On the discrete-time \(\mathrm{Geo}/\mathrm{G}/1\) queue with vacations in random environment
Summary: A discrete-time \(\mathrm{Geo}/\mathrm{G}/1\) queue with vacations in random environment is analyzed. Using the method of supplementary variable, we give the probability generating function (PGF) of the stationary queue length distribution at arbitrary epoch. The PGF of the stationary sojourn time distribution is also derived. And we present the various performance measures such as mean number of customers in the system, mean length of the type-\(i\) cycle, and mean time that the system resides in phase \(0\). In addition, we show that the \(\mathrm{M}/\mathrm{G}/1\) queue with vacations in random environment can be approximated by its discrete-time counterpart. Finally, we present some special cases of the model and numerical examples.
scientific article; zbMATH DE number 7026882
scientific article