A single-server queueing system with background customers

A single-server queueing system with a Markov flow of primary customers and a flow of background customers from a bunker containing unbounded number of customers, i.e., the background customer flow is saturated, is studied. There is a buffer of finite capacity for primary customers. Service processes of primary as well as background customers are Markovian. Primary customers have a relative service priority over background customers, i.e., a background customer is taken for service only if the buffer is empty upon completion of service of a primary customer. A matrix algorithm for computing the stationary state probabilities of the system at arbitrary instants and at instants of arrival and completion of service of primary customers is obtained. Main stationary performance indexes of the system are derived. The Laplace–Stieltjes transform of the stationary waiting time distribution for primary customers is determined.