Theory and application of BCMP networks

Abstract

The joint equilibrium distribution of an open queueing network containing N nodes and I different classes of customers is derived. The joint distribution of all the nodes is the product of marginal distributions of individual nodes. There are five different types of service (node) disciplines namely the first-come first-served, processing sharing, last-come first-served preemptive resume, infinite and finite number of servers. The service time distribution of the first service discipline must be independent identically exponentially distributed for all classes while the remaining four must be Coxian arbitrarily distributed. Each customer belongs to a single class of customers while waiting or receiving service at a particular node but customers may change their classes as they move from one node to another according to fixed probabilities. The concept of symmetric service discipline is considered in detailed. Unfortunately, the symmetric service excludes service discipline as the FCFS but it does include all the other four types of service disciplines. In particular, the concept of quasi-reversibility is discussed in detail as it has a fundamental role in the BCMP theory. Quasi-reversibility is an input-output property of queues and it implies that when a system is in stochastic equilibrium, the future arrival processes, the current state of the system, and the past departure processes are independent. An application dealing with the orders of drugs from the Government Pharmaceutical Services (GPS) is considered.