Multi-Source Multi-Sink Stochastic-Flow Networks Reliability under Time Constraints

Objectives: This study is centered on four issues related to the reliability evaluation in multi-source multi-sink networks. Each issue discusses the reliability evaluation under different condition. These conditions play an important role in determining the quickest paths used in transmitting data between source and sink nodes, with the condition that the transmission time of the quickest path does not surpass a predetermined upper bound T. Methods/Statistical Analysis:


Introduction
The quickest path problem is to obtain a routing path in a network with a minimum time to ship σ units of data from the source to the sink 1 . In 2 , the proposed method is targeted towards the situation where multi-commodities are conveyed through all disjointed minimal paths (MPs) in a network. In 3 , distributed algorithms are developed for the quickest path problem in any a synchronous communications networks. In 4 the problem is supposed as a criteria path problem, allowing the use of a very efficient algorithm, which solves the quietest path problem for all possible values of the amount of data that has to be transmitted.
The system reliability of stochastic-flow networks under time constraint is defined as the probability of sending d units of data from the source to the sink through the network within T units of time, denoted by R d,T [5][6][7] . The problem of determining the optimal routing policy with the highest system reliability discussed in 8 and 9 . Network reliability has been evaluated in the case of sending units of data through a number of MPs simultaneously under both time and budget constraints 10 . Moreover, network reliability according to the spare routing was evaluated 11 . In order to reduce transmission time, the problem of simultaneously transmitting data through multiple disjoint minimal paths was presented in 12 .
A multi-source multi-sink stochastic-flow network is an extension of the concept to multiple sources and sinks on the same network. Evaluating the system reliability of multi-source multi-sink stochastic flow networks has furthermore been addressed in [13][14][15][16] . In 13 , the optimal resource allocation problem subject to reliability maximization has been formulated and presented an algorithm to solve it. For more than one resource 14 , the optimal resource flow allocation problem has been studied and a GA was proposed to solve it. In 15 , the flow allocation problem subject to transportation cost was studied and solved using GAs. In 16 , the author modified and solved the formulation of the flow allocation problem subject to the probability of the capacity vector and transmission cost. Further, system reliability was evaluated by searching for the optimal lower boundary points In this paper, we will extend the quickest path problem to multi-source multi-sink flow networks. The presented problem has been studied under the following cases: 1. Each source node (s i ) sends the specified demand d wj (demand for resource w at sink node t j ) separately to each sink (t j ). 2. Each source node (s i ) sends the specified demands to all sink nodes (t j , j=1,2,…,through different paths that do not share any common arcs (disjoint paths). 3. Each source node (s i ) sends the specified demands to all sink nodes (t j , j=1,2,…,through joint paths that share some arcs. 4. All source nodes (s i , i=1,2,3,… ) sends multiple demands to all sink nodes (t j , j=1,2,…,) through joint paths, the general case, simultaneously transmitting.
The rest of the paper is organized as follows. Section 2 presents notations and assumptions. Section 3 presents Case A: transmitting demands separately. Section 4 describes Case B: transmitting demands through disjoint paths. Section 5 provides Case C: transmitting demands through joint paths. Case D: the general case, when transmitting multiple demands from all sources to all sinks given in Section 6. Section 7 offers our conclusions. T {t 1 ,…,t θ }: set of sink nodes.

Case A: Transmitting Demands Separately from One Source to One Sink
The following subsections describe how to calculate i.e. the reliability of transmitting a single demand from the source node to the sink . Vol

Definition of Lower Boundary Points for (d wj , T)
If X is a minimal capacity vector such that the network can send d wj units of data from the source to the sink within T units of time, then X is called a lower boundary point for (d wj , T).

Generate All Lower Boundary Points for (d wj , T).
In the following steps, for the k th MP, MP i,j,k from s i to t j , MP i,j,k = {, , …,}, we will show how to find the minimal capacity vector j i X , = (x 1 , x 2 ,…, x e ,…, x n ) such that the network sends d wj units of data within T units of time from the source s i to the sink t j .
1. For each MP ijk , determine the smallest integer v such that, 2. If , generate the system capacity vector Where x e is an element of X i,j and u is the minimal capacity of a i .

Evaluation of
are the collection of all (d wj , T)-MP i,j , and then the system reliability T d wj R is defined as follows: Several methods [16][17][18][19][20] can be used to evaluate (3), in this paper, we will use to evaluate

Illustrative Example
As an example, we consider the network in Figure 1, which has two source and two sink nodes. The arcs are numbered from a 1 to a 14 ; their capacities, corresponding probabilities and lead-time of each arc taken from 15,16 . In the following steps, we will show how to calculate R d11,T , i.e., the reliability from the source node s 1 to the sink node t 1 , where d 11 = 11 and T = 9, i.e. evaluate R 11,9 . There are three MPs from s 1 to t 1 : MP 1,1,1 = {a 1 , a 5 }, MP 1,1,2 = {a 1 , a 6 , a 9 },MP 1,1,3 = {a 2 , a 7 , a 9 }.

Case B: Transmitting Demands through Disjoint Paths
We study how to calculates 2 , wj T D R , the reliability from the source node s 1 to the sink nodes t 1 and t 2 ,where ( ) 11 12 , u wj There are five MPs from s 1 to both t 1 and t 2 shown in Table 2-3. In this case, the following constraint for each bandwidth should be satisfied when is sent through.   Where is the upper bound for bandwidth of, it is given by: Finally, each in the capacity vector X, X =(), is constructed by eq. (7), (2).
The following algorithm is used to evaluate the reliability for Case B. Given = (10, 10) and T = 11, withthe consumed capacity shown in Table 2. Table 4,5 summarizes the values of , the set of disjoint paths and respectively.  Finally, Table 6 summarizes the candidate vectors. The corresponding reliability is =0.999988. The reliability values for other sources and sinks are shown in Table 7.

Case C: Transmitting Demands Via Joint Paths from One Source to all Sinks
We study how to calculates T D w j R , 2 , the reliability from the source node s 1 to the sink node t 1 and t 2 , where. There are five MPs from s 1 to both t 1 and t 2 shown in Table 8 and 9. In this case, the following constraint for each bandwidth should satisfy Eqs.   where a * is a commonly-used arc of two or more MPs .
(7). Table 12 summarizes the candidate vectors, the corresponding = 1 1 , 2 D R 0.999988 the reliability value from s2 to both t1 and t2 shown in Table 13.

Case D: Transmitting Multiple Demands Via Joint Paths from all Sources to all Sinks
The system reliability R D, T of the multi-source multi-sink flow network can be calculated by using the inclusionexclusion rule according to the generated set of all lower boundary points for (D, T). Given D 4 = (5, 5, 5, 5) and T = 11, with the consumed capacity W 11 (a l )= W 12 (a l )=1,W 21 (a l )=W 22 (a l )=2.. Table 14

Conclusions
The study is successfully evaluated the reliability in multisource multi-sink stochastic flow networks in different situations. Given the network information (arcs capacities, probabilities, lead times), required demand single or multiple based on the source and sink nodes, and the time constraints T. Situation A, transmitting with no restriction on determining the relationship between the group of paths, joint or disjoint. Situation B, transmitting the required demands via disjoint paths situation C, transmitting demands through a group of joint paths, share one or more common arc. Finally, the general case when all sources send the required demands to the sink nodes via joint paths, evaluating, of a multi-source multi-sink flow network.