A state-dependent queueing system with asymptotic logarithmic distribution

A Markovian single-server queueing model with Poisson arrivals and state-dependent service rates, characterized by a logarithmic steady-state distribution, is considered. The Laplace transforms of the transition probabilities and of the densities of the first-passage time to zero are explicitly evaluated. The performance measures are compared with those ones of the well-known M/M/1 queueing system. Finally, the effect of catastrophes is introduced in the model and the steady-state distribution, the asymptotic moments and the first-visit time density to zero state are determined.