Abstarct: In this thesis, we investigate in Stochastic Queue Core (SQC) problem. The problem of finding the core of a network with a queue deals with finding a path on a network such that the average travel time of particles along the network for service is minimized. First of all, we formulate the problem by using the classic queues. We suppose that all customers are located at the nodes of a tree network and there is a server on an optimal path which services the demands according to an M/M/1 or M/G/1 queue using the FIFO discipline. The time of service includes the travelling time of the server on the optimal path and time the customer travels on the tree network to reach the optimal path. The construction cost in our model includes the construction or maintenance cost of the optimal path. And then, we report the results for a given example. Evacuation network can be an example of the application of this problem. So, we focus on one of the traffic flow model, i.e., state-dependent model. By determining the evacuation destination(s), we will look for an optimal path such that average travel time for evacuation all vehicles is optimal. Several algorithms are developed when there is blocking and where there is none. If there is significant blocking during the evacuation time, then the capacities of the roadway lixnks are improved so as to alleviate the blocking. We also compare and contrast the performance of the state dependent models with two well-known formulas for calculating the travel time of a vehicle along a network.