Infinite- and finite-buffer Markov fluid queues: a unified analysis

Abstract

In this paper, we study Markov fluid queues where the net fluid rate to a single-buffer system varies with respect to the state of an underlying continuous-time Markov chain. We present a novel algorithmic approach to solve numerically for the steady-state solution of such queues. Using this approach, both infinite-and finite-buffer cases are studied. We show that the solution of the infinite-buffer case is reduced to the solution of a generalized spectral divide-and-conquer (SDC) problem applied on a certain matrix pencil. Moreover, this SDC problem does not require the individual computation of any eigenvalues and eigenvectors. Via the solution for the SDC problem, a matrixexponential representation for the steady-state queue-length distribution is obtained. The finite-buffer case, on the other hand, requires a similar but different decomposition, the so-called additive decomposition (AD). Using the AD, we obtain a modified matrixexponential representation for the steady-state queue-length distribution. The proposed approach for the finite-buffer case is shown not to have the numerical stability problems reported in the literature.

Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier

“…In implementing our algorithms, we chose an error tolerance of 10 −15 in the computation of the required (·) matrices and an error tolerance of 10 −13 in the transform inversion algorithm. Table 3 demonstrates that at all points x, the computed transient distribution does converge to the steady state distribution as t → ∞ and that the steady state distributions computed using the two different methods of [7] and [8] agree to the precision that is achievable with our choice of the error tolerance values.…”

“…6 in Akar and Sohraby [8] for which steady state results computed using the methods of [8] were provided to us by Akar. A common buffer receives multiplexed input from 10 identical traffic sources, and the buffer is drained at a constant rate.…”

“…We assumed the fluid buffer to start empty with the environmental Markov chain in a state of S 1 chosen according to the steady state vector of Q restricted to S 1 and renormalized. We present in Table 3 the complementary distribution of the buffer size at various time points t and buffer levels x; for x = b, where b is the buffer size, we have the ccdf at b− so that our results are comparable to those provided by Akar. In that table, the column marked t = ∞, AR provides the steady state results obtained using the algorithms in [7] while those in the column marked t = ∞, AS provide those communicated by Akar based on [8]. In implementing our algorithms, we chose an error tolerance of 10 −15 in the computation of the required (·) matrices and an error tolerance of 10 −13 in the transform inversion algorithm.…”

“…With regard to MMFFs, several approaches to them exist in the literature and are based on matrix Wiener-Hopf factorizations [18,40], Siegmund duality [13], spectral decomposition [11,29], sign function iterations [8], and series expansions [43]. The approach followed here is the one initiated by Ramaswami [36] built on arguments that parallel those in the matrix-geometric method [30,34] to quasi birth and death processes (QBDs) that are discrete analogues of MMFFs; for a discussion of the various advantages of this approach, refer to cited references.…”

Multiclass Markovian fluid queues

“…For example, recent studies on Markovian fluid queues are found in [1,2,5,11,14]. More general models are also studied in [3,9,10,15].…”