Bunches of individual customers approach a single servicing facility according to a stationary compound Poisson process. The resulting waiting line process is studied in continuous time by the method of the imbedded Markov chain, cf. Kendall , , and of renewal theory, cf. Blackwell , Feller , and Smith . Busy period phenomena are discussed, cf. Theorem 1, in which the transform of the joint d.f. of busy period duration and the number of departures in that duration is expressed as the root $x_1(s, z)$ of a functional equation, a generalization of a result of Takacs . In terms of $x_1(s, z)$ "zero-avoiding" transition probabilities are characterized. A simple model for "instantaneous defection" is analyzed. Using renewal theory, ergodic properties of waiting line lengths and waiting times are discussed for the "general" process, in which idle and busy periods recur.
"Imbedded Markov Chain Analysis of a Waiting-Line Process in Continuous Time." Ann. Math. Statist. 30 (3) 698 - 720, September, 1959. https://doi.org/10.1214/aoms/1177706200