Bicriteria shortest path in networks of queues

In this paper, I propose a new methodology to find the bicriteria shortest path from the source to the sink node of a network of queues under the steady-state condition. I assume the number of servers in each service station settled in a node of the network is either one or infinity. Moreover, the arc lengths are mutually independent random variables. First, I propose a method to transform each node, containing a service station, into a proper stochastic arc corresponding to the waiting time in the particular service station. Then, the stochastic network is transformed into a bicriteria network by computing the mean and the variance of the waiting time in each service station and augmenting those to the transformed arc. Finally, by defining a proper utility function, dynamic programming is used to obtain the bicriteria shortest path. The time complexity of the proposed algorithm is O(b), considering b as the number of service stations. The proposed method is suitable to find the shortest rout from the origin to the destination in stochastic routing problems. Moreover, this method is useful to approximate the mean and the variance of project completion time in dynamic PERT networks.