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The Distribution of the Line Length in a GI/G/1 Queue Using Distribution Little Laws and Roots Methods

Banff International Research Station for Mathematical Innovation and Discovery

In this talk, we provide a method, called L, based on distributional law of Little (DLL) to determine the equilibrium distribution of the number of elements in a discrete-time GI/G/1 queueing system. We also clarify a number of issues, and provide a number of new results. We assume that the inter-arrival times range from 1 to g+1 and the service times from 1 to h+1. The majority of authors have formulated the system in question as a quasi birth and death process, which can be solved by the matrix iterative methods pioneered by Neuts, methods that are cubic in the number of phases, and in fact, if g=h, all matrix analytic methods we found in literature are cubic in g, or worse. In contrast, our method L, which finds the distribution of the number of elements in the system in quadratic time. This implies that for large enough g and h, our algorithm will outperform all cubic algorithms, a claim verified by numerical tests. In particular, for realistic values g and h, algorithm L is more than 50 times faster than the different algorithms based on matrix analytic methods we found in literature.

Mathematics; Mathematics

video/mp4

Author affiliation: University of British Columbia, Okanagan

2021-02-19

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/77342

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/