Solving Multi-Regime Feedback Fluid Queues

Abstract

In this paper, we study Markov fluid queues with multiple thresholds, or the so-called multiregime feedback fluid queues. The boundary conditions are derived in terms of joint densities and for a relatively wide range of state types including repulsive and zero drift states. The ordered Schur factorization is used as a numerical engine to find the steady-state distribution of the system. The proposed method is numerically stable and accurate solution for problems with two regimes and 2 10 states is possible using this approach. We present numerical examples to justify the stability and validate the effectiveness of the proposed approach.

The workload-dependent MAP/PH/1 queue with infinite/finite workload capacity

“…In this paper, we assume that R (k) is invertible for each regime, i.e., S (k) 0 = ∅, ∀k, which does not lead to any loss of generality from the viewpoint of workload-dependent buffers with non-zero service speeds. Moreover, in each state of the modulating process, the sign of the service speed remains the same for all regimes, which allows us to simplify the set of boundary conditions listed in [10] to obtain…”

“…The following generalization is based on [20,10]. A multi-regime Markov fluid queue has more boundaries than the usual two terminal boundary points, 0 and B, that a single-regime Markov fluid queue has.…”

“…The entries of the block-banded matrix are obtained by using Schur decomposition and Sylvester equations, as in [10]. We also show by numerical examples that the computational complexity of the proposed algorithm depends linearly on the number of regimes used for discretizing the CFMFQ.…”

“…The approach we take in the current paper is to use the well-established theory of Markov fluid queues (MFQs) and in particular multi-regime MFQs (MRMFQs) [10] to study workload-dependent MAP/PH/1 queues. This approach saves us from the burden of rederiving the integral equations that would arise, and we can readily use the already existing numerically stable and efficient methods that have been proposed for the steady-state solution of MRMFQs.…”

“…In MRMFQs, which are also called ''level-dependent'' [16], ''multi-layer'' [17,18] or ''multi-threshold'' [19] fluid queues, the buffer space is partitioned into a finite number of non-overlapping intervals which are called the regimes of the MRMFQ. In MRMFQs, the infinitesimal generator of the background CTMC as well as the drift of the buffer process are allowed to depend on the regime in which the buffer level resides; see [16,20,10] for more detailed description of MRMFQs as well as the boundary conditions that arise in the solution of the steady-state distribution of the buffer level. In continuous-feedback Markov fluid queues (CFMFQs), both the infinitesimal generator of the background CTMC and the drift of the buffer process are allowed to continuously depend on the instantaneous buffer level [21].…”