On transient queue-size distribution in the batch arrival system with the N-policy and setup times

In the paper the M-X/G/1 queueing system with the N-policy and setup times is considered. An explicit formula for the Laplace transform of the transient queue-size distribution is derived using the approach consisting of few steps. Firstly, a "special" modification of the original system is investigated and, using the formula of total probability, the analysis is reduced to the case of the corresponding system without limitation in the service. Next, a renewal process generated by successive busy cycles is used to obtain the general result. Sample numerical computations illustrating theoretical results are attached as well.