Approximations for the departure process of the GI/G/1 queue with Markov-modulated arrivals

The interarrival time random variables at the nodes of queueing networks are typically correlated. The analysis, by ignoring the correlations, may not be very accurate in many instances, especially when the correlations are high. The GI/G/1 queue with Markov-modulated arrivals is one of the few models that can be used to study queues with non-renewal arrival processes. The waiting time of the Markov-modulated GI/G/1 queue has been studied widely in the literature. In this paper, we study its departure process. We derive the MacLaurin series for the moments and the lag-1 autocorrelation of the departure process. The coefficients of the MacLaurin series are calculated through simple recursive equations, which are then used to approximate the moments and the lag-1 autocorrelation based on Padé approximation. Several numerical examples are provided which show that our method gives very good estimates. Our results on the departure process can be potentially used to develop better approximation methods for general queueing networks.