# Mdpli Mitigation of Routing Congestion on Data Networks: A Quantum Game Theory Approach
## 2 Modeling the Congestion Problem  
Our goal is to minimize the total transmission time, which is formed by the sum of the routing time and the traveling time. The routing time measures how long it takes for a packet to find a path for going from origin to destination. More precisely, in our model, the routing time is a quantity proportional to the number of games a packet must play before finding its final path. The greater the number of possible paths that a packet considers, the longer the routing time. The traveling time, for its part, measures how long it takes for a packet to travel from origin to destination when the path is chosen. This means, the sum of the weights of all the edges that a final path has.  
Therefore, \(TotalTime_{i}=K_{G}G_{i}+K_{E}\sum_{e=1}^{E_{i}}w(e,e+1)\), where \(K_{G}\) is the time it takes to play one game, \(G_{i}\) is the number of games played by the player \(i\), \(K_{E}\) is the maximum distance between two nodes in the network, \(E_{i}\) is the number of edges of the final path of player \(i\) and \(w(e,e+1)\) is the weight of the edge connecting the nodes \(e\) and \(e+1\).  
The communication network model has \(n1\) nodes and \(n2\) packets to be sent between nodes. The net is generated using the Erdos-Renyi-Gilbert model [18] with \(G(nodes\ =\ n1\), \(probability=0.5)\) (where \(nodes\) is the number of nodes and \(probability\) the probability that two nodes are connected to each others) with the condition that every node has at least one connection to other node. The maximum capacity of every channel is assumed to be one packet, with linearly declining performance from there on. This decline will be reflected in our model by increasing the corresponding weight, \(w(e,e+1)\), of each channel proportionally to the number of packets going through it. An example of a \(n1=10\) nodes network is shown in Figure 1 where the packets traveling through are represented with different colors.  
In a congested channel it will take much longer traveling time for a given packet than in an idle one. In other words, as the number of packets traveling through the same channel increases, the traveling time for each packet increases. This situation is analogous to city transport where the time a vehicle takes to travel a route increases as the number of vehicles on that route increases. If this situation is not handled well, this may result in heavy traffic congestion [19].  
Every packet is thought as a player in a game that prefers the shortest path channel. The traveling channel choice is made by playing a game. Whenever many players choose the same channel, the traveling time for that channel increases. Channel congestion makes traveling time much longer. Finally, on every congested channel packets play game, each of them has two options: to travel through the preferred channel or to search another path.  
If every player decides to choose their shortest path, some network channels will be overloaded with all the packets that are disputing for it, slowing down the network. Therefore, if players behave in a selfish way, always choosing their shortest path, the whole communication network will be harmed. On the other hand, if every player decides to look for idle channels, the routing time increases and the network would be also slowed down. This simple model reflects the massive problem faced in modern communication technologies which poses a severe challenge to the existing routing strategies [20]. This phenomenon is called congestion dilemma, as we will observe in the following sections, the routing time may decrease by increasing traveling time or traveling time may decrease by increasing routing time.