Serbia & Montenegro, Belgrade, November 22-24, 2005
EUROCON 2005
Algorithm for Dynamic Analysis of Multi-State System by Structure Function Elena Zaitseva, Stefan Kovalik, Vitaly Levashenko and Karol Matiasko
Abstract - The reliability of the Multi-State System is investigated in this paper. The new class of Reliability Indices is proposed. They are Dynamic Reliability Indices. These indices estimate influence upon the Multi-State System reliability by the state of a system component. The MultipleValued Logic mathematical tools are used for calculation of Dynamic Reliability Indices. Keywords - Multi-State System, Reliability, Structure Function, Dynamic Reliability Indices
1. INTRODUCTION
Interest in network reliability, particularly telecommunications network reliability, has increased substantially in resent years [1, 2]. Discrete probability models are typically employed in network reliability analysis. In the most commonly studied model to which are investigated, network component (nodes and edges) can take on one of two states: failure or functioning. Similarly, the networks model itself is in one of two states too. This model is named Binary System. Many problem of the Binary System have been settled. But this approach fails to describe many situations where the system can have more than two distinct states [2, 3]. In a Multi-State System (MSS), both the system and its components may experience more than two states, for example, completely failed, partially functioning and perfect functioning. A MSS reliability analysis is a more flexible approach to evaluate system reliability [3, 4]. A simple, but very general, MSS model of network is structure function [2, 3]. The structure function model is developed in this paper. The minimal cut or minimal path sets is one of the major and fundamental tools for evaluating MSS reliability by structure function. Different measures are obtained over there that are important topics in the planning, design and control of network. But well know the structure function model has a disadvantage. It does not allow to investigate the dynamic behavior of MSS. This work was partially supported by grant Belorus/Slov/ZU/05 VV_MVTS 13 of MS SR (Slovakia). E.Zaitseva is with the Department of Informatics, University of Zilina, Moyzesova 20, 01026, Zilina, Slovakia (phone: 421-41-5134157; fax: 421 -41 -5652044;e-mail:
[email protected]) S.Kovalik is with the Department of Informatics, University of Zilina, Moyzesova 20, 01026, Zilina, Slovakia (phone: 421-41-5134157; fax: 421 -41 -5652044;e-mail:
[email protected]) V.Levashenko is with the Department of Informatics, University of Zilina, Moyzesova 20, 01026, Zilina, Slovakia (phone: 421-41-5134177; fax: 421-41-5652044; e-mail: Vitaly.Levashenko @fri.utc.sk) K.Matiasko is with the Department of Informatics, University of Zilina, Moyzesova 20, 01026, Zilina, Slovakia (phone: 421-41-5134177; fax: 421-41-5652044; e-mail:
[email protected])
Here it is necessary to emphasize two decisions of this problem. Firstly, in [5] authors had proposed to combine the Markov processes and the structure function for estimation of dynamic properties of MSS. Secondly, the interesting decision of this problem was offered in [6]. R. Boedigheimer and K. Kapur proposed the customer's structure function methodology that has permitted to analyze the changing of system states conditioned by a change of the component states. Then the results that had presented in [6] were developed in [7 - 9]. The Logical Differential Calculus was used to examine the dynamic proprieties of MSS in [7 - 9]. In the paper [7] authors had substantiated basic conceptions to apply the Direct Partial Logic Derivatives (it is part of Logical Differential Calculus) for the reliability analysis of MSS. The Direct Partial Logic Derivatives reflect changing the value of investigation function when the values of its variables change. In [8] the new class of reliability indices was determined and was named Dynamic Reliability Indices (DRI). The dynamic character of DRI consists to determine states of the system failure or, in other case, to the repair system is caused by a change of state the system component. In other words, these indices define the boundary states of MSS. In paper [9] three groups of DRI were obtained. There are the group of deterministic and two groups of probabilistic indices. The deterministic indices define the sets of the boundary states of MSS. The other group of indices reveals the probability of system failure or it's repairing. Indices of third group examine the influence of modifications of every system component to the system reliability. But two probabilistic groups are applied only. The deterministic indices have a very high dimension for real application. However it is necessary to note, DRI were determined by structure function only and the components states were not taken into account in [7 - 9]. This lack removes in this paper and DRI are calculated by the probability of components states also. So the algorithm for calculation DRI improves. II. BASIC CONCEPTIONS
A. Structure Function of the MSS We use the MSS model for reliability analysis of network of n nodes (components). MSS and its components have m reliability states (level to performance a task) from the complete failure (it is 0) to the perfect functioning (it is m-1) [10]. The system components are denoted as xi (i = 1, ..., n).
1-4244-0049-X/05/$20.00 (C2005 IEEE 1224
The system reliability (system state) is depended of its components state and is defined by the structure function [3, 10]:
i(xl,
..., x) =
0(X): {0, ..., m-1}_In
{0, ..., m-1}. (1)
We investigate the coherent MSS in this paper therefore [2, 3]: 0 (s, ..., s) = s, 0 (0, ..., 0) = 0 and x < x',
A(x) < AX').
The structure function (1) is interpreted as the Multiple-Valued Logic (MVL) function [7, 10]. This admission needs to use the MVL mathematical tools. In particular, the Differential Calculus allows investigating the dynamic properties of the MVL function. B. Direct Partial Logic Derivatives
The dynamic properties of MVL function are revealed through Logic Deferential Calculation and Direct Partial Logic Derivatives in particular. A Direct Partial Logic Derivative a /K-*k)Daxi(a-*b) of a MVL function sb(x) of n variables with respect to variable xi reflects the fact of changing of function fromj to k when the value of variable xi is changing from a to b [1 1 ]:
a O-k) ax i (a-b)=
m - 11 if 0(a,, x) = j & 0(b,, x) =
0,
in the other case
k(2)
Derivative aD5(o-k)Daxj(o-m-l). Secondly it is increase of component state, that is described in terms as ao5(0-l)1ax)(c-*c+l). However, the first variant is the more important for application. So, for analysis of the MSS dynamic behavior is used Direct Partial Logic Derivative
ab(0-*O)IDxi(a-ma-i), ace {1, 2, ..., m-I },
a sa(0ok)1axj(O->m-1 ) ke { 1, 2, ..., ml}-1.
Direct Partial Logic Derivatives allow to analyze dynamic properties of the MSS, which is submitted as structural function (1). III. DYNAMIC RELIABILITY INDICES
A. Component Dynamic Reliability Indices
Component Dynamic Reliability Indices (CDRI) are a probability evaluation of influence of i-th a system component on possibility of system failure or repair. A point of view of system reliability the unstable components are determined by these indices. Definition 1. CDRI are probabilities of MSS failure or repairing at a modification of a state of i-th system component:
where i(si, x) i(xl,..., xi-,,s,,xi+1,.. ., xn), se {1, .. .,Im-1}. For example, the Direct Partial Logic Derivative of the MVL function sb(x) (m=3, n=3) ax(0-*1)1Dxi(0-*2) (which corresponds to description of MSS 2-out-of-3) lists in Table 1. =
000 00 1 002 010 011 012 020 021 022
bX) a 0 (O-1)
)x 1 (0->2)
0 0 0 0 1 1 0 1
2
0 2 0 2 0 0 0 0 0
X1 X2 X3
100
101
102
1 10 1 1 1 112
120 121 122
bX) a 0 (O-1)
ax 1 (0->2)
0 1 1 1 1 1 1 1 2
0 0 0 0 0 0 0 0 0
X1 X2 X3
200 201 202 210 211 212 220 221 222
t(X) 0 (->)I
ix 1 (0>2'
0 1 2 1 1 2 2 2 2
Pfi)
m-1 =
a=l
P(i) ao-l,
(5)
m-1
pr(i) =
0SPi)Om-l k=l
(6)
S
where P(i)"7i. 1is probability of the system failure if i-th component state changes from a to (a-i) and P(i)8im_- is probability of the system repairing if i-th component is replaced, which calculate by next equations:
TABLE 1: THE EXAMPLE OF THE STRUCTURE FUNCTION OF MSS 2-OUT-OF-3 AND THE DIRECT PARTIAL LOGIC DERIVATIVE X1 X2 X3
(3) (4)
0 0
P(i)aXa-l = P
0 0 0 0 0 0 0
p(io
