Computing the reliability of a complex network using two techniques

To find the reliability of any complex network, we need to convert it into a simple network to facilitate the process of reliability calculation. In this paper we simplify a complex network into a simple network by using two techniques, in the first technique “reduction method” the complex network was converted into a parallel network, while in the second technique “delta - star method”, the complex network has been transformed into a series network. The 2nd technique is based on the 1st technique in finding the reliability of a subnetwork. At the end of the research, a comparison between these two methods was made.


Introduction
Reliability has come to be an extra problem in latest years, because advanced technology commercial tactics with ever increasing ranges of class contain maximum engineering structures nowadays [1,8]. So, reliability engineers are referred to as upon to evaluate the reliability of the machine during the layout phase of a product. Reliability concept describes the capacity of a network to complete the task for which it's far responsible at a specific time [2,11]. Network reliability evaluation gets first rate interest for the making plans, effectiveness, and protection of many actual worldwide networks, including computers, communications, electrical circuits, aircraft, linear accelerators, and electricity networks [3,4]. The devices of a network are subject to random failures, as many groups and institutions become dependent upon networked computing packages. The failure of any component of a network might also additionally right away have an effect on the operation of a community, for that reason the probability of every component of a network is a totally important whilst thinking about the reliability of a network. Hence the reliability consideration is an essential aspect in networked computing [5,6]. There are number of ways to calculate the reliability of complex network. Such are, for examples, reduction to parallel elements technique and delta-star transformation which are depend on the graph of a network. The above-mentioned methods reduce complex networks and convert them into simple networks for easy reliability calculation, as will be evidenced by our discussion of those methods [7,8]. In this article, although we refer to our previous work [8][9][10][11][12][13][14] and [17], the emphasis is different from other authors'. We use reduction and delta star techniques to determine the reliability of a given network. The aim of this paper is to reduce the complex network in order to simplify the network to the simplest method, and these strategies help one to deal with a given network as the simplest way to learn certain things that are connected to the work properly or optimize the system, such as reliability allocation, reliability value, redundancy, and so on, as potential work.

Minimal path
A path is a chain of lines which connects the beginning node of the network to its end node. A minimal path is a path from which no line can be eliminated without disconnecting the link it creates between the start node and the stop node [6,7].

Structure function
Consider a network with components, each component of them could have two feasible states success and failure. The state of component is given by the binary variable The state of this network can be described by the binary function where which is called the structure function of the network [7][8][9][10].
The network that is working if and only if all of its components are success, is called a series structure, and the structure function of it is given by: Assume that is the reliability of a component . Then the reliability of the network with components in series is (2) While, the network that is working whenever at least one of its components is success, is called a parallel structure. The structure function of this network is given by: So that the reliability of the network with components in parallel is given by [11,13]: Now we'll use two methods to convert the complex network in figure 1 into a simple network to find the structure function and reliability of it

Reduction to parallel elements Footnotes
In this technique we systematically replace each series minimal path by an equivalent single path, and finally reduce the given network to another with two parallel elements [10][11][12]. We can convert the complex network to a series-parallel one depending on minimal path sets. From figure 1 the minimal paths of our complex network are , and . So, our network can be becoming as in figure 2 below The complex network becomes simple as in figure 3.   (7) and (8) we get: Apply equation (3) on the right diagram of figure 4 to get the structure function of our network: which is means (11) So, the reliability of our complex network is given by:

Delta-star method
To transform a delta subnetwork to an equivalent star subnetwork we want to derive a metamorphosis method for equating the different components to each different between the diverse terminals [7,15]. Consider figure 5 below.

Figure 5. Delta to Star transformation
Compare the three components of delta with three components of star, then applying the equations and to get three relations: Assume that and Now, solving three equations above to get delta-star relationships (16) Thus, our complex network in figure 1 becomes a simple network as in the figure 6 2nd  Figure 6. Simple network Now applying the reduction method on the subnetwork resulting from deletion from figure 6. The minimal paths of this subnetwork are and , so that this subnetwork converts to a series-parallel subnetwork and can drown as in figure 7

Figure 7. Simple series-parallel subnetwork
Note that in 1 st method we assuming that Then and as in equations (5) and (6) respectively and the subnetwork can drown as in figure 8

Figure 8. The simpler subnetwork
Now if we apply equation (1) to simplify the network on the left of figure (8), we will get an equivalent network as on the right of the figure above and we have and So, we obtain the structure function by apply the equation (3) on the network on the right as following which in turn leads to: So, the reliability of simpler subnetwork is (18)

Comparison between two techniques
In this section we are able to do a little computations with a purpose to make a comparison between these methods and discover which is better with the aid of substituting random values in equations (12) and (20)

Discuss the results
From table 1, we are able to see that the very best fee of reliability of complex network in figure 1 is 0.97 in (case 2) also, the best value of reliability of complex network is 0.85 in (case 1), while the two values are equal in (case 3). So, the best value of reliability of complex network is by using "reduction to parallel elements method".

Conclusions
In 1st technique "reduction to parallel elements", we reduced all parallel minimal paths to make the given network easier, that is the complex network has been transformed into parallel form and we noticed that the reliability was calculated simply. While in the 2nd technique "delta-star", the complex network has been transformed into series form, the network simplified techniques have been applied to calculate its reliability. We saw the ease of dealing with complex networks by using these two methods. On the other hand, in the first and second cases, the reliability value is better by using 1st technique "reduction to parallel elements", while in the third case, the reliability value is the same in both techniques.