Reliability Equivalence Factors for Some Systems with Mixture Weibull Failure Rates

In this article, the failure rates of the system's components are functions of time t. We study two cases (i) the mixture of two stages of life time distribution with Weibull failure rates, (ii) the mixture of two stages failure rates with Weibull distribution. The reliability equivalence factors of some systems with identical components are obtained. Two different methods are used to improve the given systems. Numerical examples are presented to interpret how one can utilize the obtained results. Some special cases are obtained from our results.


Introduction
In reliability analysis, sometimes different system designs should be compared based on a reliability characteristic such as the reliability function or mean time to failure in case of no repair. The concept of reliability equivalence factors has been introduced by [10]. The reliability equivalence factors for a single component and for two independent and identical component series and parallel systems calculated in [10,11], the author assumed that the reliability function of the system can be improved by three different methods as: (1) improving the quality of one or several components by decreasing their failure rates; (2) adding some hot redundant components to the system; and (3) adding some cold redundant component to the system.The survival function has used as the performance measure of the reliability system see [10,11]. In [12], the reliability equivalence factors of n independent and non-identical components series system obtained, the author used the survival function and mean time to failure as characteristics to compare different system designs [5,9,13 -18] have applied the concept of reliability equivalence on a parallel (series) and series-parallel (parallel-series) systems with independent and identical (non-identical) components and [6] has studied the simple system of two non-identical components when the system improved by add only one component to the system by using the improving techniques.
The previous articles in reliability equivalence techniqueassumed that the system components have one type of constant failure rate [8], assumed a system components have three types of constant failure rates and made a mixture of these types; see [2,3].
All articles mentioned above are about components of exponential distribution. But [19] applied the reliability equivalence factor of a parallel system with n independent and identical components with Gamma life time distribution. Gamma distribution has a failure rate of function of time [7] applied the concept of reliability equivalence factor on some system with linear increasing failure rates. We hope to discuss more life time distributions, failure rates and apply the mixture approach on them to obtain the general systems.
To derive the reliability equivalence factors of a system, we use need the following definition Definition 1. ( [13])A reliability equivalence factor is a factor by which a characteristic of components of a system design has to multiplied in order to reach equality of a characteristic of this design a different design.
The reliability function and mean time to failure will be used as characteristics of the system performance. In this case the reliability equivalence will be referred as survival reliability equivalence factor, Shortly SREF, and mean reliability equivalence factor, shortly MREF, respectively.
In the current study, we shall calculate the SREF and MREF for some systems, consisting of independent and identical components. These components are assumed to be having two stages of failure rates or failure life times. The reliability of the system can be improved according to one of the following different methods: (1) Reducing the failure rates of some of the system components. This method will be referring by the reduction method.
(2) Assuming hot duplication of some of the system components. This means that each component is duplicated by a hot redundant standby component. This method will be called the hot duplication method.
(3) Assuming cold duplication of some of the system components. This means that each component is duplicated by a cold redundant standby component connected with perfect (imperfect) switch. This method will be called the cold (imperfect switch) duplication method.
The methods of cold and imperfect switch duplication contain some problems in the integrations and Maple program cannot compute it. So we shall improve the study systems by using only two methods.
This paper is organized as follows. Section 2 presents the one component system with mixture of Weibull life time distributions. Section 3 gives n-components series system with mixture of two stages of Weibull failure rates. Special cases of our works are introduced.

Mixture of Life Time Distributions
We consider a system whose components fail if they enter either of two stages of failure mechanisms. The first mechanism is due to excessive voltage, and the second is due to excessive temperature. Suppose that the failure mechanism enters either the first stage with probability θ 1 or the second stage with probability θ 2 . Let the probability density function (p.d.f.) of the first stage is f 1 (t) and the p.d.f. of the second stage is f 2 (t). Hence the failure of a component occurs at the end of either the first or the second stage. Therefore, the p.d.f. of the failure time for a component is The reliability function of the component is The hazard (failure) rate function of the component is Where θ 1 + θ 2 =1.See [2].

The Original System
In this section, we consider a simple system consists ofone component has two stages with Weibull failure rates as where α i , β i > 0, t ≥ 0, i =1, 2. See [1] and [2]. The life time of the stage i, has the Weibull distribution, from equation (2), the reliability function is given as follows 1 2 Let MTTF be the system mean time to failure, which is given by

Reduction Method
In this method, we can reducing the failure rate of the stage i by the factor i ρ , 0 be the reliability function of the improved system when we reduce the failure rate of the stage i by the factor ρ i . One can obtain the function ) (t R ρ as follows From equation (7), the mean time to failure of the improved system say MTTF ρ becomes be the reliability function of the improved system assuming hot duplication. The function (5), one can easy find ) (t R H , see [1]. Let H F MTT denotes the mean time to failure of the system improved in this case. Using equation (9), that can be obtained by

γ-Fractiles
This section presents the γ -fractiles of the original and improved systems. Let L(γ) be the γ -fractiles of the original system and ) (γ H L , the γ -fractiles of the improved system assuming hot duplication method.
The γ -fractilesL(γ) and ) (γ H L are defined as the solution of the two following equations, respectively, It follows from equation (5) and the first equation of (11) that L= L(γ) satisfies the following equation From the second equation of (11), and equation (9), one can verify that ) (γ H L L = satisfies the following equation Equations (12) and (13) have no closed form solution and can be solved using numerical method technique.

Reliability Equivalence Factors
Now we are ready to derive the reliability equivalence factors of the system. We will deduce the survival reliability equivalence factor, say SREF and mean reliability equivalence factor, say MREF of the underlying system as follows.
The first type of reliability equivalence factor, SREF say ) (γ ρ H , can be obtained by equating the reliability function of the improved system that obtained by improving the system according to reduction method with the reliability function of the system improved by improving the system according to hot duplication method at the level γ. Hence from equations (7) and (9) Let us now explain how one can deduce the second type of reliability equivalence factor. This type is MREF say H ξ that can be obtained by equating the mean time of the improved system that obtained by improving the system according to reduction method with the mean time to failure of the system improved by improving the system according to hot duplication method. It means that, H ξ can be derived from equations (8) and (10) as follows.
The mean time to failure of the original system is MTTF=3.95857and MTTF H =4.98433, then MTTF<MTTF H .
The γ-fractiles, L(γ), L H (γ) and the values of ρ H (γ) are calculated using Mathematica Program system according to the previous theoretical formulae. In these calculations the level γ is chosen to be 0.1, 0.2, …, 0.9.
The γ-fractile and the survival reliability equivalence factor are given in table 1 at some values of γ.
Based on the results presented in table 1: 1. One can recognize that L(γ) < L H (γ) for all studied cases, which confirms the results obtained for MTTF. 3. In the same manner one can read the rest results. 4. The notation NA in table 1 means that the value of SREF is not available and therefore there is possible equivalence between the system improved by reduction method and that system improved by using the redundancy method at this level.
The mean reliability equivalence factors are 1 .5771, then one can conclude that, the improved system that can be obtained by improving the system according to hot duplication method, has the same mean time to failure of that system which can be obtained by doing one of the following: (i) Reducing the failure rate of the first stage by the factor ξ =0.4360, (ii) reducing the failure rate of the second stage by the factor ξ =0.2836, (iii) reducing the failure rate of the first and second stage by the factor ξ =0.5771.

Special Cases
We can calculate the reliability equivalence factors for the special cases from the present system as follows 1. if β i =0, equation (4), can be reduced to In this case, the stages have the Exponential distribution with parameter 1 i α , see [3].
2. if β i =2, equation (4), can be reduced to In this case, the stages have the Rayleigh distribution with parameter 2 2 i α ,see [4].
Therefore, the reliability equivalence factor for these systems can be obtained as special cases from the studied system in this section.

Mixture of Failure Rates
In this section, we consider a system whose components fail if they enter either of two stages of failure mechanisms. The first mechanism is due to excessive voltage, and the second is due to excessive temperature. Suppose that the failure mechanism enters the first stage with probability θ 1 , and the failure rate is of the failure time is h 1 (t). It enters the second stage with probability θ 2 and the failure rate of its failure time h 2 (t). The failure of a component occurs at the end of either first stage or the second stage. Hence the failure rate of the component i is The reliability function of the component i is given as follows be the cumulative failure rate of the stage i, i=1, 2.See [2]. In this section, we consider a simple system with n-identical components in series system. Each component has two stages of the hazard (failure) rate functions.

The Original System
In this section, we consider the stages for component i, have theWeibull failure rates as follows and the reliability function for the component i becomes Therefore, the reliability function of n-independent and identical components in series system is given as Let MTTF be the system mean time to failure, which is given by

Reduction Method
In this method, we can reduce the mixture failure rate by reducing the stages of the failure rates by the factor ρ i , 0<ρ i <1, i=1, 2. Let ) ( , t R r ρ be the reliability function of the system improved when reducing the failure rates of r components. One can obtain the function as follows From equation (24), the mean time to failure of the improved system, say MTTF ρ, r, becomes The mean time to failure in (23, 25, 27) can be calculated numerically.

The γ-Fractiles
In this section, we present the γ-fractiles of the original and improved systems. Let L(γ), be the γ-fractiles of the original and improved system assuming hot duplication method. The γ -fractilesL(γ) and ) (γ Equations (29, 30) have no closed form solution and can be solved using numerical method technique.

Reliability Equivalence Factors
Now we are ready to derive the reliability equivalence factors of the system. We will deduce survival reliability equivalence factor, say SREF and mean reliability equivalence factor, say MREF of the underlying system as follows. The

Numerical Results
To explain how one can apply theoretical results obtained in the previous subsections, we introduce a numerical example. In this example, we assume three components series system with identical components, α 1 =4, β 1 =3, α 2 =3, β 2 =4, θ 1 =0.45 and θ 2 =0.55, that the failure for the stages are   In these calculations the level γ is chosen to be 0.1, 0.2, …, 0.9.
The γ-fractile of the original and improved system are given in table 3 at some values of γ.
Based on the results presented in  The values of the SREF are given in (i) table 4, when we reduce the second stage by the factor ρ 2 , (ii) table 5, when the first and second stages by the same factor ρ, at some values of γ.
3. The notation NA in tables 4 and 5 means that the value of SREF is not available and therefore there is possible equivalence between the system improved by HDM and that system improved by using the redundancy method at this level. Table 6 introduces the values of the mean reliability equivalence factor, when we reduce the second stage by H 2 ξ and the two stages by the same factor H ξ .