New LST of inter-departure times in PH/G/1 queue, and extensions to ME/G/1 and G/G/1 queues

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.