A novel method for generating the optimal routing matrix of queuing networks with batch service

In this paper, we consider a large-scale open queuing network. The arrival process in the queueing network is Poissonian. Сustomer transitions between nodes are described by the routing matrix. Each node consists of a single server and an infinite waiting queue. Customers are served as a unique batch of a given size with exponentially distributed service time. After the completion of service, customers are routed between nodes one at a time, independently of each other. We assume that, at any node, the number of destinations is much larger than the batch size. We also assume that the transition probabilities of customers between nodes are comparable. In this paper, we propose a method for generating the optimal routing matrix that provides the minimum average sojourn times in each node. We also provide a condition for relative arrival rates, under which the queuing network topology is radial (star-shaped), and expressions for optimal input rates to nodes. Finally, examples of the optimal routing matrix for different values of the overall input rate and in case of link failures are presented.