Complex-analytic and matrix-analytic solutions for a queueing system with group service controlled by arrivals

A bulk M/G/1 system is considered that responds to large increases (decreases) of the queue during the service act by alternating between two service modes. The switching rule is based on two ‘up’ and ‘down’ thresholds for total arrivals over the service act. A necessary and sufficient condition for the ergodicity of a Markov chain embedded into the main queueing process is found. Both complex-analytic and matrix-analytic solutions are obtained for the steady-state distribution. Under the assumption of the same service time distribution in both modes, a combined complex-matrix-analytic method is introduced. The technique of ‘matrix unfolding’ is used, which reduces the problem to a matrix iteration process with the block size much smaller than in the direct application of the matrix-analytic method.