System-theoretical algorithmic solution to waiting times in semi-Markov queues

Abstract

a b s t r a c tMarkov renewal processes with matrix-exponential semi-Markov kernels provide a generic tool for modeling auto-correlated interarrival and service times in queueing systems. In this paper, we study the steady-state actual waiting time distribution in an infinite capacity single-server semi-Markov queue with the auto-correlation in interarrival and service times modeled by Markov renewal processes with matrix-exponential kernels. Our approach is based on the equivalence between the waiting time distribution of this semi-Markov queue and the output of a linear feedback interconnection system. The unknown parameters of the latter system need to be determined through the solution of a SDC (Spectral-Divide-and-Conquer) problem for which we propose to use the ordered Schur decomposition. This approach leads us to a completely matrix-analytical algorithm to calculate the steady-state waiting time which has a matrix-exponential distribution. Besides its unifying structure, the proposed algorithm is easy to implement and is computationally efficient and stable. We validate the effectiveness and the generality of the proposed approach through numerical examples.

A bivariate two-state Markov modulated Poisson process for failure modeling

“…In this paper we consider the batch counterpart of the MMPP, the so-called Batch Markov-modulated Poisson process, noted BMMPP. This process has been already considered in the literature Akar and Sohraby, 2009; for modeling real-time multimedia communication systems and computer networks systems. However, in most of such papers, a reduced version of the BMMPP with batch probabilities independent from the states of the underlying Markov chain, is used.…”

Delay Analysis of Timer-Based Frame Coalescing in Energy Efficient Ethernet

“…Note that P (W M/G/1 > t) can be numerically obtained either by Laplace transform inversion as in [10] or using matrix analytical techniques as in [8], the latter for service time distributions with rational Laplace transforms only. On the other hand, the mean waiting time…”

On Ultra-Sharp Queueing Bounds

“…Studying such models is not only driven by measurement studies (e.g., Crovella and Bestavros [12] or Mi et al [20]) but also by the fundamentally different queueing behavior when compared to the baseline renewal models (e.g., Tin [28] or Patuwo et al [22]). The standard analysis of SM/SM/1 models is computational in nature involving, e.g., transform methods (e.g., C ¸inlar [7] or Adan and Kulakarni [1]) or matrix analytical methods (e.g., Neuts [21] or Akar and Sohraby [3]).…”