Transient analysis of permanent customers in a single-server queue with mixed traffic (Q2709778)

The paper discusses the \(M/G/1\) queue with \(M\) permanent customers in addition to the usual flow of standard customers. Each permanent customer remains in the system, re-joining the end of the queue on completion of service. Cycles are defined as the intervals between successive arrivals at the server of the same permanent customer; these divide naturally into alternating intervals in which ordinary and permanent customers are served. Laplace-Stieltjes transforms are calculated for (a) cycle lengths in the \(M/M/1\) case under various initial conditions, (b) successive departure times of permanent customers in the \(M/G/1\) case under general initial conditions. The probability distribution of the number of arrivals in the \(i\)th \(M/M/1\) cycle when \(M= 1\) is also given.