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Garima Chopra, Deepak Kumar
COMPARISON OF BRIDGE SYSTEMS RT&A, No 4 (71)
WITH MULTIPLE TYPES OF COMPONENTS Volume 17, December 2022
Comparison of Bridge Systems with Multiple Types of
Components
Garima Chopra1, Deepak Kumar2
^Department of Mathematics, University Institute of Engineering & Technology, Maharshi Dayanand University, Rohtak, Haryana, India 1 garima.choprafcgm ail. com 2dkkr111 @gmail.com
Abstract
This paper aims to compare some bridge systems with multiple types of components in stochastic, hazard rate, and likelihood ratio order. Such systems are generally used in the designing and production industries. These systems are supported by a buffer store that balances the fluctuation in two production lines during the production process. The survival signature tool and distortion function technique are employed to compare the performance of four different bridge systems. Survival signature and henceforth survival function is computed for each considered system. The findings of comparisons are facilitated with the help of tables and figures. The comparison of large size coherent systems based on the structure-function approach is quite challenging. As this study is based on survival signature, so it is not so complex and has future scope.
Keywords: survival signature; bridge system; survival function; distortion function.
1. Introduction
In today's competitive and technology-driven world, it has become consequential to develop safe, reliable and long-lasting systems. The accurate reliability assessment of components and systems is crucial, and hence the branch of reliability engineering is in very much demand. In reliability theory, the stochastic comparison of systems is an imperative concept and has been explored by many researchers. It is quite challenging to compare complex systems, and most realistic cases generally have complex structures. Birnbaum et al. [2] and Barlow and Proschan [1] compared the same order coherent systems based on component lifetime using the structure-function approach. But these methods involve analytical complexities while comparing complex manufacturing systems. Recently, system signature and survival signature have emerged as advanced and promising tools in reliability analysis. These tools have suitable applications in studying system reliability and comparing various coherent systems.
A system having monotonic structure function with each of its components being relevant is known as coherent system. Samaniego [12] introduced the concept of system signature for the systems having independent and identically distributed (iid) components, with common distribution function F. For such coherent systems, Samaniego [12] derived an explicit expression of the failure rate in terms of components' failure rate and F. The IFR closure theorem for fc-out-of-n system is also discussed by researcher. Kochar et al. [6] further derived the expression of system signature for fc-out-of-n systems with component-wise and system-wise redundancy. Samaniego
COMPARISON OF BRIDGE SYSTEMS WITH MULTIPLE TYPES OF COMPONENTS
[11] extended the concept of signature for preservation, characterisation, and system reliability. The applications of network reliability and economical reliability to systems having shared components are also presented. Navarro et al. [9] defined a joint signature for coherent systems with shared components. They discussed the sufficient condition for bivariate stochastic ordering between the joint lifetimes of two pairs of the systems.
Coolen and Coolen-Maturi [3] extended the concept of system signature to systems with multiple types of components, and they coined the new term 'survival signature'. The survival function of the coherent systems having iid and exchangeable components is evaluated using the survival signature tool. Coolen et al. [4] further adopted this technique and developed non parametric predictive inference for studying the reliability of systems. Krpelik et al. [8] introduced the formula for computing system survival signatures by means of merging survival signatures of multiple subsystems. They also introduced a decomposition method that allows decoupling the dependencies among subsystems. Huang et al. [5] analysed the reliability of the phased mission systems having identical components in each phase using survival signature.
Several authors have worked on the stochastic comparison of coherent systems. Kochar et al. [6] compared various systems on the basis of stochastic, hazard rate, and likelihood ordering using the notion of system signature. Authors derived an important theorem on hazard rate ordering of the system based on its components' hazard rate ordering. Coolen and Coolen-Maturi [3] compared some coherent systems with iid and non-iid components based on a novel technique of survival signature. Koutras et al. [7] stochastically compared two systems having exchangeable components. They further provided a necessary and sufficient condition for examining hazard rate ordering and reverse hazard rate ordering. Samaniego and Navarro [13] presented the methodology to compare some systems having heterogeneous components in different modes (stochastic, hazard rate, and likelihood ratio ordering) using survival signature and distortion function.
The bridge systems are broadly used in system designing in addition to the series and the parallel systems. Such systems are found in the production process in various industries. The production system having two parallel production lines connected by a buffer store to balance their productivity variation is investigated as a bridge structure system [10]. The analytical evaluation of the lifetime of the bridge system is too dense. Therefore, the comparison among such systems becomes more complicated. The present study compares the lifetimes of the bridge systems having multiple types of components at different positions. The survival signature technique is used to compare these complex systems. This paper investigates some bridge systems having two/three types of components shown in Figure 1, Figure 2, and Figure 3. The comparative analysis of considered systems is done using the survival signature approach [13].
The present section includes prevalent concepts, definitions, and theorems. For 'm' components system, the state vector x = (x1,x2,.... xm) E {0,1}#, where
for all i = 1,2,3,.. ,m. Thus, the set [0,l}m represents all the possible state vectors of m-order binary coherent system. Barlow and Proschan [1] defined the structure function 0 mapped from the set {0,1}m to {0,1} as follows
2. Definitions and Notations
1, when ith component of system is working 0, when ithcomponent of system is not working
_ (1,if systems works (p^ xxm) -[o^f systems faus.
As compared to structure function, system signature [12] is less general but more significant. For the coherent system of order 'm', the system signature is a probability vector such that some ith component causes system failure. Mathematically, the ith element 's$', of the system signature s — (sx, s2,.., sm) is expressed as
s$ — P (T — Xi:m) — #
where T denotes the lifetime of the system, Xi:m represents the ith order statistic of the failure time of the m-components and mi is number of those orderings corresponding to which system fails on failure of ith component. It is evident that V i, s$ >0 and Yih s$ — 1.
For a coherent system with m iid components having a continuous lifetime distribution, the survival signature 0(1) for I — 0,1,2, ...,m is defined as the probability of functioning of system, provided that its exactly l components are working [3]. Mathematically, the survival signature of coherent system is given by
0(i) = -$— ={l) I.x£si4M
where s4 is the set of all such state vectors whose exactly l components (x$ ) are 1 and remaining are 0. The system reliability FT(t) in terms of survival signature for iid components is
FT(t) = P (T>t)= 0(1) Q [F(t)]m~l [F(t)]4
where F(t), F(t) be the distribution and survival function respectively of components.
Coolen and Coolen-Maturi [3] considered the coherent system of order m, with K > 1 types of independent components. All the components of certain type are assumed to be identically distributed. Considering mk components of type k, the survival signature O(ll, l2, —., l8) is given by
0(li,l2......lK) =
mr1 x ™
xesh.....i*
where lk (k = 1,2, ...,K) is the number of functioning units of type k. In the above expression, x is a state vector given by x = (x!,x2,.... ,xK), where xk = (x7,x7,..., x7*). In case lk (k = 1,2, ...,K) units of type k are working, then the vector xk has precisely its lk components (xk) as 1 and remaining
are 0. The set of all such state vectors is denoted by sl± l(.....4*. The reliability function FT(t) of such
systems in terms of survival signature as given by Coolen and Coolen-Maturi [3] is
m& m*
FT(t)—p(t>t)—x —. X 0(i!,1"......i8)UG;M Fi(t)mi~iipi(t)1
i1=o i+=o i=l
where F$(t), Fi(t) be the distribution and survival function of the ith component.
Garima Chopra, Deepak Kumar
COMPARISON OF BRIDGE SYSTEMS RT&A, No 4 (71)
WITH MULTIPLE TYPES OF COMPONENTS Volume 17, December 2022
Some results on stochastic order properties which appeared in [14] are discussed below. Let Ti,T2 be the random variables with the distribution functions Fi(t),F2(t) and reliability functions Fi(t), F2(t) respectively, then
• Ti is smaller than T2 in usual stochastic order, i.e. Ti <ST T2 if F1 (t) < F2 (t)for all t;
• Ti is smaller than T2 in the hazard rate order, i.e. Ti <HR T2 if F^(t)/Fi(t) is increasing in t;
• Ti is smaller than T2 in the likelihood ratio order, i.e. Ti <>R T2 if f2(t)/fi(t) is increasing in t; where f-i(t) and f2(t) are probability density functions (pdfs) of Ti and T2 respectively.
Samaniego and Navarro [13] also derived a result for the comparison of two systems having mk independent type k components with distribution function Fk for k E { 1,2,..,r}. The following theorem appeared as Theorem 2.1. in Samaniego and Navarro [13] .
Theorem 1. If Ti, T2 be the lifetimes and Oi,O2 be survival signatures of two systems A and B respectively and if for all vectors (li,..., I?), with lk = 0,..., mk and k = 1,....,r, the inequality
......I?) < ®2(k.....I?)
holds, then it follows that Ti <ST T2 for all distribution functions Fv ...,F?.
Samaniego and Navarro [13] further proved a theorem, which aids in the comparison of two systems having different orders. For such comparisons, some irrelevant components are considered and added to the systems. The following proved result appeared as Theorem 3.1 in Samaniego and Navarro [13] .
Theorem 2. Let O be the survival signature of m-order coherent system, having r types of components and suppose it has to be compared with some system of order m+1. An irrelevant component of type-k is added to m-order coherent system, and let O* be the survival signature of resulting new m+1 order system. Considering mj components of type j, Samaniego and Navarro [13] established following relations for survival signatures O and O*
(i) For 0 < lj < mj, j = 1,2,,..,k — 1,k,k + 1, ...,r,
0*(li, ... lk-i, 0, tk+l, . , I?) = @Gl, . lk-1, 0,1-k+i, ., I?)
(ii) For 0 < lj < mj,j = 1,2,,.. ,k — 1,k,... ,r, and for 1 < lk < mk, <P*(li, ...Ik-iAk.....I?)
I lk \ imk — lk + 1\
= LM) *&,..-ik-i,ik—1.....o + (-tkfH wi,..-1^.....o
(iii) For 0 < lj < mj, j = 1,2,,..,k — 1,k,k + 1 ...,r,
0*(li, ... lk-i, mk + 1, tk+i, . , I?) = ^Oi, . lk-i, mk, tk+i, ..■, I?).
Samaniego and Navarro [13] also adopted a generalized distorted distribution technique for comparing two systems. They employed a dual distortion function, Q(ui,u2, ...,u?) and distortion function Q(ui, u2,..., u?) in this technique. These functions satisfy the following properties:
(i) Qi^, u2,..., u?) is a continuous increasing function
(ii) Q(ui, u2,..., u?) = 0 if u$ = 0 V i E {1,2, ..,r}
(iii) Q(ui,u2, ...,u?) = 1 if u$ = 1 V i E {1,2,..,r}.
(iv) Q(ui,u2,...,u?) = 1 — Q(1 — Ui,1 — U2,... ,1 — u?)
The survival function FT (t) of coherent system having r types of components can be expressed as-
FAt)= Q(Fi(t),F2(t).....F? (t)),
where Ft is the reliability function of components of type l. The lifetimes Ti and T2 of the two coherent systems with r types of components can be compared using the distortion function as discussed below. The following proved result appeared as Theorem 4.1. in Samaniego and Navarro [13].
Theorem 3. Let Fl,F2, ...,Fr be the distribution functions of the components of type 1, type 2,..., type r respectively. Samaniego and Navarro [13] proved that if Ql and Q2 be the dual distortion functions of two considered systems, then
(i)
(ii)
(iii)
Tl <ST T2 holds for all Fl,...,Fr if and only if Ql < Q2 in (0,1)r;
Tl <HR T2 holds for all Fl,...,Fr if and only if Q2/Qlis decreasing in (0,1)r;
Tl <LR T2 holds for all F-l,...,Fr, if the distributions of Tl and T2 are absolutely continuous,
and if y(ul,u2,...,ur,v2,..., vr) is decreasing in ul,u2,...,ur and increasing (decreasing) in vr
in (0,1)r x (0,ra)r~l and Fl <LR f$ (>lr) for i = 2, ...,r where
V(u1,u2,...,u?,v2, ...,v?) =
D1Q2(U1,U2.....Ur)+Yj[=2 VjDiQ2ÎU1,U2.....Ur)
DiQiiu1,u2.....Ur)+'Z^=2 ViDiQ1(u1,U2.....Ury
D$Qj denotes the partial derivatives of Qj about ith component for i £ {1,..., r} and j e {1,2} and ur denotes components' reliability function of type r and vr denotes the ratio of pdfs of components of type r to the type 1.
3. Analysis and Discussion
The purpose of this article is to compare the bridge systems having multiple types of components. The survival signature tool is used to compare the considered systems in three different senses (stochastic, hazard rate, and likelihood ratio ordering). The bridge system as shown in Figure 1 has two units xll, x2l of type 1 and three components namely xl2, x22, x32 of type 2. The second considered system as shown in Figure 2 has again two components of type 1 and three components type 2, but at different positions. The bridge system (Figure 3) having three types of components is also investigated in this study.
Figure 1: System A (five-component bridge system)
Figure 2: System B (five-component bridge system with changed positions of components)
Figure 3: System C (five-component bridge system containing three types of components)
3.1. Comparison of two bridge systems with two types of components at different positions
Theorem 4. Consider two bridge systems of order five with two types of components at different positions. Let T1, T2 be the lifetimes of bridge systems A and B (Figure 1 and Figure 2) respectively. Then, T1 is smaller than T2 in usual stochastic order i.e., T1 <ST T2.
Proof. Let O1 (l1, l2) and O2 (l1, l2) be the survival signatures of the systems A and B respectively. These systems have two and three components of type 1 and type 2 respectively. The general expression of the survival signature O(l1, l2) for considered systems is as follows
O(k,l2) = {l)~1 iff T.Xesh:i2 MX) where s4 l2 is set of all state vectors of the system.
Table 1: Survival signature <P1 of the system A
h=0 l2 = 1 l2=2 l3 =3
l1 =0 0 0 0 0
ll = 1 0 1/3 1 1
Il =2 0 2/3 1 1
Table 2: Survival signature <P2 of the system B
l2=0 l2 = 1 h =2 l3 =3
l1 = 0 0 0 1/3 1
ll = 1 0 0 2/3 1
Il = 2 1 1 1 1
As discussed in Theorem 1, the survival signatures O1(l1, l2) and O2(I1, l2) given in Table 1 and Table 2 are non-comparable because 0X(0,2) < 02(0,2) and O1(1,2) > O1(1,2). Thus, the domination of survival signature is not possible for the considered systems. To compare these systems, we need to do further analysis. Let FTl(t), FT2(t) be the survival functions of the bridge systems A and B with components distribution function F1(t) and F2(t). The difference between survival function FT2(t) and FTl (t) is given by
ft2 (t) - FTi(t) = h2=0 CO" (h,l2) - Oí(lí, l2)) ("&) (Í2) F1(t)2-'1 j°1(t)'1 F2(ty~'2F2(t)'2. (1)
Figure 4: The difference function D(x1,x2) .
To simplify the system's comparison, we consider the variable Fl(t) — 1 — Fl(t) as xl and F2(t) — 1 — F2(t) as x2. So, the pair (xl,x2) belongs to unit square as t £ [0,ro). The above difference in Equation (1), is taken as D(xl,x2) and can be represented as-
D(x1,x2) — xl2(1 — x2)3 — 2x1x2(1 — xl)x2(1 — x2)2 + x12x2(1 — x2)2 + x22(1 — x-)2(1 — x2) —
2XlX22(1 — Xl)(1 — X2) + X23(1 — Xl)2 .
The difference function D(xl,x2) illustrated in Figure 4 has clearly non-negative values for each value of xl and x2. i.e., D(xl,x2) > 0, V xl,x2 £ [0,1]. This implies that FT( (t) > FT& (t). Hence the lifetime Tl is smaller than lifetime T2 in usual stochastic order i.e., Tl <ST T2 holds for all Fl(t),F2(t).
3.2. Comparison of bridge systems using distortion functions
In this part, the systems A and B are compared as per stochastic, hazard rate and likelihood ratio ordering, by using their distortion functions.
Theorem 5. Let Tl,T2 be the lifetimes of the bridge systems A and B (Figure 1 and Figure 2) respectively. These systems have two types of components with the distribution functions Fl(t),F2(t) and reliability function Fl(t),F2(t). The lifetime of system A is smaller than the lifetime of system B in usual stochastic order but not in hazard rate and likelihood ratio order.
Proof. Let Ql and Q2 be dual distortion functions of systems A and B respectively. We have,
Q2(XI,X2) — QI(XI,X2)
— (1 — x2)3xl2 — 2x1x2(1 — x2)2(1 — xl) + x12x2(1 — x2)2 + (1 — xl)2x22(1 — x2)
— 2X-X22(1 — Xl)(1 — X2) + X23(1 — X1)2.
Figure 4 indicates that Q2(xl,x2) > Ql(xl,x2) V xl,x2 £ [0,1]. Using Theorem 3, we can say that the system lifetime T1 is smaller than system lifetime T2 in usual stochastic order. i.e., T1 <ST T2 hold for all Fl(t),F2(t).
Let R be the ratio of Q2 to Q1 i.e.,
Q2
R(XuX2) — —
( 1 2) Ql
Figure 5 exhibits that the ratio R(x1,x2) is neither increasing nor decreasing in xl, x2 in (0,1)2. Data presented in Table 3 confirms the same. Using Theorem 3, we can say the system lifetime T1 is not smaller than system lifetime T2 in hazard rate order i.e., T1 SHR T2. Figure 6, further shows that these two bridge systems are not hazard rate ordered when the components of type-1 and type-2 follow exponential and Weibull distribution respectively.
Table 3: The ratio fi(x1, x2)
x± ^ x2 i 0.00010 0.09999 0.19988 0.29977 0.39966 0.49955 0.59944 0.69933
0.00010 1.000 499.908 999.330 1498.767 1998.220 2497.687 2997.169 3496.667
0.09999 458.761 1.000 1.235 1.635 2.086 2.566 3.068 3.588
0.19988 861.762 1.220 1.000 1.076 1.236 1.436 1.663 1.912
0.29977 1239.023 1.567 1.073 1.000 1.039 1.129 1.251 1.397
0.39966 1611.871 1.939 1.216 1.037 1.000 1.024 1.085 1.172
0.49955 1998.480 2.331 1.390 1.121 1.023 1.000 1.017 1.063
0.59944 2417.177 2.754 1.588 1.231 1.081 1.017 1.000 1.013
0.69933 2889.461 3.225 1.812 1.365 1.162 1.061 1.013 1.000
To compare the hazard rate ordering of system A and component of type 1, the ratio R!& (x1,x2) is computed. We get
0 (x x )
R.& = U = 2x2(1 - x2)2(1 - xj + 6x22(1 - x^(1 - x2) + 2x23(1 - xj + 2xix2(1 - x2)2 xi
+ 3x1x22(1- X") + x±X23.
Figure 7 indicates that the ratio R.& (xr, x2) increases with increase in x2, but it decreases with increase in x-l in (0,1)2. Therefore, the lifetimes of system A and type 1 components are not comparable in the hazard rate order, i.e., T± SHR X1 where X1 indicates the type 1 component's lifetime. Similarly, for hazard rate order comparison of system A and the type 2 components, the ratio R.2 is evaluated. We obtain
, q1(^,x2) 2 2 2 2
R.2 =-= 2xi(1 - x2)2(1 - x-t) + 6xix2(1 - x^(1 - x2) + 2x1x22(1 - x-J + 2xi2(1 - x2)2
X2
+ 3x±2X"(1 - X") + x12x22.
Here, the ratio R.2 (xu x2) decreases with increase in x^, but it neither increases nor decreases with increase in x2 in (0,1)2. Therefore, T± SHR X2, where X2 is type 2 component's lifetime. In the same manner, system B is compared with type 1 and type 2 components in hazard rate order by evaluating the ratios R"& and R.2 respectively. We have
r2& = qáxi,x2) =X2_ (i - X")(i - x^2 +X"_ (i- ^)2 + 4(i - xi)x22(i - x2) + 2x23(1 - xj + xx(1 - x2)3 + 3X±X"(1 - X2)2 + 3x±(1 - X")X"2 + X±X23
and
Q (x , x )
R"2 = -'-= x2(1 - x2)(1 - xi)2 + (1 - xi)"x22 + 4x1x2(1 - xi)(1 - x2) + 2x1x22(1 - xj
x2
2 (1 _ X^ ) 3
+---+ 3x12(1 - x2)2 + 3x12x2(1 - x-¡) + x12x22.
x2
Figure 5: The Graphical interpretation of the function R(xx2)
Here, the ratio (xl,x2) increases with x2 but it is not monotonic in xl in (0,1)2. Thus, T2 SHR Xl. The ratio R.2 (xl,x2) increases with increase in xl but decreases with increase in x2 in (0,1)2. Therefore, T2 SHR X2.
Figure 6: Hazard rate functions of the bridge systems (A (dash), B (dot)) and their components (dark lines). Type 1 and Type 2 components follow exponential and Weibull distribution (a = 2, b = 1) respectively for t > 0
Let Xl, X2 be the lifetimes of the components of type 1 and type 2 with respective pdfs fl(t), f2(t). The components of Type 1 and type-2 are assumed to be exponentially (mean = 1) and Weibull (a — 2,b — 1) distributed respectively. The ratio J2is increasing in t as shown in Figure 8. Hence, we get that X1 is smaller than X2 in likelihood ratio ordering i.e., X1 <LR X2. For likelihood ratio ordering comparison of systems A and B, as per Theorem 3, we have function Y ('xl, x2, as-
v(2X22(XI — 1)2 — 2XIX22(XI — 1) — 2X2(X2 — 1)(X- — 1)2 + 8XIX2(XI — 1)(x2 — 1)) _ —2Xl(X2 — 1)3 + 6XlX2(X2 — 1)2 — 2XlX22(X2 — 1) + 2X22(Xl — 1)(X2 — 1)
fl(t)) V(2X12(X2 — 1)2 — 2X12X2(X2 — 1) — 2Xl(Xl — 1)(x2 — 1)2 + 8XlX2(Xl — 1)(X2 — 1)) —2x2 3(xi — 1) + 2xlx2(x2 — 1)2 — 2x2(xl — 1)(x2 — 1)2 + 6x2 2(x- — 1)(x2 — 1)
Y{X1,X2,W)) —
where — v.
fi(t)
Figure 7: The Ratio R^, R^, R!2,R"2
Figure 8: Likelihood ratio ordering of the components of type 1 and type 2
Table 4 indicates that the function Y(x1tx2, 0.0001) is increasing in x1 for the particular value of x2. But we can see the function is neither increasing nor decreasing in x2 for any particular values of x1. In Table 5, the function Y(x1,0.09999,v) is increasing in x1 for the particular values of v. Table 5 further shows that the function is increasing in v for x1 = 0.0001, but it is decreasing in v for x1 = 0.09999,0.19998,0.29977. Hence, we get that the function Y (x1, x2, y^ is increasing in x but not monotonic in yjy in set (0,1)" x (0, <x>). Therefore, these considered bridge systems are not likelihood ratio ordered. i.e., T1 SLR T2.
Table 4: The function Y(x1, x2,0.0001)
X1 ^ X2 ^ 0.00010 0.09999 0.19988 0.29977 0.39966 0.49955 0.59944 0.69933
0.00010 1.000 908.936 1665.738 2306.254 2855.383 3331.387 3747.970 4115.610
0.09999 0.083 1.000 1.965 2.985 4.062 5.203 6.412 7.697
0.19988 0.138 0.549 1.000 1.497 2.049 2.664 3.355 4.135
0.29977 0.173 0.414 0.687 1.000 1.361 1.783 2.283 2.885
0.39966 0.193 0.349 0.530 0.744 1.000 1.312 1.701 2.199
0.49955 0.200 0.304 0.428 0.579 0.764 1.000 1.307 1.725
0.59944 0.193 0.264 0.349 0.454 0.587 0.762 1.000 1.343
0.69933 0.173 0.219 0.276 0.346 0.438 0.561 0.735 1.000
Table 5: The function Y(x1,0.09999, v)
X1 ^ X2 ^ 0.00010 0.09999 0.19988 0.29977 0.39966 0.49955 0.59944 0.69933
0.00010 0.083 1.000 1.965 2.985 4.062 5.203 6.412 7.697
0.09999 0.175 1.000 1.715 2.349 2.923 3.451 3.945 4.413
0.19988 0.266 1.000 1.540 1.963 2.312 2.611 2.876 3.116
0.29977 0.358 1.000 1.410 1.704 1.932 2.119 2.279 2.422
0.39966 0.449 1.000 1.311 1.519 1.672 1.795 1.898 1.989
0.49955 0.541 1.000 1.232 1.379 1.484 1.565 1.634 1.694
0.59944 0.633 1.000 1.168 1.270 1.340 1.395 1.440 1.479
0.69933 0.724 1.000 1.115 1.182 1.228 1.263 1.291 1.316
3.3. Comparison of two bridge systems with different number of components
Theorem 6. Suppose T1t TK be the lifetimes of the bridge systems A and D shown in Figure 1 and Figure 9 respectively. The system D has six components, where type 1 components are x11, x21 and x31 and type 2 components are x12, x22 and x32. Then the lifetime TK is smaller than T1 in usual stochastic order. i.e., TK <ST 7\.
Proof. Let 0K(l-i, I2) be the survival signature of the system D. The survival signature 01(l1, l2) of system A is already discussed and given in Table 1. An independent irrelevant component of type k = 1 is added to system A, and let us suppose that 01(^,1^ be the survival signature of new resulting system of order 6. Using Theorem 2, we have
(i) For 0 <l2<m2
(O,l2) = 01(O,l2)
(ii) For 1 < l1 <m1 and 0 < l2 <m2
( U N /m* -U +1\
01 (l1,12) = (zr+rr) 01(l1 -1,12) + ( ' +1 ) 01(l1,12)
\m1 + 1) \ m1 + 1 )
(iii) For 0 < l2 <m2
0l(m1 + 1,12) = 01(m1,l2).
Figure 9: System D (six-component bridge system)
Tables 6 and 7 indicate that the survival signature O1 is greater than OK for all values of l^, l2 i.e., O*1(l1,l2) — ®Si,h) ^ 1-1,1-2 e {0,1,2,3}. Thus, the lifetime TK is smaller than T1 in usual stochastic order, i.e., TK <ST 7-.
Table 6: Survival signature 01 of the bridge system A with irrelevant component of type-1
l2 =0 I" = 1 l2 =2 l2 =3
l± = 0 0 0 0 0
k = 1 0 2/9 2/3 2/3
= 2 0 4/9 1 1
l± = 3 0 2/3 1 1
Table 7: Survival signature <P$ of the bridge system D
O4(k,l") l2 =0 l2 = 1 l2 =2 l2 =3
l± = 0 0 0 0 0
k = 1 0 0 0 0
= 2 0 2/9 2/3 2/3
l± = 3 0 2/3 1 1
3.4. Comparison of lifetimes of two bridge systems with two and three types of components
Theorem 7. Consider two bridge systems A and C, shown in Figures 1 and 3. Let T1 and T3 be the respective lifetimes of systems A and C. Type 1, type 2 and type 3 components are assumed to be iid with reliability functions F-1, F2 and F3 respectively. Then T1 <ST T3 if F2(t) < F3(t).
Proof. Let O1(l1,l2) and O3(l1, l2, l3) be the survival signature of bridge systems A and C respectively. Here, system C contains two components of type 1, two components of type 2, and one component of type 3. The survival signature O3(l1,l2,l3) can be written as O3(l1,l2,l3) =
i2) (2) i1) ^xeshhh and is given in Table 8. For comparison of bridge systems A and
C, we have added an irrelevant component of type 3 (k = 3) to system A. Using Theorem 2, we have survival signature O^ (lv l2, l3) of resulting 6-components system as:
(i) For 0 <lA < mj; j = 1,2
&l(l1,l2,0) = Q(l1,l2,0)
(ii) For 0 <lj < mj ; j = 1,2
O1(li, l",m3 + 1)= O(li,l2,m.3).
Similarly, we have added one component of type 2 (k = 2) which is irrelevant in nature to system C. Using Theorem 2, the survival signature O3 of the resultant 6-components system is given by
(i) For 0 <lj < mj ; j = 1,3
(l1,O,l3) = 03(l1,O,l3)
(ii) For 0 < lj < mj ; j = 1,3 and 1 < l2 <m2
O*3(li, 12, h) = 3O3(li, l2 - 1, l3) + 3—2O3(li, l2, l3)
(iii) For 0 < lj < mj ; j = 1,3
®3(h,™2 + 1,13) = &3(li,m2,h).
Table 8: The survival signature i^ft, l2,1%) of the system C
k I2 I3 ®3(k, 12,13)
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 0
0 2 0 0
0 2 1 0
1 0 0 0
1 0 1 0
1 1 0 1/2
1 1 1 1
1 2 0 1
1 2 1 1
2 0 0 0
2 0 1 0
2 1 0 1
2 1 1 1
2 2 0 1
2 2 1 1
Table 9 shows that the survival signatures 01 and 01 are identical for all the combinations of 1-l, l2, l3 except two cases. The survival signature 01 and 01 are not dominated in any sense since 01 (1,1,1) < 01 (1,1,1) but 01 (1,2,0) > 01 (1,2,0). So, the comparison of systems A and C needs further analysis. Let FTl (t), FT. (t) be the respective reliability functions of systems A and C. We have
2 3 1
Ft. (t)-FTi (t) =XX X«01 (l1,l2,li) ll=0 l2=0 l3=0
- 01 (l1,l2,l3)](l)(l)(l) F1(t)2~l&F1(t)l&F2(t)3~l2F2(t)12Fi(t)1-1.Fi(t)1. Using survival signature given in Table 9, we get
Ft. (t) - Fn(t) = -2 F1(t)F1(t)F2(t)F2(t)2F3(t) + 2 F1(t)F1(t)F2(t)2F2(t)F3(t)
(2)
Table 9: The survival signature and <P% of systems after adding an irrelevant component of type-3 and type-2
respectively to system A and system C
ii ^2 ¿3 o3 Gi,M3)
0 0 0 0 0
0 0 1 0 0
0 1 0 0 0
0 1 1 0 0
0 2 0 0 0
0 2 1 0 0
0 3 0 0 0
0 3 1 0 0
1 0 0 0 0
1 0 1 0 0
1 1 0 1/3 1/3
1 1 1 1/3 2/3
1 2 0 1 2/3
1 2 1 1 1
1 3 0 1 1
1 3 1 1 1
2 0 0 0 0
2 0 1 0 0
2 1 0 2/3 2/3
2 1 1 2/3 2/3
2 2 0 1 1
2 2 1 1 1
2 3 0 1 1
2 3 1 1 1
To simplify the comparison process, we have taken variable F1(t) =1 — Fx(t) as x1, F2(t) =1 — F2(t) as x2 and F3(t) =1 — F3(t) as x3. The 3-tuple (x1,x2,x3) lies in the unit cube as t varies from 0 to For t £ [0, ro), the difference FTs (t) — FT& (t) given in Equation (2) can be written as the multivariable function D(x1, x2,x3) as
D(xi,x",xi) = —2xi(1—xi)(1 — X2)(1 — X3)y2 + 2X1X2X3(1 — Xi)(1 — X2)2 = 2XiX2 (1 — Xi)(1 — X2)(X3 — X2).
If x2 < x3 or x3 = 1 then D(xi,x2,x3) > 0. In addition, D(xi,x2,x3) = 0 if xi,x2 = 1 or x2 = x3. This implies that the system's lifetime Ti is smaller than T3 in usual stochastic order if the components lifetime of type 2 is less than the component lifetime of type 3. i.e., Ti <;T T3 if F2 (t) < F3 (t) .
4. Conclusion
The bridge structures are generally used in the design and production industry. The comparative study of such systems is crucial to ensure system productivity and to distinguish the system that performs well. Comparing bridge systems having iid and multiple types of components without knowing their component's distribution is very challenging. In this paper, we have seen that the lifetime of bridge system A is smaller than the lifetime of bridge system B in usual stochastic order.
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However, the lifetimes of these systems (Figure 1 and Figure 2) are not found to be hazard rate and likelihood ratio ordered. Further, coherent systems A (five order) and D (six order) are compared stochastically by adding irrelevant components. It is found that the lifetime of system D is smaller than A in usual stochastic order. For stochastic comparison of lifetimes of bridge systems A and C, a result has been derived by imposing some conditions on the survival function of its components. This study compares bridge systems by considering different cases with the aid of survival signature. There is further scope to analyse the reliability characteristics and compare the combination of higher-order multi-state bridge systems with different types of components.
Conflict of Interest Declaration: The authors have no conflicts of interest to declare. References
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