A genetic algorithm for the arrival probability in the stochastic networks
 Gholam H. Shirdel^{1}Email author and
 Mohsen Abdolhosseinzadeh^{1}
Received: 7 February 2016
Accepted: 3 May 2016
Published: 20 May 2016
Abstract
A genetic algorithm is presented to find the arrival probability in a directed acyclic network with stochastic parameters, that gives more reliability of transmission flow in delay sensitive networks. Some subnetworks are extracted from the original network, and a connection is established between the original source node and the original destination node by randomly selecting some local source and the local destination nodes. The connections are sorted according to their arrival probabilities and the best established connection is determined with the maximum arrival probability. There is an established discrete time Markov chain in the network. The arrival probability to a given destination node from a given source node in the network is defined as the multistep transition probability of the absorbtion in the final state of the established Markov chain. The proposed method is applicable on large stochastic networks, where the previous methods were not. The effectiveness of the proposed method is illustrated by some numerical results with perfect fitness values of the proposed genetic algorithm.
Keywords
Mathematics Subject Classification
Background
Established connections in networks should be reliable to transmit flow from a source node to a destination node especially in delay sensitive networks. Determination of the best connection is one of the most important problems to avoid traffic congestion in networks. So, the arrival probability is used to evaluate the reliability of an established connection and it has been considered as an optimality index of the stochastic shortest path length (Bertsekas and Tsitsiklis 1991; Fan et al. 2005; Kulkarni 1986). The stochastic shortest path problem (SSP) is defined in a network with stochastic parameters. Liu (2010) produced the SSP where arc lengths were assumed to be uncertain variables. Pattanamekar et al. (2003) considered the individual travel time variance and the mean travel time forecasting error. Also, Hutson and Shier (2009) and Rasteiro and Anjo (2004) supposed two criteria, mean and variance of path length. Fan et al. (2005) assumed that each link to be congested or uncongested, with known conditional probabilities for link travel times. Wu et al. (2004) modeled a stochastic and timedependent network with discrete probability distributed arc weights. The considered model in this paper is a directed acyclic stochastic network with known discrete distribution probabilities of leaving or waiting in nodes.
Our criterion to evaluation of the connections from the source node toward the destination node in the network is presented as the arrival probability, which is obtained by the established discrete time Markov chain (DTMC) in the network. Liu (2010) applied models according to the decision criteria and converted them into deterministic programming problems. Hutson and Shier (2009) and Rasteiro and Anjo (2004) determined the path with maximum expected value of a utility function. Fan et al. (2005) proposed a procedure for dynamic routing policies. Nie and Fan (2006) formulated the stochastic ontime arrival problem with dynamic programming, and Fan et al. (2005) minimized the expected travel time. In this paper, the maximum arrival probability from a given source node to a given destination node is computed according to known discrete distribution probabilities of leaving or waiting in nodes, and a DTMC stochastic process is used to model the problem rather than dynamic programming or stochastic programming.
Kulkarni (1986) developed a method based on a continuous time Markov chain (CTMC) to compute the distribution function of the shortest path length. Azaron and Modarres (2005) applied Kulkarni’s method to queuing networks. Thomas and White (2007) modeled the problem of constructing a minimum expected total cost route as a Markov decision process. They wanted to respond to dissipated congestion over time according to some known probability distribution. From the static viewpoint, our studied model is related to the dynamic timeevolving networks recently studied by Shang (2015a, b), however problems addressed are from a different field.
The stochastic topology of networks motivated us to consider the arrival probability from the source node toward the destination node. So, the arcs of the network could be congested probably, that it is commonly happen in communication and transportation networks where the connecting arcs of some nodes are unable to transmit flow. The stochastic topology of the network causes several unstable connections between nodes; however, the physical topology of the network determines possible and impossible connections between pairs of nodes. The leaving distribution probability from one node toward another node is known as the probability that their connected arc to be uncongested. A DTMC with an absorbing state is established and the transition matrix is obtained. Two conditions at any state of the established DTMC are assumed: departing from the current state to a new state when a larger labeled node is visited in the original network, or waiting in the current state with expecting better conditions. Then, the probability of arrival the destination node from the source node is computed as the multistep transition probability from the initial state to the absorbing state in DTMC.
The proposed method applies DTMC and a genetic algorithm is produced to obtain an acceptable solution, that it applies small locally created state spaces instead of the original large state space. The computed arrival probability describes the overall situation of the network to transmit flow from the source toward the destination; while, the previous works focused on a specific path Liu (2010), Hutson and Shier (2009), Rasteiro and Anjo (2004).
The remain of the paper is organized as follows. Section “The stochastic topology of the network” consists of some preliminary definitions and assumptions for the considered model of the stochastic network. The established DTMC and computational method for the arrival probability is presented in “The established discrete time Markov chain” section. In “A genetic algorithm to find the arrival probability” section, a genetic algorithm is presented to approximate the arrival probability. In section “Numerical results”, some implementations of the proposed method on the networks with large size of nodes and arcs are provided.
The stochastic topology of the network
Directed acyclic networks are considered for various applications, some of typical examples are related in the following: citation networks in information sciences, phylogenetic networks in biology, data structures in computer science and engineering, acyclic graphs in pure mathematics, random graphs and Bayesian networks in statistics, and etc. (for more details see Ahuja et al. 1993; Karrer and Newman 2009). Recently, Shang (2014) considered group consensus problems in generic linear multiagent systems with directed information flow under directed fixed interaction topology and randomly switching, where the underlying networks are governed by a continuoustime Markov process.
Let network \(G=(N,A)\), with node set N and arc set A, be a directed acyclic network. Then, we can label nodes in a topological order such that for any \((i,j)\in A\), \(i<j\) (Ahuja et al. 1993). The physical topology for any \((i,j)\in A\) shows the connection of nodes \(i,j\in N\). Actually, the physical topology shows the possibility of communication between nodes in the network. To model the stochastic topology of a network think about the transportation networks, where there are some physical connections between nodes but we cannot traverse anymore toward the destination node because of probable congestion. Network G has a stochastic topology if there are some facilities in the network but it is not possible to use them continuously. So, the existence of any arc \((i,j)\in A\) does not mean there is a stable communication between nodes \(i,j\in N\) all the time (it could be probably congested). For any node i, it is supposed that the uniform distribution probabilities of leaving arcs (i, j) to be uncongested are known.
Figure 1 shows the example network with its topological ordered nodes and it is the initial physical topology of the network. The numbers on arcs show the leaving probabilities \(q_{ij}\). Node 1 is the source node and node 5 is the destination node. It is not possible to traverse arc (1, 4) because it does not exist in the physical topology of the example network. However, the arcs in the physical topology could be congested according to the known distribution probabilities.
The established discrete time Markov chain
The state space of the example network
State space  Current nodes 

\(S_1\)  {1} 
\(S_2\)  {1, 2} 
\(S_3\)  {1, 3} 
\(S_4\)  {1, 2, 3} 
\(S_5\)  {1, 2, 4} 
\(S_6\)  {1, 3, 4} 
\(S_7\)  {1, 2, 3, 4} 
\(S_8\)  {1, 2, 3, 4, 5} 
 1.:

By arriving the destination node, the process can traverse neither any node nor any arc (i.e. the absorbing state)
 2.:

The new state is created, if a new node is added to the current state nodes
 3.:

According to the current state, it is allowed to reach only one node during transition to a new state.
The transition and the wait probabilities

\(0\le p_{kl}\le 1\,\,\) for \(k=1,2,...,S\) and \(l=1,2,...,S\)

\(\sum _l{p_{kl}}=1\,\,\), for \(k=1,2,...,S\).
Theorem 1
If \(p_{kl}\) is kl element of matrix P, that \(k \ne l\), \(l < S\) and \(S_{k}=\{v_{0}=1,...,v_{m}\}\) is the current state, then the transition probability from state \(S_{k}\) to state \(S_{l}\) is computed as follow
Proof
Since, it is not allowed to traverse from one state to the previous states (Assumption 2), then necessarily \(p_{kl}=0\), for \(l\ <\ k\). Otherwise, suppose \(l\ >\ k\), during transition from the current state \(S_k\) to the new state \(S_l\), it should be reached just one node other than the nodes of the current state, so \(S_l\backslash S_k=1\), \(v\in S_k\), and \(w\in S_l\backslash S_k\) are held by Assumptions 2 and 3. Two components of \(p_{kl}\) formula should be computed.
In the last node \(v_m\) of the current state \(S_k\), it is possible to wait in \(v_m\) with probability \(q_{v_mv_m}\). Notice, it is not possible to wait in the other nodes \(v\in S_k\backslash \{v_m\}\) because it should be leaved to construct the current state, however it is not necessary for node \(v_m\) with the largest label (leaving \(v_m\) leads to a new node, and therefore results in a new state). If \(w\in S_l\backslash S_k\), then one or all of events \(E_{vw}\) (i.e. traversing a connecting arc between a node of the current state and another node of the new state) can happen for \(\left( v,w\right) \in \Psi\), and the arrival probability of node \(w \in S_l\) from the current state \(S_k\) is equal to \(Pr[\bigcup _{\left( v,w\right) \in \Psi }{E_{vw}}]\). The collection probability should be computed because of deferent representations of the new state (for example see Fig. 2). Then, the nodes of the current state \(v\in S_k\backslash \{v_m\}\) (while waiting in \(v_m\)) should be prevented from reaching other nodes \(u\notin S_k\) and \(u\ne w\) (Assumption 3), so arcs \(\left( v,u\right)\) are not allowed to traverse and they are excluded simultaneously, thus it is equal to \(\prod _{\left( v,w\right) \in \Psi }{(1\sum _{\begin{array}{c} \left( v,u\right) \in A \\ u\ne w,u\notin S_k \end{array}}{q_{vu}))}}\). The other possibility in node \(v_m\) that is leaving it toward the new node \(w\in S_l\backslash S_k\) with probability \(q_{v_mw}\). \(\square\)
For example, in the established DTMC of the example network, the transition probability \(p_{47}\) is computed by the constructed components as shown in Fig. 4; and it is \(Pr(E_{14}\cup E_{24})\times (1q_{15})(1q_{25})\times q_{33}+q_{34}\), where \(Pr(E_{14}\cup E_{24})= q_{14}+q_{24}q_{14}q_{24}\), however \(q_{14}=q_{15}=q_{25}=0\) as shown in Fig. 1, then \(p_{47}=q_{33}\times q_{24}+q_{34}\). It is possible to wait in node 3 but not other nodes of the current state \(S_4=\{1,2,3\}\); where, by traversing arc (2, 4) or (3, 4) the new state \(S_7=\{1,2,3,4\}\) is revealed.
Theorem 2 describes the transition probabilities to the absorbing state \(S_{S}\), and they are the last column of the transition matrix P.
Theorem 2
Proof
To compute the transition probabilities \(p_{kS}\), for \(k=1,2,...,S1\) it should be noticed the final state is the absorbing state \(S_{S}=\{1,2,3,...,N\}\) containing all nodes of the network, and the stochastic process does not progress any more (Assumption 1). So, it is sufficient to consider leaving arcs \((v,v_n)\) from \(v\in S_k\), the nodes of the current state, toward the destination node \(v_n\in S_{S}\). Then, one or all of events \(E_{vv_n}\) (i.e. traversing a connecting arc between a node of the current state and the destination node of the absorbing state) can happen and the transition probability from the current state \(S_k\) to the absorbing state \(S_{S}\) is totally equal to \(Pr[\bigcup _{v\in S_k,(v,v_n)\in A}E_{vv_n}]\). The collection probability should be computed because of deferent representations of the states (for example see Fig. 2). \(\square\)
For state \(S_4\), transition probability \(p_{48}\) is obtained by \(Pr(E_{15}\cup E_{25}\cup E_{35})\), however \(q_{15}=q_{25}=0\), then \(p_{48}=q_{35}\). The wait probabilities, those are the diagonal elements of the transition matrix P, are obtained by Theorem 3.
Theorem 3
Proof
The wait probabilities \(p_{kk}\) are the complement probabilities of the transition probabilities from the current state \(S_k\), for \(k=1,2,...,S1\), toward the all departure states \(S_j\), for \(j=k+1,k+2,...,S\). Then, we have \(p_{kk}=1\sum ^{S}_{j=k+1}p_{kj}\), for \(k=1,2,...,S1\), in other word, they are the diagonal elements of matrix P, those are computed for any row \(k=1,2,...,S1\) of the transition matrix (see Ibe 2009). The absorbing state \(S_{S}\) does not have any departure state, so \(p_{SS}=1\) as the transition matrix P.□
The arrival probability
For the example network, we want to obtain the probability of the arrival node 5 from node 1. The arrival probability \(p_{18}(r)\) is obtained as shown by solid line in Fig. 5 after seven transitions. For r sufficiently large, the probabilistic behavior of DTMC becomes independent of the starting state i.e. \(Pr[X_r=S_{S}X_0=S_1]=Pr[X_r=S_{S}]\), that is the multistep transition probability (Ibe 2009). For the example network, it is 0.9994.
A genetic algorithm to find the arrival probability
The implementation of the proposed genetic algorithm
Node number  Arc number  Transition component size  Almost value  Fitness probability  Arrival space size  State 

20  5  0.9981  0.9771  127  
10  26  5  9  1.0000  0.9977  130 
29  5  0.9902  0.9249  253  
82  15  0.9973  0.9973  126,593  
20  98  15  13  0.9998  0.9998  200,593 
105  15  0.9999  0.9947  122,385  
208  22  0.9999  0.8492  
30  216  22  13  1.0000  0.9623  
228  22  1.0000  0.9488  
363  40  1.0000  0.8927  
40  372  60  11  1.0000  0.9754  
386  60  1.0000  0.7571  
609  87  1.0000  0.9719  
50  614  87  11  1.0000  0.8559  
630  75  1.0000  0.6983 
The proposed genetic algorithm needs to obtain some initial feasible solutions, so the first step is to extract some subnetworks from the original network with local source nodes and destination nodes. Extracting the subnetworks is applied by all operators of the proposed genetic algorithm and it is presented as a basic operator. We need to define three types of operators as described by Sivanandam and Deepa (2008) and Dréo et al. (2006). The initial population operator extracts some subnetworks and their required complement components to establish connections between the original source and destination nodes. Then, the crossover operator tries to obtain new populations from the current initial populations and to replace the worst ones with the best ones. To avoid probable local optimality, the mutation operator extends the search area of selecting local source and destination nodes. Therefore, the subnetworks inherit all characteristics of the original network; they are directed, acyclic and all nodes are reachable from the local source node and the local destination node is reachable form all nodes. A connection is a union of some subnetworks, those the local source node of one is the local destination node of another one and it can transmit flow from the original source node toward the original destination node.
Extracting a subnetwork from the original network

If node i is a local source node for a subnetwork, then the leaving probabilities are changed as below$$\begin{aligned} \forall v<i,\quad \forall w\in N: {\left\{ \begin{array}{ll} q_{vv}:=q_{vv}+q_{vw},&\quad q_{vw}:=0\\ q_{ww}:=q_{ww}+q_{wv},&\quad q_{wv}:=0. \end{array}\right. } \end{aligned}$$(1)

If node j is a local destination node for a subnetwork, then the weights of arcs are changed as below$$\begin{aligned} \forall v>j, \quad \forall w\in N: {\left\{ \begin{array}{ll} q_{vv}:=q_{vv}+q_{vw},&\quad q_{vw}:=0\\ q_{ww}:=q_{ww}+q_{wv},&\quad q_{wv}:=0. \end{array}\right. } \end{aligned}$$(2)
Extracting subnetwork operator
 1.
For any node v satisfying (1) and (2) omit node v from the node set and row v and column v form the arc weight matrix.
 2.
While there exists any zero row k (column k) except the row and the column of the local destination node, omit node k from the current node set and row k and column k from the current arc weight matrix.
The initial population operator
To accomplish a connection between the original source and destination nodes, it is required to produce some complement components. Obviously, there should exist such components those could connect together, otherwise there is not feasible solution for the problem. The proposed genetic algorithm starts with some randomly created initial components; then, midcomponents are obtained such that a connection between the original source node and the original destination node is established. For producing complement components, the local destination node of a subnetwork and the local source node of another one are selected according to their labels, then a midcomponent is created with these successive local source and destination nodes; especially, the original source node and the original destination node are considered as components themselves.
The initial population operator
 1.
Choose a local source node and a local destination node randomly.
 2.
Extract the subnetwork with local source and destination nodes (initial component).
 3.
Put the initial local source node as \(s_0\), and the initial local destination node as \(d_0\).
 4.
Create a midcomponent between the original source node and \(s_0\): put \(s_0\) as the local destination node, then select randomly the local source node i for \(i<\ s_0\); if \(i\ne s\), then put \(s_0=i\) and repeat.
 5.
Create a midcomponent between \(d_0\) and the original destination node: put \(d_0\) as the local source node, then select randomly the local destination node i for \(i>\ d_0\); if \(i\ne d\), then put \(d_0=i\) and repeat.
The crossover operator
To produce new populations of the current populations and to improve the current optimality, we define the crossover operator similar to the initial population operator, except that the local source and the destination are selected form the nodes belonging to subnetworks. To produce distinct populations, the algorithm chooses one node from the set \(\pi _1=\{\text {the nodes belonging to the initial population}\ i\}\{\text {the nodes belonging to the initial population}\ j\}\) and another node from the set \(\pi _2=\{\hbox {the nodes belonging to the initial population } j\}\{\hbox {the nodes belonging to the initial population }i\}, i\ne j\), if none of them is empty; otherwise, one of the obtained initial populations is the subset of the other one and the algorithm extends the search area by the mutation operator.
The crossover operator
 1.
If none of sets \(\pi _1\) and \(\pi _2\) is empty then randomly choose one node belonging to \(\pi _1\) and one node belonging to \(\pi _2\), as the local source node and the local destination node, according to their labels.
 2.
Extract a subnetwork and apply the initial population operator on the new extracted subnetworks.
The mutation operator
To avoid local optimality the mutation operator extends the search area to other parts of the network. So, if the current populations do not change during the crossover operator (one of sets \(\pi _1\) and \(\pi _2\) is empty), the mutation operator extends the search area of the local source and destination nodes detection, and it tries to change the current populations and improves the arrival probability.
The mutation operator
 1.
Select two nodes randomly belonging to \(N(\{\hbox {initial population }i\}\bigcup \{\hbox {initial population}\, j\})\) as the local source node and the local destination node.
 2.
Extract a subnetwork and apply the initial population operator on the new extracted subnetworks.
Fitness function
The selection operator through the algorithm is a ranked selection operator. Connections are sorted from the worst one to the best one, then the proposed genetic algorithm replaces the worst one with the new connections of the improved fitness values. After the multistep transition probabilities of the all components contained in any population were computed, a path is constructed between the original source node and the original destination node through the local destination nodes of the initial population. Then, by the similar process to the original network, the arrival probability is computed for the path and it is recorded as the fitness value of the population. In any iteration, the above process is repeated for the optimal populations with the maximum arrival probabilities. Output of the proposed genetic algorithm for the example network is shown by dash line in Fig. 5. The obtained arrival probability is 0.9975 and it is given after 7 transitions, with 5 initial populations and its fitness value is 1.0000.
Numerical results
Some implementations of the proposed method on large networks are presented in this section (see Table 2). The networks are acyclic directed networks and there is a path from each node to the destination node. The leaving and waiting probabilities of nodes are random numbers produced by the uniform distribution probability. Then, the arrival probability is computed for the established DTMC. All of the experiments are coded in MATLAB R2008a and they are performed on Dell Latitude E5500 (Intel(R) Core(TM) 2 Duo CPU 2.53 GHz, 1 GB memory).
The implementations of the proposed genetic algorithm according to the arc numbers are shown in Fig. 7. Results show the numbers of the components containing in the constructed subnetworks and the numbers of transitions, both are two most important parameters to obtain the arrival probability by the proposed genetic algorithm.
Conclusion
We considered an established discrete time Markov chain stochastic process over directed acyclic networks. The arrival probability from a given source node to a given destination node was computed according to the probability of transition from the initial state to the absorbing state by multistep transition probability in DTMC. A genetic algorithm was proposed for large networks, where the state space of the established DTMC grew as rapidly as exponentially. However, the proposed genetic algorithm applied small locally created state spaces instead of the gigantic large state space. Numerical results showed efficiency of the proposed method to obtain the multistep transition probability that the destination node is accessible for the first time. Extension of described model on the continuous time varying networks, and using the discrete nature of the proposed model to apply metaheuristic methods and reducing the computations can be considered as future works guidelines. Also, the shortest path problem with recourse where some local decisions are made during routing process and they could be evaluated by the proposed method.
Declarations
Authors' contributions
GS participated in the design and conceived of the study. MA worked out the algorithms and helped to draft the manuscript. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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