service (RS) blocking.