A Gambler's Ruin Type Problem in Queuing Theory

Abstract

The Takács process, X ( t ) describing the virtual waiting time or server backlog for a single-server queue with Poisson arrivals and general service time distribution, is discussed with two absorbing boundaries. The process terminates at x = 0 when the server becomes idle or at x = T when a given backlog level is exceeded. The probabilities Γ T ( x 0 ) that absorption will occur at x = 0 if the process starts at x 0 , and Γ T ( x 0 , t ) that absorption will occur at zero before time t , are exhibited. The process is also of interest to the theory of collective risk.