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DETERMINATION OF MEAN TIME TO FAILURE OF A NETWORK CONSISTING OF
IDENTICAL NON-REPAIRABLE ELEMENTS
•
О.А. Tkachev
Institute of mathematics and informatics Of Moscow City Pedagogical University. Russia, 127521, Moscow, Sheremetyevskaya st, 29. e-mail: [email protected]
ABSTRACT
It is suggested the analytical model permitting to get expression for determination of mean time to failure of a network consisting of identical non-repairable elements that fail independently of one another and have exponential distribution of time to failure. To determine values of obtained expressions it is necessary to determine probability of network failure in failure of defined quantity of its elements. This factor may be determined exactly with analysis of all possible combinations of failed elements or approximately with Monte-Carlo method.
1. INTRODUCTION
In assessment of reliability of networks consisting of non-repairable elements, determination of mean time to failure is of great interest.
In suggested work it is considered Markov process, states of which are characterized with quantity of failed elements and the state of network. The analysis of this process permits obtaining of analytical expression for mean time to failure of networks consisting of identical non-repairable elements. It is supposed that network elements fail independently one on another and have exponential distribution of time to failure.
In considered examples we will use connectivity of a network as criterion of its operability, however, obtained expressions are true for other criterion of its operability.
2. RELIABILITY MODEL OF NETWORK CONSISTING OF IDENTICAL NON-REPAIPABLE ELEMENTS
We supposed that network nods are absolutely reliable, and edges are identical in reliability, fail independently of one another and have exponential distribution of time to failure.
We will use probability of network failure due to i elements failure as main network parameter permitting us to determine values of network reliability factor under consideration. We will denote this factor as - Zi.
Value of Zi is equal to ratio quantity of non-operable states of the network in case failure of i elements (Yi) to the total quantity of possible combinations of i elements of n, where n is the quantity of network elements.
Z =t\ (!)
I n I
V i j
Let us determine values Yi for network presented in Figure 1. Since considered network is biconnected, therefore one edge moving off cannot break its connectivity. Hence Yi=0. For definition of Y2 and Y3 we consider all possible states of the network when 2 and 3 edges are moved off respectively. It is possible to define that 2 cutsetss of capacity 2, and 14 cutsetss of
capacity 3 are within the considered network, hence Y2=2, Y3=14. Any combination of i edges at i
> 3 is a cutsets, hence Y =
v1J
for i>3.
Values Zi for this network: Zi=0, Z2=2/21, Z3=14/35, Z4=Zs=Z«=Z7=1. More detailed analysis of this network is described in article Tkachev (2011).
For small n the values of Yi can be determined by means of enumeration of all possible
network states. For great values of n it is necessary to use Monte-Carlo method.
t
Figurel: Example of a network
Let us denote with Z*i the probability of network failure in failure of i element, if the network was operable for i-1 failed elements. Correlation of values Z*i h Zi was established in article Tkachev (1983).
Let us consider Markov chain describing change of network states at the moments of its elements failure.
If for state of i-1 failed elements the network is operable, then for failure of i-element it transforms with probability Z*i to non-operable state, or with probability 1- Z*i it will remain in operable state.
If in presence of i-1 failed elements the network was non- operable, than for the failure of i-element it will remain with probability of 1 in non-operable state.
Transition diagram of considered process is shown in Figure 2. States i' correspond to network operable states, and states i" correspond to non-operable states of network for failure of i elements.
-1——-——
*{£)------►(m)— ' »(T)------H^r)"
Figure 2: Network state transition diagram
Let us denote Pi' - probability of operable state, and Pi" -probability of non-operable state of network in the case of failure of i elements. From definition of Zi it follows
P = 1 - Z
(2)
P = Z (3)
In accordance with diagram (Figure 2), the network state transition can be written as
P,= P,iy(1 - Zi)
(4)
wherefrom
P - P
Z* (/■-!)' 1 (■)'
P
(i-1)'
Substituting in (5) values Pi' h Pi" from (2) and (3) we obtain
Z *
Besides, from expression (4) it follows
Z, - Zi-i 1 - Z.--1
P' = Ui-Z*j _ 1 - Zi
(5)
(6)
(7)
j _i
For determination of mean time to failure we will consider continuous Markov process, describing system behavior in time. Process states are preset with quantity of failed elements and states of network. State transition diagram of this process is shown in Figure 3.
_ _ _ _ -^rr^-in-iiii^i^ _ _ _ _ ^
(n-i+1)X(z i)
—►(m)——Hn-D'
nX(1-z j) (n-1)X(1-z 2) (n-i+1)X(1-z i)
Figure 3: The Markov process of network state transition.
Let us denote elements failure rate as X. Let in some moment of time to be i failed elements and the network to be operable. In infinitely small time interval At it can occur one of the following events:
- network will remain in operable state.
Probability of this event (l-(n-i)X, At);
- One more element will fail and the network will transit to non-operable state.
Probability of this event (1-(n-i)X Z*i+1 At);
- One more element will fail but the network will remain in operable state.
Probability of this event (1-(n-i)X (1-Z*i+i) At).
To simplify analytic calculations we will substitute a set of non-operable states with one absorbing state Figure 4.
Figure 4: The Markov process of network state transition with absorbing state
0
Probabilities of process being in different states in arbitrary moment of time Pi(t) can be found by means of solving the following differential equation system:
dP (t)
0() = -A0 P0(t) - X P0(t) = -A0 P0(t)
dt dp (t) dt
Ai - (t) - ^ (t) - Ap (t) = A - (t) -A p (t) (8)
^ — X A P (t)
dt i—0
Here Xi=(n-i)X X'i=(1- Z*i+i) X,
xMi=z*i+i
k - is the maximum quantity of elements, after the failure of which the network can be operable.
Let us solve differential equation system (8) using Laplace transformation under initial conditions P0 (0)=1, Pi(0)=0 V; ^ 0. Let us designate
F (5) = J P(t )e -st dt
0
Then differential equation system (8) is deduced into algebraic equation system relatively
to F(s).
sF0(s) = 1 - X0 F0( s)
sF (s) = a;-1 Fi-i(s) - ¿F (s) (9)
F (s) = £ Xf (s)
;=0
Wherefrom we have
F)( s) = — s + X0
F(s) = X-F-i(s) (10)
s + X
- k
Fn (s) — - X A' F (s)
k
s)
s ;=0
This solution permits obtaining of expression for mean time to failure. According to definition Kozlov&Ushakov (1975).
T — £ F (s)|s — 0 (11)
i—0
Expanding (10) we obtain
Fo(0) = -1 = \ A nA
(12)
A n(1 -Z*)
F (0) = -f1 F _i(0) = ' 71
A (n - i)A
(13)
Using (7) the expression (13) may be simplified
n (1 -Z *)
F (0)- ;-1 -
1 - z.
(n - i)A (n - l)A
Then from (11) it follows
(14)
T - 1 y (1 - Zj)
A j-0 n -1
(15)
and X.
Expression (15) permits determine the value of mean time to failure for known values Zi
Let us determine value T for network, shown in Figure 1, for X=0,01 (1/hour)
11 19 1 21 1 269
T = 100(— + - + (—) - + (—) -) = 100*-* 64,047(hour)
7 6 21 5 35 4 420
3. COMPARISON WITH KNOWN RESULTS
To check obtained expression let us determine mean time to failure of a system, for which there are known analytical assessments Figure 5.
Figure 5: Series-parallel system
In Gnedenko&Belyayev&Solovyov (1965) there can be found following expressions. Probability of failure-safe operation of system consisting of 2 parallel identical elements.
P) = 1 - (1 - *"At )2 = 2e
(t)
-At -2 At
e
Probability of failure-safe operation of system consisting of 3 parallel identical, series-connected subsystems.
Pc(t) = (2e"At - e"2At )3 = 8e"JAt - 12e"4At + 6e"3At - e
Pc(t) - (P(t))
-3 At
-4 At
-5 At
-6 At
c (t)
3
Mean time to failure is equal to
r =f p ,t ,d t = 1(8 -12+6 -1)=I(42)=0,71
0 c(t) X 3 4 5 6 X 60 X
For determination of mean time to failure with use of expression (15), it is necessary to determine values Z,. For considered system they are equal to: Z1=0,0; Z2=3/15; Z3=12/20; Z4 = Z5 = Z6=1.
Substituting these values into expression (15), we obtain:
r = 1(1+1+(12)!+(±)1) = 1(42) = 0,71
X 6 5 1y 4 2a 3 X 60 X
This example also shows that suggested method can be used for determination of mean time to failure of complicated series-parallel and parallel-series systems consisting of identical non-repairable elements.
4. EXAMPLE OF NETWORKS RELIABILITY ANALYSIS
Let us denote the number of nodes by m and the number of edges by n. Let us consider network with parameters m=20, n=24, Figure 6. Values Zi of this network can be determined by means of enumeration of all possible network states. Calculation results are shown in table 5.
Table 5. Values Yi and Zi for network on Figure 6.
i 1 2 3 4 5 i>=6
Yi 0 12 328 4082 29960
Zi 0 0,043478 0,162055 0,384252 0,704875 1
Suggested method can be used for other criteria of a network operability. For example: the network is operable, if number of connected nodes >= m-k.
Obtained results can also be used in this case, but it is necessary to make some corresponding corrections in determination algorithm of values Zi. In table 6 there are listed values Zi for different values of k for network shown in Figure 6. Determination of values Zi for k>0 was carried out with Monte-Carlo method. Number of tests amounted 106.
Table 6. Values of Zi for different values of k for network on Figure 6.
i / k 0 2 4 6 8 10
1 0,0000000 0,00000 0,00000 0,00000 0,00000 0,00000
2 0,0434783 0,00000 0,00000 0,00000 0,00000 0,00000
3 0,1620553 0,01581 0,00000 0,00000 0,00000 0,00000
4 0,3841521 0,07980 0,01498 0,00435 0,00296 0,00000
5 0,7048748 0,25247 0,08813 0,03329 0,01644 0,00000
6 1,0000000 0,56194 0,27850 0,13070 0,05976 0,00324
7 1,0000000 0,86648 0,56942 0,33252 0,17912 0,04445
8 1,0000000 1,00000 0,82642 0,58123 0,36764 0,16783
9 1,0000000 1,00000 0,96544 0,79849 0,57894 0,34029
10 1,0000000 1,00000 1,00000 0,93211 0,76173 0,52593
11 1,0000000 1,00000 1,00000 0,98807 0,89262 0,69336
12 1,0000000 1,00000 1,00000 1,00000 0,96470 0,82566
13 1,0000000 1,00000 1,00000 1,00000 0,99360 0,91743
14 1,0000000 1,00000 1,00000 1,00000 1,00000 0,97046
15 1,0000000 1,00000 1,00000 1,00000 1,00000 0,99403
a 0,214850615 0,29501 0,354027 0,411813 0,476755 0,571528
a=± ^Z)
,=0 n -, T= a*X
It should be noticed that obtained results can be used in the case when nods failed, though edges are absolutely reliable. Node failure can be modeled by removal of all edges coming from this node.
5. CONCLUSION
It was obtained the analytical expression for determination of mean time to failure of networks consisting of identical non-repairable elements that fail independently of one another and have exponential distribution of time to failure. And as the nods, so the edges can be assumed as absolutely reliable.
Suggested method can also be used for reliability assessment of complicated series-parallel systems.
REFERENCES
1. B. V. Gnedenko, Y. K. Belyayev, A. D. Solovyov (1965). Mathematical methods in reliability theory, Nauka, Moscow. (In Russian).
2. Kozlov B.A. , Ushakov I.A. (1975). Handbook on Reliability of Radio and Automation Systems. Moscow: Soviet Radio . (In Russian).
3. Tkachev O. A. (1983). Application of Markov chains for the reliability analysis of systems with a complex structure. "Cybernetics" magazine, No. 5 1983. (In Russian). Cybernetics and Systems Analysis Volume 19, Number 5, 709-716, DOI: 10.1007/BF01068770. (In English).
4. Tkachev O. (2011). Analytical estimation of reliability of networks consisting of identical elements. Reliability - Theory & Application 2(21): 9-20.
