Abstract
The three node Jackson queueing network is the simplest acyclic network in which in equilibrium the sojourn times of a customer at each of the nodes are dependent. We show that assuming the individual sojourn times are independent provides a good approximation to the total sojourn time. This is done by simulating the network and showing that the sojourn times generally pass a Kolmogorov-Smirnov test as having come from the approximating distribution. Since the sum of dependent random variables may have the same distribution as the sum of independent random variables with the same marginal distributions, it is conceivable that our approximation is exact. However, we numerically compute upper and lower bounds for the distribution of the total sojourn time; these bounds are so close that the approximating distribution lies outside of the bounds. Thus, the bounds are accurate enough to distinguish between the two distributions even though the Kolmogorov-Smirnov test generally cannot.
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Kiessler, P.C., Melamed, B., Yadin, M. et al. Analysis of a three node queueing network. Queueing Syst 3, 53–72 (1988). https://doi.org/10.1007/BF01159087
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DOI: https://doi.org/10.1007/BF01159087