Discrete random bounds for general random variables and applications to reliability

doi:10.1016/j.ejor.2005.09.043

European Journal of Operational Research

Volume 177, Issue 1, 16 February 2007, Pages 378-405

https://doi.org/10.1016/j.ejor.2005.09.043 Get rights and content

Abstract

We here propose some new algorithms to compute bounds for (1) cumulative density functions of sums of i.i.d. nonnegative random variables, (2) renewal functions and (3) cumulative density functions of geometric sums of i.i.d. nonnegative random variables. The idea is very basic and consists in bounding any general nonnegative random variable X by two discrete random variables with range in $h N$ , which both converge to X as h goes to 0. Numerical experiments are lead on and the results given by the different algorithms are compared to theoretical results in case of i.i.d. exponentially distributed random variables and to other numerical methods in other cases.

Introduction

We are interested here in bounding different quantities from the reliability field:

First, we propose bounds for the cumulative density function (c.d.f.) of the sum of m independent nonnegative random variables with general distributions (nonnecessarily identical). Such a quantity represents for instance the unreliability of a m-units cold-standby redundant system. Indeed, assuming all components to be independent and unrepairable with respective times to failure U₁, U₂, … , U_m, the time to failure of the whole system then is U₁ + U₂ + ⋯ + U_m and its unreliability at time t is $P (U_{1} + U_{2} + \dots + U_{m} ⩽ t)$ , namely the c.d.f. of U₁ + U₂ + ⋯ + U_m. Such quantities are also encountered in some reliability softwares for the evaluation of the reliability of large Markov systems (as in the software Figseq for the French society of electricity supply EDF for instance). Indeed, such an evaluation may be done by exploration of the different possible sequences, retaining only those which are beyond some probability threshold (see [6] or [20] for instance). Once such sequences selected, the problem of reliability evaluation then reduces to the computation of quantities like $P (U_{1} + \dots + U_{m} ⩽ t)$ where U₁, … , U_m are independent and exponentially distributed. More generally, in case all the U_i’s admit a probability density function (p.d.f.) f_i towards Lebesgue measure, a p.d.f. for U₁ + U₂ + ⋯ + U_m then is f₁∗f₂∗ ⋯ ∗f_m (where ∗ stands for convolution) and the numerical computation of $P (U_{1} + U_{2} + \dots + U_{m} ⩽ t)$ requires m successive integrations. According to the probability distributions of the U_i’s (e.g. very disparate values for the different rates in the exponential case, see [20] e.g.), the numerical evaluation of such quantities sometimes leads to different numerical errors, which are not always that easy to bound.

The second quantity we are interested in, is the renewal function associated to some eventually delayed renewal process, where we recall that the renewal function represents the mean number of renewals for the process on [0, t]. Such renewal functions typically arise in reliability theory to evaluate the mean number of replacements on [0, t] for some unit submitted to replacement in case of failure, or to other replacement strategies (see [1] e.g.). They also arise naturally in queueing theory, inventory control and networks (see [18] e.g.). Such renewal functions generally have no closed form and their numerical evaluation has made the object of numerous previous papers (see [7], [9], [14], [15], [21], [25] or [12] with references therein e.g.).

Finally, the third quantity we are interested in, is the cumulative distribution function (c.d.f.) of some geometric sum of independent and identically distributed (i.i.d.) random variables (r.v.). Such geometric sums are used in reliability theory, but also in risk analysis, queueing theory,… (see [3] or [19] e.g.). Similarly as for renewal functions, such c.d.f. of geometric sums generally have no closed form and accurateness of their numerical approximation often is difficult to evaluate.

For the three quantities described above, namely c.d.f. of sums, renewal functions and c.d.f. of geometric sums, we propose a very simple method for their bounding: for each U_i involved in such quantities, we construct some discrete random approximation $U_{i}^{h}$ and $U_{i}^{h +}$ (where h > 0) bounding U_i in the sense that $U_{i}^{h} ⩽ U_{i} < U_{i}^{h +} .$ The range of both $U_{i}^{h}$ and $U_{i}^{h +}$ is included in $h Z$ (or $h N$ in case U_i is nonnegative) and both $U_{i}^{h}$ and $U_{i}^{h +}$ converge everywhere towards U_i when h → 0⁺ (see Section 2 for details). Such bounds for each U_i easily provides us with bounds for the three quantities to evaluate, which are similar quantities as the initial ones, with U_i substituted by $U_{i}^{h}$ or $U_{i}^{h +}$ . The problem then reduces to the computation of similar quantities where all r.v. now have the same discrete support $h N$ , instead of general r.v. with general support. For each of the three quantities we are interested in, algorithms very easy to implement are given for their computation. Beside, all bounds are shown to converge towards the right quantity when h goes to zero so that bounds can be made as tight as wanted.

The idea of approximating continuous quantities by discrete ones of course is not new, especially because implementation of formulae often is simpler with discrete quantities than with continuous ones. However, contrary to what is usually done, we do not approximate directly the real quantities to evaluate but the different random variables which appear in such quantities. This immediately provides us with bounds expressed in terms of the discrete random bounds. The only real point then is to compute such bounds.

The paper is organised as follows: In Section 2, we introduce our discrete random bounds for some general random variable. Sections 3 C.d.f. of sums of random variables, 4 Renewal functions, 5 C.d.f. of geometric sums are respectively devoted to c.d.f. of sums, renewal functions and c.d.f. of geometric sums. In each of these section, theoretical results such as convergence of the bounds are proved. Then, numerical algorithms are provided and tested: first, in case of i.i.d. exponentially distributed random variables, where theoretical results are available; secondly, in other cases, where the results are compared to those by other numerical methods. We conclude in Section 6.

Section snippets

Discrete random bounds for general random variables

Let symbol ⌊⋯⌋ stands for the floor function. We recall that, by definition $(⌊ y ⌋ = n) \Leftrightarrow (n ⩽ y < n + 1)$ for all $y \in R$ , $n \in Z$ . For all $x \in R$ and h > 0, we then have $⌊\frac{x}{h}⌋ ⩽ \frac{x}{h} < ⌊\frac{x}{h}⌋ + 1 .$ Now, let $x^{h} = h ⌊ \frac{x}{h} ⌋$ and $x^{h +} = h (⌊ \frac{x}{h} ⌋ + 1) = x^{h} + h$ . We get $x^{h} ⩽ x < x^{h +} = x^{h} + h$ and $x - h < x^{h} ⩽ x .$ We derive that, for all $x \in R$ $\lim_{h \to 0^{+}} x^{h} = \lim_{h \to 0^{+}} x^{h +} = x .$ Now, let U be a random variable with general distribution. For any h > 0, we set $U^{h} = h ⌊\frac{U}{h}⌋$ and $U^{h +} = h ⌊\frac{U}{h}⌋ + h = U^{h} + h$ (such notations are used in all the following).

Both of U^h and U^h+ are discrete random variables with support in $h Z = {hk : k \in Z$

Theoretical results

Let U₁, U₂, … , U_m be random variables with general distributions (m ⩾ 2), not necessarily independent and t > 0. We are interested here in bounding $P (U_{1} + \dots + U_{m} ⩽ t)$ . Using the discrete random bounds introduced in the previous section we get

Proposition 2

Let U₁, U₂, … , U_m be random variables with general distributions. Then $P (U_{1} + U_{2}^{h +} + \dots + U_{m}^{h +} ⩽ t) = P (U_{1} + U_{2}^{h} + \dots + U_{m}^{h} ⩽ t - (m - 1) h) ⩽ P (U_{1} + \dots + U_{m} ⩽ t) ⩽ P (U_{1} + U_{2}^{h} + \dots + U_{m}^{h} ⩽ t)$ for all $t \in R$ , all h > 0, where $U_{i}^{h}$ and $U_{i}^{h +}$ are the discrete approximations of U_i given by (4), (5), where U_i is substituted to U.

Theoretical results

We consider a renewal process with an eventual delay V₀: the first arrival time is V₀ and the other inter-arrival times are U₁, … , U_m, … where V₀ and the U_i’s are assumed to be nonnegative and independent, all the U_i’s with the same distribution. We are interested here in bounding the associated renewal function, namely the function $R_{V_{0}, U_{1}}$ such that $R_{V_{0}, U_{1}} (t) = P (V_{0} ⩽ t) + \sum_{n ⩾ 1} P (V_{0} + U_{1} + \dots + U_{n} ⩽ t) = E (1_{{V_{0} ⩽ t}} + \sum_{n ⩾ 1} 1_{{V_{0} + U_{1} + \dots + U_{n} ⩽ t}})$ for all t ⩾ 0. We recall that $R_{V_{0}, U_{1}} (t)$ represents the mean number of renewals on [0, t].

Theoretical results

We finally consider geometric sums of nonnegative i.i.d. random variables with an eventual “delay”: let V₀, U₁, … , U_m, … be nonnegative independent random variables with U₁, … , U_m, … identically distributed and G a random variable independent from V₀ and the U_i’s, with geometric distribution G(p) (0 < p < 1) $P (G = n) = q^{n - 1} p$ for all $n \in N^{*}$ , with q = 1−p. We are interested here in bounding the c.d.f. of $V_{0} + 1_{{G > 1}} \sum_{i = 1}^{G} U_{i}$ , namely the function $F_{G, V_{0}, U_{1}}$ such that $F_{G, V_{0}, U_{1}} (t) = P (V_{0} + 1_{{G ⩾ 2}} \sum_{i = 1}^{G - 1} U_{i} ⩽ t)$ for all $t \in R$ . Note that in

Conclusion

As a conclusion, we have presented in this paper some new algorithms to bound different quantities useful in reliability theory (and also in other fields, see Section 1). The algorithms are very simple to implement. The results have been tested and compared to theoretical results in case of i.i.d. exponentially distributed r.v.. The results have also been compared to those by other methods and, in each case, we have seen that the algorithms presented here could stand for such comparison, even

Acknowledgements

The author thanks an anonymous referee for suggesting interesting comparisons and Prof. M. Tortorella for allowing her to use the material from [26].

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Stochastics and StatisticsDiscrete random bounds for general random variables and applications to reliability

Abstract

Introduction

Section snippets

Discrete random bounds for general random variables

Theoretical results

Theoretical results

Theoretical results

Conclusion

Acknowledgements

Statistical Theory of Reliability and Life Testing

An approximate solution to the integral equation of renewal theory

Journal of the Royal Statistical Society B

Some estimates of geometric sums

Statistics & Probability Letters

Numerical comparisons of approximations of geometric sums

Applied Stochastic Models in Business and Industry

Bounds, inequalities, and monotonicity properties for some specialized renewal processes

Annals of Probability

An algorithm in order to implement reliability exponential approximations

Reliability Engineering & System Safety

The Weibull renewal function for moderate to large arguments

Computational Statistics & Data Analysis

Some normal approximations for renewal function of large Weibull shape parameter

Communications in Statistics—Simulation and Computation

An approximate solution of the integral equation of renewal theory

Journal of Applied Probability

Solution of the Volterra equation of renewal theory with the Galerkin technique using cubic splines

Journal of the Statistical Computation and Simulation

Renewal processes and their computational aspects

Algorithms for computing the distributions of sums of discrete random variables

Mathematical and Computer Modelling

Stochastics and Statistics
Discrete random bounds for general random variables and applications to reliability