A Gambler's Ruin Type Problem in Queuing Theory

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(x₀) that absorption will occur at x = 0 if the process starts at x₀, and Γ_T(x₀, t) that absorption will occur at zero before time t, are exhibited. The process is also of interest to the theory of collective risk.