Yukio Takahashi

GEOMETRIC DECAY OF THE STEADY-STATE PROBABILITIES IN A QUASI-BIRTH-AND-DEATH PROCESS WITH A COUNTABLE NUMBER OF PHASES

A sufficient condition is proved for geometric decay of the steady-state probabilities in a quasi-birth-and-death process having a countable number of phases in each level. If there is a positive number η and positive vectors x = (x i) and y = (y j ) satisfying some equations and inequalities, the steady-state probability π mi decays geometrically with rate η in the sense π mi ∼ cη m x i as m → ∞. As an example, the result is applied to a two-queue system with shorter queue discipline.

Transient analysis of fluid model for ATM statistical multiplexer

Abstract We analyze the transient behavior of a fluid model of ATM multiplexer, in which the input process is generated by a Markov modulated rate process (MMR Process). This model is a natural extension of the on-off-type multi-entry queueing model which has been widely used for modeling ATM multiplexers with bursty inputs. The Laplace transform of the joint distribution of the buffer content and the state of the input process is obtained by solving a system of partial differential equations. The unknown functions included in the solution are determined from the boundary conditions by using eigenvalues and eigenvectors of a key matrix. By taking a limit of the solution, the state probabilities in the steady-state are written explicitly. An effective lower bound of the relaxation time is also presented. In case of an on-off-type input fluid model with phase-type on/off period distributions, the equations for the eigenvalues and eigenvectors of the key matrix are reduced to more concrete ones using the Laplace transforms of the on/off period distributions. The lower bound of the relaxation time is also reduced to the maximum among the relaxation times of phase-type renewal processes governing the on/off periods of the inputs.

Steady-state analysis of ATM multiplexer with variable input rate through diffusion approximation

Abstract We propose an approximation formula for the stationary distribution of the buffer content Qt of a fluid model for an ATM multiplexer in which the input rate process Rt is given as a Gaussian process of the form Rt = m + ∝t−∞ h(t − s) dws, where ws is a standard Wiener process. To derive the approximation formula, we approximate the behavior of Qt by a diffusion process on R (−∞, ∞) having no reflecting barrier. Simulation results show that the formula provides good approximate values. The formula is similar to the one derived by H. Kobayashi and Q. Ren for multiple On-Off types of traffic, but probabilistically it seems more reasonable. Our model is fairly general and it covers almost all variable input rate processes proposed so far for ATM multiplexer. In this sense, the new approximation formula provides a useful tool to analyse and estimate the congestion in the buffer of the ATM multiplexer.

Upper bound for the decay rate of the joint queue-length distribution in a two-node Markovian queueing system

AbstractThis paper studies the geometric decay property of the joint queue-length distribution {p(n1,n2)} of a two-node Markovian queueing system in the steady state. For arbitrarily given positive integers c1,c2,d1 and d2, an upper bound n

Tail probability of a Gaussian fluid queue under finite measurement of input processes

overline{eta}(c_{1},c_{2})

Bounds of performance measures in large-scale mobile communication networks

nof the decay rate is derived in the sense

Asymptotic end-to-end stochastic evaluation for tandem networks with many flows

expBigl{limsup_{nrightarrowinfty}n^{-1}log p(c_{1}n+d_{1},c_{2}n+d_{2})Bigr}leqoverline{eta}(c_{1},c_{2})<1.

Sliding window protocol with selective-repeat ARQ: performance modeling and analysis

It is shown that the upper bound coincides with the exact decay rate in most systems for which the exact decay rate is known. Moreover, as a function of c1 and c2, n