Departures from queues with changeover times (Q913383)

The one-channel queueing system with Poisson input (with the parameter \(\lambda)\), FCFS discipline and no lost customers is considered. The type of customer is chosen from a Markov chain with the transition matrix \(\| P_{ij}\|\), where \(P_{ij}\) is the probability that the current type customer is j, if the previous customer was of type i. The conditional distribution function of the service time is \(P_{ij}(1- e^{-\mu_{ij}x}).\) Throughout is assumed \(\mu_{ij}<\lambda\) for all i,j. Let \(T_ n\), \(N_ n\), \(Z_ n\) be the n-th departure epoch, the queue length at \(T^+_ n\), and the type of the customer that departs at \(T_ n\), respectively. It is proved that the process \((T_ n\), \(N_ n\), \(Z_ n\), \(n\geq 0)\) is equivalent to a renewal process if and only if \(\mu_{ij}\equiv const\). The expression for E \(\{\) \(M_ j(t)M_ k(t+r)\}\) is found, where \(M_ j(t)\) is the number of type j departures in (0,t]. The expression is used for the computation of the cross- covariance and the cross-correlation.