Decomposition of the M/M/1 transition function
The authors investigate transition probabilities of the queue length process in the classical M/M/1 queue. With this goal two auxiliary stochastic processes are introduced. The first process corresponds to the presence of the absorbing barrier at the origin (equivalently, it describes the queue length process during a busy period). The second process corresponds to the absence of barrier at the origin at all (equivalently, it describes the difference between two Poisson processes). By simple probabilistic arguments some relations between transition probabilities of these processes are established. They imply several inequalities for the transition probabilities and asymptotic behaviour as \(t\to \infty\). An asymptotic formula for the transition probabilities of the queue length process in the case of a geometric initial distribution should be noted. This formula describes dependence of the relaxation time on the parameter of the initial geometric distribution.
dependence of the relaxation time on the parameter of the initial geometric distribution
scientific article; zbMATH DE number 4209263
scientific article