A Note on the Relationships among the Queue Lengths at Various Epochs of a Queue with BMAP Arrivals

A Note on the Relationships among the Queue Lengths at Various Epochs of a Queue with BMAP Arrivals

For a stationary queue with BMAP arrivals, Takine and Takahashi [8] present a relationship between the queue length distributions at a random epoch and at a departure epoch by using the rate conservation law of Miyazawa [6]. In this note, we derive the same relationship by using the elementary balance equation, ‘rate-in = rate-out’. Along the same lines, we additionally derive relationships between the queue length distributions at a random epoch and at an arrival epoch. All these relationships hold for a broad class of finite-as well as infinite-capacity queues with BMAP arrivals.