Transient solution for the queue-size distribution in a finite-buffer model with general independent input stream and single working vacation policy

A single-channel finite-buffer queueing model with a general independent input stream of customers, exponential processing times and a working vacation policy is considered. Every time, when the server becomes idle, an exponentially distributed single working vacation period is being initialized, during which the processing is provided with another (slower) rate. After the completion of the vacation period, the service is being continued normally, with the original speed. Using the idea of an embedded Markov chain, the systems of Volterra-type integral equations for the time-dependent queue-size distributions, conditioned by the initial buffer state and related to each other, are built for models beginning the operation in normal and working vacation modes, separately. The solutions of the corresponding systems written for the Laplace transforms are obtained in compact forms using the linear algebraic approach. The numerical illustrative examples are attached as well.