Distributions of unexpected services in a D-node Markov-renewal network distinguished by customer class

摘要

Customer units from any one of K distinct classes arrive at random but in a fixed, known order for processing in a D-node Markov-renewal network that contains at least one absorbing node and no proper closed subsets of nodes. The joint probability function of counts of units in nodes, distinguished by customer class, is derived for the transient case. A numerical method for routinely computing all possible marginal distributions is demonstrated for implementation on off-the-shelf spreadsheets or database systems. The model permits explicit computation of counts by units, by class and node, for arrivals that are not Poisson distributed.Populations of units arriving randomly in time, classes being arbitrarily mixed but in known order, represent a generic model applicable in many disciplines including biology, engineering, demography, and operations research.