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Gertsbakh I., Shpungin Y. RT&A, No3 (42) RELIABILITY OF HETEROGENEOUS (k,r)-out-of-(n,m) SYSTEM_Volume n, September 2016
Reliability Of Heterogeneous (k,r)-out-of-(n,m) System
Ilya Gertsbakh
Ben Gurion University e-mail: elyager@bezeqint.net
Yoseph Shpungin
Sami Shamoon College of Engineering e-mail: yosefs@sce.ac.il
Abstract
We consider a coherent binary system consisting of m components of a-type and n components of b-type. The a-type and b-type components have i.i.d lifetimes with cdf Fa(t) and Fb(t), respectively. The a-type and b-type components are stochastically independent. Our system is UP if at least k atype components are up and at least r components of b-type are up.We present a simple formula for this system lifetime cumulative distribution function.
Keywords: heterogeneous system; k-out-of-n system; survival signature.
We consider the following generalization of a"standard" k-out-of-n system. Our system has two types of components: n components of a-type and m components of b-type. The components of each type have iid lifetimes denoted as Fa(t) and Fb(t), respectively. The a-type and b-type components are stochastically independent. The standard k — out — of — n system is operational (i.e. in state UP) iff at least k of its components are operational, i.e. are up. Our system is defined to be operational if and only if at least k components of a-type and at least r of b-type components are up. Formally, our system can be viewed as a series connection of two k — out — of — n-type subsystems.
Suppose, without loss of generality, we number the a-type components by numbers 1,2, ...,m and components of b-type by numbers m + 1,... ,m + n. System state is therefore a binary vector x = (x1,...,xm,...,xm+n, where xt = 1 or xfi if component i is up or down, respectively.
System state is a binary function ç(x) which takes values 1 or 0 if the system is UP or DOWN, respectively.
If ^(x1) = 1, then x1 is called an UP-vector or an UP-set.If the state vector x is not an UP-vector, we call it a DOWN-vector or DOWN-set
According to the above description of our system, an UP-set must have at least k ones on the first m positions of vector x and at least r ones on the last n positions. For example, for m = 5 and n = 6, k = 4,r = 4, the vector x = (0,1,1,1,1; 1,1,1,0,1,0) is an UP-vector.
Denote by NU(v,w) the number of UP-vectors which have exactly v ones on the first m positions and w ones on the last n positions. Obviously, v >k,w >r.
The following Lemma follows from the above description:
Gertsbakh I., Shpungin Y.
RELIABILITY OF HETEROGENEOUS (k/r)-out-of-(n/m) SYSTEM
RT&A, No3 (42) Volume 11, September 2016
Lemma
NU(v, w) = C»-C™=-—-. (1)
v J m n vi(m-vy.wi(n-wy. v '
The proof is obvious: there are ways to locate v ones on the first n positions of the state vector x and C™ ways to locate w ones on the last n positions of this vector.#
For sake of brevity, an UP-vector with v and w components of a-type and b-type, will be called an (v, w)-UP-vector.
Now everything is ready to write the formula for system UP probability. Let us take an arbitrary time instant t and denote by qa = Fa (t) the probability that an a-type component is down at time instant t.Smilarly, qb = Fb(t) is the down probability that component of b-type is down at time instant t. Denote pa = 1 - qa and pb = 1 - qb.
By independence of all a-type and b-type components and by independence of a-type components from b-type components,the probability that a state vector x is an (v, w)-UP-vector equals
P(U(v,W)) = VZqna-vVbqb-w- (2)
Now we arrive at
Theorem 1
P( system is UP at time t) = Ev>k,w>r NU(v,w) ■ P(U(v,w). (3)
If the system is UP at time instant t its lifetime ts is greater or equal t, Therefore,
P(TS >t) = C™ ■ Cn ■ [Fa(t)]v[1 - Fa(t)]n-v[Fb(t)]w[1 - Fb(t)]m-W. # (4)
Remark 1. The central role for deriving formula (4) is played by the expression for UN(v,w), see (1). Let us note thatWU(v,w) depend only on system structure function and they are, therefore, system structural invariants. It is quite obvious how to generalize the above derivation for the case when the system has more than two, say K > 2 types of components. By definition, this system is UP iff it has at least vt up components of each type, i = 1,2,... K.#
Remark 2. A system consisting of several k - out - of -n subsystems is, to the best of our knowledge, the only lucky case where we can find in a simple form (like in (1)) an explicit formula for the number of system UP-state vectors having exactly Vi components of i-th type in up state, i = 1,2, ...,K.
Samaniego and Navarro suggested to call the collection of all NU(v,w) values survival signature, see [1]. If ND(u,w) is the number of system DOWN states with exactly v a-components and w b-components down, then it would be natural to call the collection of all ND (v, w) failure signature. #
Remark 3. There is a simple relationship between the values of ND(v, w) and NU(v, w):
ND(v,w) + NU(n - v,m - w) =-^-. (5)
Indeed, let us chose v components of a-type and w components of ¿-type and let them be down. Then we will obtain either a DOWN state or an UP state vector for the system. But having v,w components down, means having the remaining components up, which proves (5). From practical point of view, (5) shows that the knowledge of the survival signature provides us the knowledge its dual failure signature.!
Gertsbakh I., Shpungin Y.
RELIABILITY OF HETEROGENEOUS (k,r)-out-of-(n,m) SYSTEM
RT&A, No3 (42) Volume 11, September 2016
Remark 4. Let us return to coherent binary systems consisting of one type iid components. Crucial role in its reliability evaluations play so-called signature f = (f1, f2,..., fn), see [2]. Let F(j) = YJk=i fj, J = 1,2,...,n be the so-called cumulative signature or system D-spectrum [3,4]. F(j) is the probability that the system is DOWN if j of its components are down, i.e. the probability that system failure appeared after x components have failed, x = 1,2,... ,j. If we know the D-spectrum of the system, we can find the number ND(r)-the number of system failure or DOWN states with exactly r components down and n — r components up, by using the following simple formula, see [3,4]:
For systems of real size, having n > 8 — 10 components, there are efficient Monte Carlo algorithms for fast and accurate estimation of F(j), see [4]
In our opinion, in case of coherent systems having two types of independent and identical components, reliability calculations must be based on the knowledge of a two-dimensional analogue of the cumulative D-spectrum. It should be a function G(k,r) expressing the probability that a random permutation of n and m components of both types contains a failure set with k and r down components of a- and b-type, respectively.#
References
[1]. Samaniego, F.J. and J. Navarro (2016). On comparing coherent systems with heterogeneous components, Adv. Appl. Prob., 48, 1-24.
[2] Samaniego, F.J.(1985).On closure of the IFR under formation of coherent systems. IEEE Trans. on Reliability,34: 69-72.
[3] Gertsbakh, I. and Y. Shpungin (2012). Stochastic models of network survivability. Qual. Tech. Qualit. Management, 9(1), 45-58.
[4] Gertsbakh, I. and Y. Shpungin (2011). Network Reliability and Resilience. Springer Briefs in Electrical and Computer Engineering, Springer.
ND(r) = F(r)n\/(r\ (n - r)\).
(6)
