Rate conservation laws: A survey

We survey the rate conservation law, RCL for short, arising in queues and related stochastic models. RCL was recognized as one of the fundamental principles to get relationships between time and embedded averages such as the extended Little's formulaH=G, but we show that it has other applications. For example, RCL is one of the important techniques for deriving equilibrium equations for stochastic processes. It is shown that the various techniques, including Mecke's formula for a stationary random measure, can be formulated as RCL. For this purpose, we start with a new definition of the rate with respect to a random measure, and generalize RCL by using it. We further introduce the notion of quasi-expectation, which is a certain extension of the ordinary expectation, and derive RCL applicable to the sample average results. It means that the sample average formulas such asH=G can be obtained as the stationary RCL in the quasi-expectation framework. We also survey several extensions of RCL and discuss examples. Throughout the paper, we would like to emphasize how results can be easily obtained by using a simple principle, RCL.     

Asymptotic distribution of maximal autoregressive process with weight tending to 1

The maximal operator plays the similar role as the summation operator in the sense of stability of operation. So, we could discussARMA processes in the maximal operation by the same way as in the summation operation. However, many papers already treated with moving order statistics. In this paper, we discuss asymptotic behaviors of maximal autoregressive (MAR) processes with the weight tending to 1.     

On the distribution of the number of packets in the fluid flow approximation of packet arrival streams

Fluid flow approximations are widely used for approximating models of communication systems where packet arrival streams are generated in a regular manner over certain intervals (constant rate). The appropriate mathematical model for describing those bursty arrival streams in the fluid flow framework are the well-known Markov modulated rate processes (MMRP). The paper deals with the distribution of the numberN(t) of packets in the interval [0,t] of MMRP. For two-state MMRPs and their superpositions we derive formulas for the distribution ofN(t) and its density. Further we give asymptotic results. The presented numerical results and simulation studies illustrate the goodness of the fluid flow approximation and show that the proposed numerical algorithms work well even in the case of multiplexing a large number of burst silence sources.This work was partially supported by a grant from the Deutsche Bundespost TELEKOM.     

Joint availability of systems modelled by finite semi–markov processes

Consider a repairable system at the time instants t and t + x, where t, x ≥0. The joint availability of the system at these time instants is defined as the probability of the system being functional in both t and t + x. A set of integral equations is derived for the joint availability of a system modelled by a finite semi–Markov process. The result is applied to the semi–Markov model of a two–unit system with sequential preventive maintenance. The method used for the numerical solution of the resulting system of integral equations is a two–point trapezoidal rule. The system of implementation is the matrix computation package MATLAB on the Apple Macintosh SE/30. The numerical results obtained by this method are shown to be in good agreement with those from simulation.     

A law of the iterated logarithm for processes with independent increments

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The multifractal spectrum of statistically self-similar measures

We calculate the multifractal spectrum of a random measure constructed using a statistically self-similar process. We show that with probability one there is a multifractal decomposition analogous to that in the deterministic self-similar case, with the exponents given by the solution of an expectation equation.     

A PARALLEL COMPUTATION SCHEME FOR IMPLICIT RUNGE-KUTTA METHODS AND THE ITERATIVELY B-CONVERGENCE OF ITS NEWTON ITERATIVE PROCESS

In this paper, based on the implicit Runge-Kutta(IRK) methods, we derive a class of parallel scheme that can be implemented on the parallel computers with Ns(N is a positive even number) processors efficiently, and discuss the iteratively B-convergence of the Newton iterative process for solving the algebraic equations of the scheme, secondly we present a strategy providing initial values parallelly for the iterative process. Finally, some numerical results show that our parallel scheme is higher efficient as N is not so large.     

Charge storage and relaxation process of polymer film by thermally stimulated discharge current technique

Charge storage and relaxation process of the polymide film electret were investigated using the thermally stimulated discharge current technique. Homo- and heterocharges were mainly observed up to and above the polarization temperature.     

Front propagation in certain one-dimensional exclusion models

Recent results on two interacting particle systems on are summarized, the asymmetric simple exclusion process and the branching exclusion process.     

Size distribution of fractured areas in one-dimensional systems

We study a one-dimensional model for fracture, identifying fractured areas with intervals on which a stress field exceeds a threshold value. When is a diffusion process, the cumulative numberN(l) of fractured areas whose length is greater thanl obeys a power lawCl ^–p asl0 with probability one. The exponentp and the constantC are determined. The exponentp agrees with the Hausdorff dimension of the end points of fractured areas, i.e., ^–1(). Even if is self-similar with parameterH>0, i.e.,(cx)– is equivalent toc ^H {(x)–} for anyc>0, the exponentp does not depend solely onH;p=H, where(0, 1/H) is another parameter characterizing. Non-diffusion processes are given whereN(l) does not follow a power law.