Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications

Abstract

Through Laplace transforms, we study the extremes of a continuoustime Markov-additive process with one-sided jumps and a finite-state background Markovian state-space, jointly with the epoch at which the extreme is 'attained'. For this, we investigate discrete-time Markov-additive processes and use an embedding to relate these to the continuous-time setting. The resulting Laplace transforms are given in terms of two matrices, which can be determined either through solving a nonlinear matrix equation or through a spectral method. Our results on extremes are first applied to determine the steady-state buffer-content distribution of several single-station queueing systems. We show that our framework comprises many models dealt with earlier, but, importantly, it also enables us to derive various new results. At the same time, our setup offers interesting insights into the connections between the approaches developed so far, including matrix-analytic techniques, martingale methods, the rate-conservation approach, and the occupation-measure method. We also study networks of fluid queues, and show how the results on single queues can be used to find the Laplace transform of the steady-state buffer-content vector; it has a matrix quasi-product form. Fluid-driven priority systems also have this property.

First Passage of a Markov Additive Process and Generalized Jordan Chains

“…It is known that (q) is a unique (admissible) solution of a certain matrix integral equation. This result in different degrees of generality appears in [1], [6], [8], [17], [18], [20], and [21]. Alternatively, we can use an iterative method to compute (q); see, for example, [6].…”

On a generic class of two-node queueing systems

“…[3,13], one would expect that the dichotomy of the tail asymptotics carries over to a considerably larger class of inputs. The recent results in [7] may give a handle on resolving this issue.…”

“…Dieker and Mandjes [7] consider networks in which the input is a Markov additive process (that is, a Markov-modulated Lévy process), and in this sense more general than just an on-off Markov fluid source; their results are, however, considerably less explicit, and they do not consider tail asymptotics either.…”

First passage of time-reversible spectrally negative Markov additive processes

“…The problem of identifying Λ received a lot of attention, see [20,21,1,8] for modulated compound Poisson, linear drift, Brownian motion and a general spectrally one-sided Lévy process respectively. There are two (related) approaches to characterize Λ available in the literature.…”

“…There are two (related) approaches to characterize Λ available in the literature. Firstly, it is known that Λ solves a specific matrix integral equation, see [8]. Secondly, a spectral method can be used to identify Λ.…”