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.

No abstract

Abstract-In this paper, we study the blocking probabilities in a wavelength division multiplexing-based asynchronous bufferless optical packet/burst switch equipped with a bank of tuneable wavelength converters dedicated to each output fiber line. Wavelength converter sharing, also referred to as partial wavelength conversion, corresponds to the case of a number of converters shared amongst a larger number of wavelength channels. In this study, we present a probabilistic framework for exactly calculating the packet blocking probabilities for optical packet/burst switching systems utilizing wavelength converter sharing. In our model, packet arrivals at the optical switch are first assumed to be Poisson and later generalized to the more general Markovian arrival process to cope with very general traffic patterns whereas packet lengths are assumed to be exponentially distributed. As opposed to the existing literature based on approximations and/or simulations, we formulate the problem as one of finding the steady-state solution of a continuous-time Markov chain with a block tridiagonal infinitesimal generator. To find such solutions, we propose a numerically efficient and stable algorithm based on block tridiagonal LU factorizations. We show that exact blocking probabilities can be efficiently calculated even for very large systems and rare blocking probabilities, e.g., systems with 256 wavelengths per fiber and blocking probabilities in the order of 10 −40 . Relying on the stability and speed of the proposed algorithm, we also provide a means of provisioning wavelength channels and converters in optical packet/burst switching systems.Index Terms-Optical packet switching, optical burst switching, wavelength conversion, converter sharing, block-tridiagonal LU factorization, Markovian arrival process.

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.

In this paper, we study the

This paper presents a relatively efficient and accurate method to compute the moments of first passage times to a subset of states in finite ergodic Markov chains. With the proposed method, the moment computation problem is reduced to the solution of a linear system of equations with the right-hand side governed by a novel recurrence for computing the higher-order moments. We propose using a form of the Grassmann-Taksar-Heyman (GTH) algorithm to solve these linear equations. Due to the form of the linear systems involved, the proposed method does not suffer from the drawbacks associated with GTH in a row-wise sparse implementation.

Bufferless and single-buffer queueing systems have recently been shown to be effective in coping with escalated Age of Information (AoI) figures arising in systems with large buffers and FCFS scheduling. In this paper, we propose a numerical algorithm for obtaining the exact distribution of both the AoI and the peak AoI (PAoI) in the bufferless P H/P H/1/1 and P H/P H/1/1/P -LCFS queues as well as the single-buffer M/P H/1/2 and M/P H/1/2 * queues, the latter one involving the replacement of the packet in the queue by the new arrival. The proposed exact models are based on the well-established theory of Markov fluid queues and the numerical algorithms rely on numerically stable and efficient vector-matrix operations. Moreover, the obtained exact distributions are in matrix exponential form making it amenable to calculate the tail distributions and the associated moments straightforwardly. We validate the proposed algorithms with simulations and we also comparatively study the AoI performance of the four queueing systems of interest as a function of the system load as well as the squared coefficient of variation (scov) of the service time. A similar study is also pursued for assessing the impact of the scov of the interarrival time for the two bufferless queueing systems.

Abstract-Optical buffering via fiber delay lines is used for contention resolution in optical packet and optical burst switching nodes. This article addresses the problem of exactly finding the blocking probabilities in an asynchronous single-wavelength optical buffer. Packet lengths are assumed to be variable and modeled by phase-type distributions, whereas the packet arrival process is modeled by a Markovian arrival process that can capture autocorrelations in interarrival times. The exact solution is based on the theory of feedback fluid queues for which we propose numerically efficient and stable algorithms. We not only find the packet blocking probabilities but also the entire distribution of the unfinished work in this system from which all performance measures of interest can be derived.Index Terms-Fiber delay line (FDL); Buffer; Optical packet switching; Optical burst switching; Feedback fluid queues.