New LST of inter-departure times in PH/G/1 queue, and extensions to ME/G/1 and G/G/1 queues. (September 2019) - Sagron, Ruth; Kerner, Yoav; Rabinowitz, Gad; Tirkel, Israel
Highlights: Analyzing inter-departure times distribution via Laplace–Stieltjes Transform (LST). Presenting a new approach to express the LST of inter-departure times in PH/G/1 queues. The new approach significantly reduces the computational complexity. Introducing an exact expression of the LST of inter-departure times in ME/G/1 queues. Introducing an approximated expression of the LST of inter-departure times in G/G/1 queues, as good as one wants. Abstract: In this paper, we provide a new approach to model the inter-departure times distribution in a PH / G /1 queue. This approach enables to further model the inter-departure times distribution in more general queues as well. Initially, we propose to express the Laplace–Stieltjes transform (LST) of inter-departure times in PH / G /1 queues by exploiting the probabilistic interpretation of phase-type distributions. Using this interpretation enables to eliminate the necessity of the matrix-geometric method, and thus significantly reduces the computational complexity. Then, we use the LST of inter-departure times distribution in a Cm / G /1 queue to express this LST in a ME / G /1 queue, where ME is a Matrix-Exponential distribution. We validate it in a few ME / G /1 examples. Finally, we propose to approximate the LST of inter-departure times distribution in a G / G /1 queue by employing the above LST of the proper PH / G /1 queue. Without loss of generality, we demonstrate our proposed approximation by using the LST as obtained in a Cm / G /1 queue, while illustrating by a few G / G /1 examples that the accuracy can be as good as one might want.