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Proceedings of the 12th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2012 July, 2-5, 2012.

On Optimal Allocation of Redundant Components for Systems of Dependent Components

F´elix Belzunce Torregrosa1, Helena Mart´ınez Puertas2 and Jos´e Mar´ıa

Ru´ız G´omez1

1 Department of Statistics and Operations Research, University of Murcia (Spain).

2 Department of Statistics and Applied Mathematics, University of Almer´ıa (Spain). emails: belzunce@um.es, hmartinez@ual.es, jmruizgo@um.es

Abstract

We consider the problem of optimal allocation of a redundant component for series, parallel and k- out of- n systems of more than two components, when all the components are dependent. We show that for this problem is naturally to consider multivariates extensions of the joint bivariates stochastic orders. However, these extensions have not been defined or explicitly studied in the literature, except the joint likelihood ratio order, which was introduced by Shanthikumar and Yao (1991).

Key words: Standby and active redundancy, multivariate extensions of the joint stochastic order.

MSC 2000: 60E15, 60K10.

1Introduction

The problem of where to allocate a redundant component in order to increase the reliability of a system is one of the important problems in reliability. Two types of redundancy, active and standby, are the most common types of redundancy. An active redundant is put in parallel with a component at the same time and a standby redundant component is put to use upon the failure of the original component. As an example consider a series system of two components with random lifetimes T 1 and T 2 and let us consider an additional component, with random lifetime S, in parallel redundancy with any of the two components. Then we have two systems U1 = min{max(T1, S), T2} and U2 = min{T1, max(T2, S)},

On Optimal Allocation of Redundant Components for Systems of Dependent Components.

and the problem is which of the two system has a larger lifetime in some probabilistic sense.

This problem has been studied along the 90’s, and more recently in Vald´es and Zequeira (2003), Romera, Vald´es and Zequeira (2004) and Vald´es and Zequeira (2006). All these papers consider the case where the components are independent. In the case of dependent components not too much work has been done, and as far as we know, only the papers by Kotz, Lai and Xie (2003), da Costa Buneo (2005) and da Costa Bueno and do Carmo (2007) deals with this problem.

The purpose is to consider this idea when all the components are dependent. Thus, let us consider a system with n > 2 components with random lifetimes T1, T2, ..., Tn and an additional component, with random lifetime S, that can be put in parallel or standby redundancy with any of the n components. Then, we can consider n diferent systems and the problem is which of the n system has a larger lifetime in some probabilistic sense. First, we discuss the case of series and parallel systems and later we extend the results to the case of k-out-of-n systems. For obtaining the results, previously has been necessary to consider multivariates extensions of the joint bivariates orders introduced by Shanthikumar and Yao (1991) and Shanthikumar, Yamazaki and Sakasegawa (1991).

F.Belzunce, H. Mart´ınez and J.M. Ru´ız.

Acknowledgements

Supported by Ministerio de Educaci´on y Ciencia under Grant MTM2009-08311 and by Fundaci´on Sen´eca (CARM 08811/PI/08).

References

[1]Boland, P.J., El-Neweihi, E. and Proschan, F. (1992). Stochastic order for redundancy allocations in series and parallel systems. Advances in Applied Probability. 24, 161–171.

[2]El-Neweihi, E. and Sethuraman, J. (1993). Optimal allocation under partial ordering of lifetimes of components. Journal of Applied Probability. 25, 914–925.

[3]Mi, J. (1999). Optimal active redundancy allocation in k-out-of-n system. Journal of Applied Probability. 36, 927–933.

[4]Romera, R., Vald´es, J.E. and Zequeira, R.I. (2004). Active-redundancy allocations in systems. IEEE Transactions on Reliability. 53, 313–318.

[5]Shanthikumar, J.G., Yamazaki, G. and Sakasegawa, H. (1991). Characterization of optimal order of servers in a tandem queue with blocking. Operations Research Letters. 10, 17–22.

[6]Shanthikumar, J.G. and Yao, D.D. (1991). Bivariate characterization of some stochastic order relations. Advances in Applied Probability . 23, 642–659.

[7]Singh, H. and Misra, N. (1994). On redundancy allocations in systems. Journal of Applied Probability. 31, 1004–1014.

[8]Singh, H. and Singh, R.S. (1997). Note: Optimal allocation of resources to nodes of series systems with respect to failure rate ordering. Naval Research Logistics. 44, 147–152.

[9]Vald´es, J.E. and Zequeira, R.I. (2003). On the optimal allocation of an active redundancy in a two-component series system. Statistics and Probability Letters. 63, 325–332.

On Optimal Allocation of Redundant Components for Systems of Dependent Components.

[10]Vald´es, J.E. and Zequeira, R.I. (2006). On the optimal allocation of two active redundancies in a two-component series system. Operations Research Letters. 34, 49–52.

Proceedings of the 12th International Conference on Computational and Mathematical Methods

in Science and Engineering, CMMSE 2012 July, 2-5, 2012.

Fixed point techniques and Schauder bases to approximate the solution of the nonlinear Fredholm–Volterra integro–di erential equation

M. I. Berenguer1, D. G´amez1 and A. J. L´opez Linares1

1 Department of Applied Mathematics, University of Granada

emails: maribel@ugr.es, domingo@ugr.es, alopezl@ugr.es

Abstract

With the aid of fixed–point theorem and biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a nonlinear Fredholm–Volterra integro–di erential equation is turned into a numerical algorithm, so that it can be solved numerically.

Key words: Biorthogonal systems, fixed–point, nonlinear Volterra–Fredholm integrodi erential equation, numerical methods.

MSC 2000: AMS codes 65R05, 45J05, 45L05, 45N05.

1Preliminaries

Denoting by C([0, 1] × R) and C([0, 1]2 × R) the Banach space of all continuous and real– valued functions defined on [0, 1] × R and [0, 1]2 × R respectively, equipped with their usual sup–sup norm, let us consider the following problem associated to the Fredholm–Volterra integro–di erential equation: given ρ R, k1, k2 C([0, 1]2 × R), and f C([0, 1] × R), find x C1([0, 1]) such that

Frequently the mathematical modelling of problems arising from de real world (see [17] and the references there are in) deal with problem (1). These are usually di cult

Fixed point techniques and Schauder bases to approximate the solution. . .

to solve analytically and in many cases the solution must be approximated (see [9], [10], [14]-[16]). The use of fixed–point techniques in the numerical study of linear and nonlinear di erential, integral and integro–di erential equations has also proven successful in some works, as [2]–[8], [11] and [12]. The purpose of this work is to develop an e ective method for approximating the solution of (1) using Schauder basis and another classical tool in Analysis: the fixed–point theorem. This algorithm generalizes the developed ones in [2], [6], [3] and [12] for linear Fredholm–Volterra integro–di erential, nonlinear Volterra integro–di erential, nonlinear Fredholm integro–di erential and nonlinear di erential equation respectively.

To establish our numerical method, we first need to review some results of a theoretical nature in section 2. We arrive at a numerical method for approximating the solution of (1) in section 3, and in order to state the results about convergence and to study the error of the proposed algorithm, we will assume that k1, k2 and f satisfying a Lipschitz condition with respect to the last variables: there exist Lf , Lk1 , Lk2 ≥ 0 such that

|f (t, y) − f (t, z)| ≤ Lf |y − z|

|k1(t, s, y) − k1(t, s, z)| ≤ Lk1 |y − z| |k2(t, s, y) − k2(t, s, z)| ≤ Lk2 |y − z|

2 Two tools of a theoretical nature.

Two fundamental tools will be used to establish the algorithm needed to solve the problem

(1). The first is the Banach fixed–point theorem (see [1]):

Banach fixed–point theorem. Let (X, · ) be a Banach space, let F : X −→ X and

unique fixed point x X. Moreover, if x is an element in X, then we have that for all

n ≥ 1,

∞

F nx − x ≤ ( γi) F x − x .

i=n

In particular, x = limn F n(x).

The second tool applied consists of biorthogonal systems in Banach spaces C([0, 1]) and C([0, 1]2) (see [13] and [18]). We will make use of the usual Schauder basis for simplicity in the exposition, although the numerical method given works equally well by replacing it with any complete biorthogonal system in C([0, 1]2). Let us consider the usual Schauder

M. I. Berenguer, D. Gamez,´ A. J. Lopez´ Linares

bases {bn}n≥1 in the space C([0, 1]) and {Bn}n≥1 in C([0, 1]2) and {Pn}n≥1 and {Qn}n≥1 the sequences of (continuous and linear) projections in C([0, 1]) and C([0, 1]2) respectively. Let it be {bn}n≥1 and {Bn}n≥1 the associated sequence of (continuous and linear) coordinate functionals in C([0, 1]) and C([0, 1]2) respectively.

3The numerical method. Study of convergence and error.

Our starting point is the formulation of (1) in terms of a certain operator L as follows: let T : C([0, 1]) −→ C([0, 1]) be the linear and continuous operator defined by

t t 1 t u

T x(t) := ρ + f (u, x(u)) du + k1(u, s, x(s)) ds du + k2(u, s, x(s)) ds du

0 0 0 0 0

(0 ≤ t ≤ 1, x C([0, 1])).

It is a simple matter to check that a function x C1([0, 1]) is the solution of (1) if and only if x is a fixed point of the operator T .

The condition assumed on M and the Banach fixed point theorem allow us to establish the existence of one and only one solution x of (1), which is

x = lim T m(x).

m

On the other hand, using Schauder’s bases introduced in the section 2, let us consider the functions ϕ C([0, 1]) and φ1, φ2 C([0, 1]2), defined respectively by

ϕ(t) = f (t, x(t)),

φ1(t, s) = k1(t, s, x(s)),

k=1

and

μ1 = φ1(t1, t1), ν1 = φ2(t1, t1)

Fixed point techniques and Schauder bases to approximate the solution. . .

and for n ≥ 2,

and for a real number p, [p] will denote its integer part.

We can then calculate iteratively using (2), at least in a theoretical way, the solution of (1). From a practical viewpoint, in general these calculations are not possible explicitly. The idea of our numerical method is to use an appropriates Schauder basis in the spaces C([0, 1]), C([0, 1]2) truncating the functions of such spaces by means of the projections of the Schauder bases. Specifically, in view of (2), we consider the sequence {xr }r≥0 defined as follows:

Let x0(t) := x(t) C1([0, 1]) and m N. Define inductively, for r {1, . . . , m} and 0 ≤ t, s ≤ 1 the functions:

ϕr−1(t) := f (t, xr−1(t))

σr−1(t, s) := k1(t, s, xr−1(s))

ψr−1(t, s) := k2(t, s, xr−1(s))

where nr N with nr ≥ 2.

The following result show that the sequence {xr } approximates the solution of (1) and

give the error. For this, let us assume that k1, k2 C1([0, 1]2 × R), f C1([0, 1] × R) such

that for each i {1, 2}, ki, ∂k∂ti , ∂k∂si , ∂k∂xi , f, ∂f∂t , ∂f∂x satisfy a global Lipschitz condition in the last variable.

Theorem. Let m N, nr N, nr ≥ 2 and, {ε1, . . . , εm} a set of positive numbers such

that for all r {1, . . . , m}

εr

Tnr ≤ 2Λr−1(β + 3β2)

T xr−1 − xr ≤ εr .

Moreover, if x is the exact solution of the integro–di erential equation (1), then the errorx − xm is given by

4A numerical example

We consider the following Fredholm-Volterra integro-di erential equation with the exact solution x(t) = t2. Its numerical results are given in following table.

and 33, the absolute errors committed for certain representative points of [0, 1] when we approximate the exact solution x(t) by means of the iteration xr (t).

Table. Absolute errors for the example

Fixed point techniques and Schauder bases to approximate the solution. . .

Acknowledgements

Research partially supported by Junta de Andaluc´ıa Grant FQM359 and the E.T.S.I.E. of the University of Granada.

References

[1]K. Atkinson and W. Han, Theoretical Numerical Analysis, 2nd Ed., Springer-Verlag, New York, 2005.

[2]M. I. Berenguer, D. G´amez, A. J. L´opez Linares, Fixed point iterative algorithm for the linear Fredholm-Volterra integro-di erential equation, Journal of Applied Mathematics (article in press).

[3]M.I. Berenguer, M.V. Fern´andez Mu˜noz, A.I. Garralda-Guillem and M. Ruiz Galn, A sequential approach for solving the Fredholm integro-di erential equation, App. Numer. Math. 62 (2012), pp. 297-304.

[4]M. I. Berenguer, A. I. Garralda-Guillem, M. Ruiz Gal´an, An approximation method for solving systems of Volterra integro-di erential equations, Appl. Numer. Math. (2011), doi: 10.1016/j.apnum.2011.03.007 (article in press).

[5]M.I. Berenguer, D. G´amez, A.I. Garralda Guillem, M. Ruiz Gal´an and M. C. Serrano P´erez, Biorthogonal systems for solving Volterra integral equation system of the second kind, J. Comput. Appl. Math. 235 (2011), pp. 1875–1883.

[6]M.I. Berenguer, A.I. Garralda Guillem and M. Ruiz Gal´an, Biorthogonal systems approximating the solution of the nonlinear Volterra integro–di erential equation, Fixed Point Theory A. 2010 (2010), doi:10.1155/2010/470149. Article ID 470149, 9 pages.

[7]M.I. Berenguer, D. G´amez, A.I. Garralda Guillem, and M. C. Serrano P´erez, Nonlinear Volterra integral equation of the second kind and biorthogonal systems, Abstr. Appl. Anal. 2010 (2010) doi:10.1155/2010/135216. Article ID 135216, 11 pages.

[8]M.I. Berenguer, M. V. Fern´andez Mu˜noz, A.I. Garralda Guillem and M. Ruiz Gal´an,

Numerical treatment of fixed point applied to the nonlinear Fredholm integral equation, Fixed Point Theory A. 2009 (2009), doi:10.1155/2009/735638. Article ID 735638, 8 pages.

[9]M. Dehghan and R. Salehi, The numerical solution of the non–linear integro–di erential equations based on the meshless method, J. Comput. Appl. Math. 236 (2012), pp. 2367– 2377.