2306
Maximization Of Network Reliability Using Ann
Under Node-Link Failure Model
N.K.Barpanda, Madhumita Panda
Abstract: The network Reliability optimization problem for any type of interconnection network is to maximize the network reliability subjected some constraint such as the total cost of the network. Even though, the problem is NP-Hard, many researchers have solved this problem in different ways but with a common assumption that nodes are perfect. But, this assumption is quite unrealistic in nature. In this paper, a new method based on artificial network is proposed to solve the network reliability optimization problem considering both the nodes and links of the interconnection network to be imperfect. The problem is mapped onto an optimization Artificial Neural Network by constructing an energy function whose minimization process drives the neural network into one of its stable states. This stable state corresponds to a solution for the network reliability problem. Some standard methods are studied and modified approximately so that they would work considering the node failure. The results of these methods are compared against the result of the proposed method. The comparison strongly supports that the proposed method provides a better maximization of network reliability than its counterparts under same working environment.
Index Terms: Interconnection network, network reliability, optimization, artificial neural network
—————————— ——————————
Notations
G Probabilistic graph of the interconnection network
N No. of nodes of graph G
L No. of links of graph G
xi,j a binary digit indicating whether the node in network
G are connected or not; xi,j =1 if nodes I and j are
connected, otherwise xi,j =0
x the node connection vector of xi,j
pl link reliability
pN node reliability
E Energy function of ANN
s state of ANN
C Cost matrix of the Interconnection network
CMax The predefined total cost of the interconnection
network
R(s) Network reliability of the network at state s
a local parameter to the ANN
vi,j output of the artificial neural network at each state
1.
INTRODUCTION
Interconnection networks play an important role in designing any type of distributed accessed or telecommunication network systems as it’s the only means of providing communication among processors and/or memory modules. The reliability and costs are the two important considerations while designing such networks. The reliability is the probability that a system works satisfactorily for a predefine period of time and environment. Out of many reliability measures the network reliability may be taken as one of the
important reliability measures for analyzing the
interconnection networks. The network reliability (or All-Terminal Reliability) is defined as the probability that each pair of nodes in the network topology can be connected by at least one physical path, subject to a failure event which is largely determined by the network topology and link as well as node reliability [1]. The exact estimation of All-Terminal Network Reliability is NP-hard [2] i.e. computational effort increases exponentially with network size. However, many researchers have attempted to find the network reliability [3]. Study of [3] reveals mainly three important methods viz. Dynamic Programming, Branch and Bound technique, and Genetic Algorithm for solving such problems. Dynamic programming is not widely used for solving such problems of large sized networks as it suffers from combinatorial explosion drawback [4]. The branch and bound method [5] are only effective for finding the exact solution of a fully connected network having up to 12 nodes. Shao et al. [6] proposed the shrinking and searching algorithm for maximizing the network reliability of a distributed access network with constraint total cost. But this algorithm is not as such applicable for large size networks, such as networks having more than 20 nodes, because the size of the generated combinatorial tree is too large, which prohibits the exact calculation of network reliability. Methods proposed in [7] and [18] effectively use the genetic algorithm to calculate and/or maximize the network reliability of some specified networks. However, for large sized networks the parameters taken in genetic algorithm based approach for approximating the optimal solution are difficult to determine
_____________________
MadhumitaPanda Asst. Prof, Dept. Of CSE, SUIIT, Sambalpur University, Burla, Odisha, India
N.K.Barpanda Reader, Dept.of Electronics, Sambalpur
University, Burla, Odisha, India
2307
which can dramatically slowdown the serving process. As artificial neural network is capable of solving some NP hard problems [8], [9], [10], [11], C. Srivaree-ratana et al. [12] apply the artificial neural network for predicting the all terminal reliability of some networks. The study of literature further reveals some other methods for solving similar type of problems [13], [14], [15]. AboEIFotoh et al. [13] proposed a neural network based approach to topological optimization of communication networks, with reliability constraint. This method is found to be a good one for estimating the optimal cost of very large size networks such as networks having 50 nodes and 1225 edges within a reliability constraint. The methods discussed so for estimating and/or maximizing network reliability assumes in common that Nodes are completely reliable. How ever, this assumption becomes quite unrealistic in actual practice as both node as well as links may fail independently of each other. Under such condition, the exact calculation and/or maximization of network reliability with existing methods becomes quite cumbersome. So, in search of a new method for solving such type maximization problems, a new method based on artificial neural network is proposed in this paper. The paper is organized as follow: Section II defines the network reliability maximization problem considering both the node as well as link failure. In Section III a new method is proposed which is based on artificial neural network. A comparison of the proposed method with some existing methods is made in Section IV. Section V concludes the papers with its future scope.
2.
FORMULATION
OF
NETWORK
OPTIMIZATION PROBLEM
The Interconnection network is represented by an equivalent probabilistic graph G (N, L), where the nodes (N) represent the processors and the edges (L) represent the links among the nodes. The network reliability of the graph G can be defined as:
R(G) Prfor every pair s,tV,there isapath from stot
The main design decisions are to select the nodes and links of the network to ensure proper and reliable operation while meeting cost constraints without distorting the topological properties of the network. Mathematically, the problem can be stated as follows:
1
1 1 , , :
) (
N
i N
i j
Max j i j i x C
C to
Subject
x R Maximize
(1)
1 , 0 pl pN
3.
PROPOSED METHOD FOR SOLVING THE
NETWORK RELIABILITY OPTIMIZATION
PROBLEM
In this section, a general idea on artificial neural network (ANN) is first presented and then, a new method based on ANN is proposed. The artificial neural network (ANN) is composed of smaller neurons that are responsible for simpler computations with numerical data. The weighted connection these neurons determine the effect of one neuron on another neuron. A general architecture of an ANN is shown in Fig.1.
The artificial neural network is having the following better characteristics than its counterpart
1. Robustness and fault-tolerance
2. Flexibility i.e. the network automatically adjusts to a
new environment with out any preprogrammed instructions.
3. Collective computation
4. Ability to deal with a variety of data situations.
The network reliability optimization problem can be mapped to an artificial neural network, where the units and connection strength are identified by comparing the objective function of the problem with the energy function of the network expressed in terms of the state values of the units and connection strengths. The solution to the problem lies in determining the state of the network at the global minimum of the energy function. Expressing the objective function and constraints of the given problem in terms of the variables of energy function of an artificial neural network and identifying the states and the weight of the network in term of the variable and parameter appearing in the objective function, the energy function can be expressed as
N
i N
i N
i j
Max j i j i
i C v C
s R E
1 1
1 1 , , )
( (2)
Where is a function defined to ensure that constraints given
in Eq. (1) are satisfied i.e. (x)= y U(y), where
U(y)=
0 0
0 1
y for
y for
(3)
Taking the partial derivative of the Eq. (2), we get
N
i
s N
i N
i j
j i j i s
N
i N
i j
j i j i
i i
i i
v C v
C
s R
s E
1
1 1
1 1
, , 1 1
1 1
, ,
0
(4)
Expressing network reliability for the state si=1 and si=0 using
[16] and [17] as
R(si)=
i j i j
i j i j i
s s
dis
j N l j N l N N
i
l p p p
p
) (
1 1
) 1 ( ) 1 (
, ,
, ,
(5)
2308 N i s N i N i j j i j i s N i N i j j i j i s s dis j N l j N l N N i l s s dis j N l j N l N N i l i i i i j i j i j i j i i j i j i j i j i v C v C p p p p p p p p s E 1 1 1 1 1 , , 1 1 1 1 , , 0 ) ( 1 1 1 ) ( 1 1 0 , , , , , , , , ) 1 ( ) 1 ( ) 1 ( ) 1 (
considering link reliability pl and node reliability pN, Eq. (4) can
be expanded to the following equation:
N i s j i N i N j ij s j i N i N j ij s s dis j N l j N l N N i l s s dis j N l j N l N N i l i i i i j i j i j i j i i j i j i j i j i C C p p p p p p p p s E 1 0 , 1 1 1 , 1 1 0 ) ( 1 1 1 ) ( 1 1 1 1 ) 1 ( ) 1 ( ) 1 ( ) 1 ( , , , , , , , , (6) Where j N l j N l N N i l
ij ij ij ij ij
p p
p
p
(1 ) (1 ), , , , 1 1 (7)
From Eq. (6), it can be observed that
0 , 0 for s E i (9)
Activation of neuron is updated by the following equation
si(t+1)=si(t)+ wi(t) (10)
Where, ) (t wi = i s E (11)
As the objective is to minimize the energy function to a value
of global minima, the local parameter should be chosen a
value 1.During each state, the wi(t) term gradually
contributes in maximizing the network reliability while
satisfying the cost constraint.The following theorem
guarantees the convergence of the energy function.
Theorem 1. The energy function
N i N i N i j Max j i j ii C v C
s R E 1 1 1 1 , , )
( converges to a
stable state for the condition 0
i s E
Proof: At each state of the artificial neural network, the energy function is updated by the Eq. (10). So, at the final state of the artificial neural network,
1 , 1 1 1 N s j i N i N j ij C = 0 , 1 1 1
N s j i N i N j ij CWhich reduces the Eq.(6) to the following Eq.
0 ) 1 ( ) 1 ( ) 1 ( ) 1 ( 0 ) ( 1 1 1 ) ( 1 1 , , , , , , , ,
i j i j i j i j i i j i j i j i j i s s dis j N l j N l N N i l s s dis j N l j N l N N i l i p p p p p p p p s Eas the 1st term of the above Eq. is always greater than or
equal to the 2nd term.
So, using Eq.(10), the energy function can be written as
) (
m in ma i N
Global R s
E (12)
Or conversely, R(siN )Maximum which proves the theorem.
4.
SIMULATED RESULTS AND DISCUSSION
2309
TABLE I
COMPARISON OF THE NETWORK RELIABILITY OF SOME FULLY CONNECTED GRAPH COMPUTED BY THE PROPOSED METHOD WITH SOME EXISTING METHODS
*Unpredictable reliability due to unexpected time i.e. more than half an hour taken by CPU $Approximation
Close observations of Table-I reveal the following facts:
- The proposed method provides a better
estimation of reliability than its counterparts
- Genetic algorithm and Shrinking and search
algorithm although provides some competent reliability prediction with the proposed method
and Optinet [15]$ are not suitable for large sized
networks such as fully connected nodes with more than 30 nodes.
The proposed method maximized the network reliability of the network with 50 nodes and 1225 links with in 10 min. while
Optinet [15]$ took around 20 min. to solve the same problem,
which further strengthens the proposed method.
V. CONCLUSION
In this paper, the network optimization problem considering the node and link failures subjected to the total cost is solved efficiently by using the neural network. The simulated guarantees the robustness and accuracy of the proposed method. The link and node failure are kept to some constant but in general practice these failure decreases exponentially with respect to time. The time is not considered during the simulation in order to compare different methods. But, the work can be extended by taking time to another important factor in estimating the network reliability.
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2 0.9 239 0.8054 0.8198 0.8237
3 0.95
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4 0.9 286 0.9306 0.9410 0.9426
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0.8107 0.8200
6 0.9 251 0.8056 0.8009 0.0.8265
7 0.9 243 0.8364 0.8264 0.0.8412
8 0.95
0.95 143 0.9078 0.9056
0.9333
0.9203 0.9352
9 0.95 268 0.9263 0.9328 0.9407
10
8
0.95
28
0.9 208 0.8306 0.8360 0.8400 0.8489 0.8500
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12 0.95
0.95
173 0.8400 0.8787 0.8789 0.8900
0.8766 0.8865
13 0.9 179 0.8406 0.8770 0.8880
14 0.95 247 0.8800 0.8789 0.9004
15
9
0.95
36
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0.8423
0.8076 0.8501
16 0.9 239 0.8200 0.8100 0.8587
17 0.95
0.95 151 0.9001 0.9210
0.9288
0.9134 0.9206
18 0.9 306 0.9108 0.9277 0.9308
19
10
0.95
45
0.9 145 0.8089 0.8345
0.8435
0.8189 0.8578
20 0.9 293 0.8109 0.8245 0.8500
21 0.95
0.95 121 0.8500 0.9008 0.9203 0.8870 0.9109
22 0.9 311 0.8608 0.8900 0.9341
23
30 0.95 435 0.95 604 * 0.9087
0.9088
* 0.9207
24 0.9 732 * * 0.9308
25
40 0.95 780 0.9 1212 * 0.8909
0.8914
* 0.9045
26 0.9 1304 * * 0.9100
27
50 0.95 1225 0.95 867 * 0.9008
0.9100
* 0.9056
28 0.9 944 * * 0.9219
2310
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