Boundedness conditions for relative error in fast simulation of reliability of non-Markovian systems

Abstract

A general model is considered that describes the functioning of non-Markovian systems. A new method is proposed for fast simulation of failure probability. The method allows obtaining estimates with a bounded relative error under some weak conditions. This method is used to evaluate the reliability of a specific class of systems. Numerical examples are considered.Keywords: fast simulation method, non-Markovian system, reliability, estimate variance, relative root-mean-square error.Among the reliability theory research areas that have recently been developed well, noteworthy are asymptotic methods of low-utilization systems analysis and fast simulation methods. Both approaches pursue the same aim -to create precise numerical tools for calculating the characteristics of high-reliable systems. For the modern state of the asymptotic systems analysis and some further results see [1][2][3][4][5]. Here, we will pay the main attention to fast simulation methods (methods of decreasing estimate variance).As is generally known, statistical simulation (the Monte-Carlo method) [6, 7] is a tool most often used in engineering practice for estimating system characteristics, including reliability. At the same time, the higher the reliability of the system, the lower the efficiency of simulation, in particular, if the reliability tends to unity, the number of realizations necessary to achieve the required accuracy increases unrestrictedly. This shortcoming of the Monte Carlo method has promoted an intensive development of methods intended to decrease estimate variance and thus to increase the efficiency of simulation. Among the most popular approaches, noteworthy are the importance sampling technique [8][9][10][11][12], the analytical-statistical method [4,[13][14][15][16][17][18][19], the stratified sample method [20-23], etc. [6, 24-32].For systems described by a Markovian or semi-Markovian process with a finite or enumerable number of states, the theory of reliability analysis has been studied in detail. The case of a non-Markovian model is much more complex. According to Kovalenko's definition, such a model is described by a multidimensional process of queuing theory, when at least two operations with nonexponentially distributed durations (i.e., with distributions that are neither exponential, nor Erlang, nor hyper-Erlang, nor a mixture of these distributions) take place with unit probability at each instant of time. The basic problem is that the theory of regenerating processes is inapplicable here.We analyze here a general model that describes the operation of non-Markovian systems. A fast simulation method for failure probability is proposed, which allows us to find, under rather weak conditions, estimates with bounded relative root-mean-square error (RRMSE) with increase in component reliability. This method is considered as applied to the reliability analysis of a specific class of restorable redundant systems. No random quantities (r.q.) describing the operation of the system are assumed exponential. Easily checked cond...

Прискорене Моделювання Ймовірності Блокування Вимог У Мережі Обслуговування З Множинним Доступом Та Періодичними Інтенсивностями Вхідних Потоків

“…Суттєвий внесок у створення сучасної теорії прискореного моделювання внесла і українська школа математичної теорії надійності на чолі з академіком НАН України І.М. Коваленком [18][19][20][21][22].…”