The Departure Process of the GI/G/1 Queue and Its MacLaurin Series

In this paper, we study the departure process of the GI/G/1 queue. We develop a simple recursive procedure to calculate the MacLaurin series of its moments and covariances with respect to a parameter in the service time. Based on this recursive procedure the explicit formulas of the coefficients of these MacLaurin series can be obtained in terms of derivatives of the probability density function of the interarrival time evaluated at zero and the moments of the interarrival time and the service time. One important application of these MacLaurin series is that they can be used to obtain the entire response curves of the moments and variances of the departure process, for example, via interpolation by polynomials or rational functions.