Duality relations in finite queueing models
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Motivated by applications in multimedia streaming and in energy systems, we study duality relations in fi nite queues. Dual of a queue is de fined to be a queue in which the arrival and service processes are interchanged. In other words, dual of the G1/G2/1/K queue is the G2/G1/1/K queue, a queue in which the inter-arrival times have the same distribution as the service times of the primal queue and vice versa. Similarly, dual of a fluid flow queue with cumulative input C(t) and available processing S(t) is a fluid queue with cumulative input S(t) and available processing C(t). We are primarily interested in finding relations between the overflow and underflow of the primal and dual queues. Then, using existing results in the literature regarding the probability of loss and the stationary probability of queue being full, we can obtain estimates on the probability of starvation and the probability of the queue being empty. The probability of starvation corresponds to the probability that a queue becomes empty, i.e., the end of a busy period. We study the relations between arrival and departure Palm distributions and their relations to stationary distributions. We consider both the case of point process inputs as well as fluid inputs. We obtain inequalities between the probability of the queue being empty and the probability of the queue being full for both the time stationary and Palm distributions by interchanging arrival and service processes. In the fluid queue case, we show that there is an equality between arrival and departure distributions that leads to an equality between the probability of starvation in the primal queue and the probability of overflow in the dual queue. The techniques are based on monotonicity arguments and coupling. The usefulness of the bounds is illustrated via numerical results.