Transient departure process in M/G/1/K-type queue with threshold server's waking up

Time-dependent behavior of departure process in a finite-buffer M/G/1/K-type queueing system with threshold server's waking up is analysed. After each idle time a new busy period is being initialized simultaneously with the Nth arrival occurrence, where the threshold value N is fixed. By applying the approach being a mixture of different theoretical techniques: the idea of embedded Markov chain, Volterra type integral equations, continuous total probability law, renewal theory and linear algebra, a closed-form formula for the mixed double transform (probability generating function of Laplace transform) of the probability distribution of the number of packets completely processed up to fixed time t is derived. The considered queueing system can be used in modeling the operation of a wireless sensor network (WSN) with power saving mechanism based on “queued” waking up of radio transmitters/receivers of nodes (sensors). An illustrating numerical example for a hypothetical network traffic is attached as well.