Time dependent analysis of multivariate marked renewal processes (Q2774448)

This article extends the work of the author [J. Appl. Probab. 35, No. 4, 976-989 (1998; Zbl 0939.60095)] on processes related to queues with vacations. Let \(A_n= N_{\tau_n}\) be observations of a compound Poisson process \(N\) at the epochs \(\tau\) of a delayed renewal process; thus \(A\) might represent the cumulative number of units arriving during consecutive server vacations. Further, let \(B_n\) be supplementary, nonnegative observations whose increments may depend on those of \(A\) and \(\tau\); \(B\) might be the cumulative weight of the aforesaid units. Finally, let \(\nu= \inf\{k\geq 0:B_k> d\}\), so that the passage time \(\tau_\nu\) could be the instant of service resumption. A variety of formulas are obtained for functionals akin to NEWLINE\[NEWLINEH(x,y,z,\theta)= \int^\infty_0 e^{-\theta t}E[x^{A_\nu} y^{B_\nu} z^{N_t}1_{\{\tau_\nu\leq t< \tau_\nu+ \Sigma\}}] dt,NEWLINE\]NEWLINE where \(0\leq \Sigma\leq\infty\) is an independent random variable. The results are then applied to \(\text{M}^X/\text{G}/1\) vacation queueing systems under \(N\)- or \(D\)-type control policies.