On time to buffer saturation in a GI/M/1/N-type queue

A GI/M/1/N-type queueing system with independent and generally distributed interarrival times and exponential service times is investigated. A system of equations for conditional distributions of the time to the first buffer saturation is built. The solution is written using a special-type sequence defined by ``input'' distributions of the system. The formula of total probability is used to derive a representation for the distribution of the time to the $k$th buffer saturation for k>=2. Moreover, special cases of the Poisson arrival process and the system with one-place buffer are discussed. Sample numerical results for the 3-Erlang and deterministic distributions of interarrival times are attached as well.