Time-dependent queue-size distribution in a finite-buffer model with server setup times

Transient queue-size distribution in a finite-buffer system with Poisson arrivals and generally distributed processing times is investigated. In the evolution of the system the server needs randomly distributed setup times preceding the service initialization in each new busy period. Applying the paradigm of embedded Markov chain and the formula of total probability, a Volterra-type system of integral equations for the transient queue-size distribution, conditioned by the number of packets being accumulated in the buffer before the opening of the system, is built. The solution of the corresponding system written for Laplace transforms is obtained algebraically in the compact explicit form. Numerical examples are attached as well.