Improving the Performance of a Series-Parallel System Based on Gamma Distribution

. The performance of a series-parallel system is improved by using the reliability equivalence factors technique. The lifetimes of the components are assumed to be gamma distributed. The system reliability is improved by using three di ﬀ erent methods: (i) Reduction method, (ii) Hot duplication method, (iii) Cold duplication method. The reliability function and mean time to failure for each method are derived. Finally, the numerical application is introduced.


Introduction
The concept of reliability equivalence factors (REF) is addressed to improve the system reliability, [14].Sarhan [17,18] improved the system reliability by: (1) Reducing the failure rates by a factor ξ, 0 < ξ < 1, is called reduction method (RM).
(2) Duplicating the system's components by hot redundant identical standby components.This method is named hot duplication method (HDM).
(3) The system's components are connected with an identical component via a perfect switch, so it is called the cold duplication method (CDM).
(4) Connecting some system's components with standby redundant component via an imperfect switch.It is called the imperfect duplication method (IDM).
The series-parallel system is one of the important systems in reliability theory and has many applications in engineering sciences.This system has many special cases such as: serial, parallel and radar system.Therefore, studying and improving the performance of this system includes improving all these systems at the same time.The series-parallel system has been studied for several lifetime distributions, such as (i) exponential, (ii) linear exponential and (iii) modified Weibull distribution, see [1,3,6,13,19,21,22,24].
The gamma distribution (GD) with parameters β, ν, has the following probability density function.
The parameters β, ν are called a shape and scale parameter, respectively.The GD has been employed in engineering to study the system reliability.
The GD has some special distributions, for some values of β and ν, as follows.
(1) The exponential distribution with constant failure rate ν can be obtained if β = 1.
(3) The GD is called an Erlang distribution, when β is an integer.
Erlang distribution is used in queuing theory to model waiting times.χ 2 distribution is used in statistical inference.
Figure 1.The h(t) for some values of β.
The rest of the paper can be organized as follows.
A brief description of the original system is introduced in Section 2. The improved systems are discussed in Section 3. The α-fractiles are obtained in Section 4. In Section 5, the reliability equivalence factors are derived.A numerical application is discussed in Section 6.The conclusion of the paper is presented in Section 7.

The Original System
A series-parallel system has m subsystems connected in series.Each of them contains n i elements connected in parallel, i = 1, 2, • • • , m, see Figure 2. Form Figure 2, if m = 1, then we will get the parallel system, while if n i = 1, then the series system will be obtained, but if m = 2 and n i = i, i = 1, 2, we will have the radar system.Consider the lifetime of the system components is independent gamma distribution.The reliability where see [23].
The RF of the subsystem i, is given as, Therefore, the RF of the original system is derived as 3) The mean time to failure (MTTF) is Some numerical techniques can be used to calculate MTTF numerically, from Equation (2.4), for given values of β, ν, n i and m.

The Improved Systems
Three different methods will be applied to improve the performance of the original system.
3.1.The RM.The failure rate, h(t), of the components of set A are reduced to r(t)h(t), 0 < r(t) < 1, where |A| = , 0 ≤ ≤ M, and M = m i=1 n i .We shall reduce h(t) by reducing the scale parameter only by the factor ξ.
where A i denotes the set of components from subsystem i whose failure rates are reduced and |A i | = i and = m i=1 i .We denote such a set by A After reducing the failure rate of component j, it has the following RF, R i j,ξ (t), The RF, R A i ,ξ (t), of the subsystem i after reducing the failure rates of A i is given by The RF of the improved system by RM, The MTTF of the reduction system, is calculated by and the RF of the improved system according to HDM, The MTTF of the improved system by HDM, from Equation (3.6), M H B , is 3.3.The CDM.In this method, the set B of system components are duplicated each one with an identical component by a perfect switch, |B| = r, 0 ≤ r ≤ M. The set B can be expressed as where Using (3.8) and (3.9), then The MTTF, M C B , is calculated by

The α-Fractiles
Let L(β, α), L D B (β, α), be the α-fractiles of the original and duplicated systems.Which can be calculated by solving the following equations, respectively.

The Reliability Equivalence Factors
Since the failure rate of GD(β, ν) is non-constant, the REFs of GD will be a function of time t.
Definition [17]: The REF is defined as the factor that must be used to reduce the failure rates of the set, A, of system components in order to obtain the reliability of the system, which is improved by improving the set, B, of system components by the duplication method.
The failure rate, h(t), of GD will be reduced by r(t), only by reducing the scale parameter from ν to ξν only.
e −ξνs ds . (5.1) We will discuss how ξ can be calculated, and r(t) can be obtained by taking ξ in Equation (5.1).
The Cold REF: where ξ = ξ C A,B (α).The systems (5.3) and (5.5) have no closed form solutions, so ξ = ξ D A,B (α) can be obtained by using some numerical techniques.

Numerical Application
Under the following assumptions the REFs of a series-parallel system are calculated: (1) There are two subsystems, m = 2.
Table 2 introduces the values of α-fractiles, L(β, α) and L D B (β, α), for D = H and C. From the results shown in Tables 1, 2 and Figures 5-7, we can conclude that: B , for all |B| = 1, 2 and 3.    Table 2.The α-fractiles.From the results presented in Tables 2 and 3, at β = 3: (1) The L(3, 0.1) is increased from 3.9990/ν to 4.5043/ν when the set B (1,0) 1 improved by HDM, see Table 2.We can get the same effect by reducing the failure rates of (i) A by ξ H = 0.8878, see Table 3.   2 and 3 can be interpreted by the same way.
(4) The symbol "-" means that there is no equivalence between both the reduction and duplication methods in this numerical study.

Conclusion
The reliability performance of a serial-parallel system based on a gamma distribution has been improved.This system is one of the important systems in reliability because it can be reduced to the series, parallel and radar systems.The system components have gamma lifetime distribution.
Lifetimes are assumed independent and identical.The gamma distribution is an important distribution that is used in engineering to study system reliability.The gamma distribution has some special distributions based on the values of its parameters.The original system was improved using three different methods.Some reliability measures are derived for each method, such as RF and MTTF.The REFs and α-fractiles were established.Numerical application was discussed to interpret how the theoretical results can be applied.The cold duplication method gives the best improvement than other methods.

Figure 2 .
Figure 2. The original system diagram.

. 4 ) 3 . 2 .
The HDM. Suppose the components are in a set B, |B| = k and 0 ≤ k ≤ M will be improved according to HDM.Each component in B is duplicated by a hot standby component.The k can be distributed such that k i components from subsystem i, where k = m i=1 k i and 0

Figure 7
Figure 7 displays the RF of the original and duplicated systems for |B| = 1, 2, 3 and different methods.

Figure 4 .
Figure 4.The RF, of the original and duplicated systems, when |B| = 1.

Table 1
views the values of M and M D B for different B, D = H and C.Table 1.The M and M D B , for different values of |B|.B

Table 3
displays the values of the REFs for different A and B.

Table 3 .
The REF, ξ D A,B (α), D = H and C

Table 2 .
We have the same effect by reducing the failure rates of (i) A C = 0.7426, see Table3.(3)The results in Tables