Stochastig modeling with continuous feedback markov fluid queues

Abstract

Markov fluid queues (MFQ) are systems in which a continuous-time Markov chain determines the net rate into (or out of ) a buffer. We deal with continuous feedback MFQs (CFMFQ) for which the infinitesimal generator of the background process and the drifts in each state are allowed to depend on the buffer level through continuous functions. Explicit solutions of CFMFQs for a few special cases has been reported, but usually numerical methods are preferred. A numerically stable solution method based on ordered Schur decomposition is already known for multi-regime MFQs (MRMFQ). We propose a framework for approximating CFMFQs by MRMFQs via discretizing the buffer space. The parameters of the CFMFQ are approximated by piecewise constant functions. Then, the solution is obtained by block-tridiagonal LU decomposition for the related MRMFQ. Moreover, we describe a numerical method that enables us to solve large scale systems efficiently. We model basically two different stochastic systems with CFMFQs. The first is the workload-bounded MAP/PH/1 queue, to which the arrivals occur according to a workload-dependent MAP (Markovian Arrival Process), and the arriving job size distribution is phase-type. The jobs that would cause the buffer to overflow are rejected partially or completely. Also, the service speed is allowed to depend on the buffer level. As the second application, we model the horizon-based delayed reservation mechanism in Optical Burst Switching networks with or without fiber delay lines. We allow multiple traffic classes and the effect of offset-based and FDL-based differentiation among traffic classes in terms of burst blocking is investigated. Lastly, we propose a distributed algorithm for air-time fairness in multi-rate WLANs that overcomes the performance anomaly in IEEE 802.11 WLANs. We also give a stochastic model of the proposed model, and provide a novel and elaborate proof for its effectiveness. We also present an extensive simulation study.