Nail Akar

An invariant subspace approach in m/g/l and g/m/l type markov chains

Let , be a sequence ofmtimes;mnonnegative matrices and let be such that A(1) is an irreducible stochastic matrix. The unique power-bounded solution of the nonlinear matrix equation has been shown to play a key role in the analysis of Markov chains of M/G/l type. Assuming that the matrix A(z) is rational, we show that the solution of this matrix equation reduces to finding an invariant subspace of a certain matrix. We present an iterative method for computing this subspace which is globally convergent. Moreover, the method can be implemented with quadratic or higher convergence rate matrix sign function iterations, which brings in a new dimension to the analysis of M/G/l type Markov chains for which the existing algorithms may suffer from low linear convergence rates. The method can be viewed as a “ bridge ” between the matrix analytic methods and transform techniques whereas it circumvents the requirement for a large number of iterations which may be encountered in the methods of the former type and the r...

Wavelength converter sharing in asynchronous optical packet/burst switching: An exact blocking analysis for markovian arrivals

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.

Matrix-geometric solutions of M/G/1-type Markov chains: a unifying generalized state-space approach

We present an algorithmic approach to find the stationary probability distribution of M/G/1-type Markov chains which arise frequently in performance analysis of computer and communication networks. The approach unifies finite- and infinite-level Markov chains of this type through a generalized state-space representation for the probability generating function of the stationary solution. When the underlying probability generating matrices are rational, the solution vector for level k, x/sub k/, is shown to be in the matrix-geometric form x/sub k+1/=gF/sup k/H, k/spl ges/0, for the infinite-level case, whereas it takes the modified form x/sub k+1/=g/sub 1/F/sup k//sub 1/H/sub 1/+g/sub 2/F/sup K-k-1//sub 2/H/sub 2/, 0/spl les/k/spl les/K, for the finite-level case. The matrix parameters in the above two expressions can be obtained by decomposing the generalized system into forward and backward subsystems, or, equivalently, by finding bases for certain generalized invariant subspaces of a regular pencil /spl lambda/E-A. We note that the computation of such bases can efficiently be carried out using advanced numerical linear algebra techniques including matrix-sign function iterations with quadratic convergence rates or ordered generalized Schur decomposition. The simplicity of the matrix-geometric form of the solution allows one to obtain various performance measures of interest easily, e.g., overflow probabilities and the moments of the level distribution, which is a significant advantage over conventional recursive methods.

A common solution to a pair of linear matrix equations over a principal ideal domain

Abstract A necessary and sufficient condition for the existence of a common solution to a pair of linear matrix equations over a principal ideal domain is obtained. The equations are of the type A i = B i XC i for i = 1,2. The solvability condition is that the equations each have a solution and a bilateral linear matrix equation made up of the matrices A i , B i , and C i has a solution.

Finite and infinite QBD chains: a simple and unifying algorithmic approach

In this paper, we present a novel algorithmic approach, the hybrid matrix geometric/invariant subspace method, for finding the stationary probability distribution of the finite quasi-birth-death (QBD) process which arises in performance analysis of computer and communication systems. Assuming that the QBD state space is defined in two dimensions with m phases and K+1 levels, the solution vector for level k, /spl pi//sub k/, 0/spl les/k/spl les/K is shown to be in a modified matrix geometric form /spl pi//sub k/=/spl upsi//sub 1/R/sub 1//sup k/+/spl upsi//sub 2/R/sub 2//sup K-k/ where R/sub 1/ and R/sub 2/ are certain solutions to two nonlinear matrix equations and /spl upsi//sub 1/ and /spl upsi//sub 2/ are vectors to be determined using the boundary conditions. We show that the matrix geometric factors R/sub 1/ and R/sub 2/ can simultaneously be obtained independently of K via finding the sign function of a real matrix by an iterative algorithm with quadratic convergence rates. The time complexity of obtaining the coefficient vectors /spl upsi//sub 1/ and /spl upsi//sub 2/ is shown to be O(m/sup 3/ log/sub 2/ K) which indicates that the contribution of the number of levels on the overall algorithm is minimal. Besides the numerical efficiency, the proposed method is numerically stable and in the limiting case of K/spl rarr//spl infin/, it is shown to yield the well-known matrix geometric solution /spl pi//sub k/=/spl pi//sub 0/R/sub 1//sup k/ for infinite QBD chain.

A simple and effective mechanism for stored video streaming with TCP transport and server-side adaptive frame discard

Transmission control protocol (TCP) with its well-established congestion control mechanism is the prevailing transport layer protocol for non-real time data in current Internet Protocol (IP) networks. It would be desirable to transmit any type of multimedia data using TCP in order to take advantage of the extensive operational experience behind TCP in the Internet. However, some features of TCP including retransmissions and variations in throughput and delay, although not catastrophic for non-real time data, may result in inefficiencies for video streaming applications. In this paper, we propose an architecture which consists of an input buffer at the server side, coupled with the congestion control mechanism of TCP at the transport layer, for efficiently streaming stored video in the best-effort Internet. The proposed buffer management scheme selectively discards low priority frames from its head-end, which otherwise would jeopardize the successful playout of high priority frames. Moreover, the proposed discarding policy is adaptive to changes in the bandwidth available to the video stream.

Exact calculation of blocking probabilities for bufferless optical burst switched links with partial wavelength conversion

In this paper, we study the blocking probabilities in a wavelength division multiplexing-based asynchronous bufferless optical burst switch equipped with a bank of tuneable wavelength converters that is shared per output link. The site of this bank is generally chosen to be less than the number of wavelengths on the link because of the relatively high cost of wavelength converters using current technologies; this case is referred to as partial wavelength conversion in the literature. We present a probabilistic framework for exactly calculating the blocking probabilities. Burst durations are assumed to be exponentially distributed. Burst arrivals are first assumed to be Poisson and later generalized to the more general phase-type distribution. Unlike 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. We propose a numerically efficient and stable solution technique based on block tridiagonal LU factorizations. We show that blocking probabilities can exactly and efficiently be found even for very large systems and rare blocking probabilities. Based on the results of this solution technique, we also show how this analysis can be used for provisioning wavelength channels and converters.

Shared-per-wavelength asynchronous optical packet switching: A comparative analysis

This paper compares four different architectures for sharing wavelength converters in asynchronous optical packet switches with variable-length packets. The first two architectures are the well-known shared-per-node (SPN) and shared-per-link (SPL) architectures, while the other two are the shared-per-input-wavelength (SPIW) architecture, recently proposed as an optical switch architecture in synchronous context only, which is extended here to the asynchronous scenario, and an original scheme called shared-per-output-wavelength (SPOW) architecture that we propose in the current article. We introduce novel analytical models to evaluate packet loss probabilities for SPIW and SPOW architectures in asynchronous context based on Markov chains and fixed-point iterations for the particular scenario of Poisson input traffic and exponentially distributed packet lengths. The models also account for unbalanced traffic whose impact is thoroughly studied. These models are validated by comparison with simulations which demonstrate that they are remarkably accurate. In terms of performance, the SPOW scheme provides blocking performance very close to the SPN scheme while maintaining almost the same complexity of the space switch, and employing less expensive wavelength converters. On the other hand, the SPIW scheme allows less complexity in terms of number of optical gates required, while it substantially outperforms the widely accepted SPL scheme. The authors therefore believe that the SPIW and SPOW schemes are promising alternatives to the conventional SPN and SPL schemes for the implementation of next-generation optical packet switching systems.

A novel computational method for solving finite qbd processes

We present a novel numerical method that exploits invariant subspace computations for finding the stationary probability distribution of a finite QBD process. Assuming that the QBD state space is defined in two dimensions with m phases and K+1 levels, the solution vector π k for level , is known to be expressible in the mixed matrix-geometric form , where R 1and R 2are certain solutions to two quadratic matrix equations, and v 1and v 2are vectors to be determined using the boundary conditions. We show that the matrix-geometric factors R 1and R 2can be simultaneously obtained irrespective of K via finding arbitrary bases for the left-and right-invariant subspaces of a certain real matrix of size 2m To find these bases, we employ either Schur decomposition or a matrix-sign function iteration with quadratic convergence rate. The vectors v 1and v 2are obtained by solving a linear matrix equation, which is constructed with a time complexity of . Therefore, the effect of the number of levels on the overall complexity is minimal. Besides its numerical efficiency, the proposed method is numerically stable, and in the limiting case of , it is shown to yield the well-known matrix-geometric solution for infinite QBD processes. We also extend the method to the cases of level-dependent and non-canonical transitions, and provide a numerical example to demonstrate its computational features in comparison with other well-known methods

Solving Multi-Regime Feedback Fluid Queues

In this paper, we study Markov fluid queues with multiple thresholds, or the so-called multi-regime 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 210 states is possible using this approach. We present numerical examples to justify the stability and validate the effectiveness of the proposed approach.