On the distribution of the time to buffer overflow in a queueing system with a general-type input stream

A finite-buffer queueing system with a generally distributed interarrival times and exponential service times is considered. A compact formula for the Laplace transform of distribution of the time to the first buffer overflow is found and written down using a special-type sequence connected with “input” distributions of the system. Moreover, applying the formula of total probability, a representation for the distribution of the time to the kth buffer overflow for k ≥ 2 is derived. Numerical examples are attached as well.