Distribution function of the shortest path in networks of queues

Abstract

In this paper, we consider acyclic networks of queues as a model to support the design of a dynamic production system. Each service station in the network represents a manufacturing or assembly operation. Only one type of product is produced by the system, but there exist several distinct production processes for manufacturing this product, each one corresponding with a directed path in the network of queues. In each network node, the number of servers in the corresponding service station is either one or infinity. The service time in each station is either exponentially distributed or belongs to a special class of Coxian distribution. Only in the source node, the service system may be modeled by an $M/G/\infty $ queue. The transport times between every pair of service stations are independent random variables with exponential distributions. In method proposed in this paper, the network of queues is transformed into an equivalent stochastic network. Next, we develop a method for approximating the distribution function of the length of the shortest path of the transformed stochastic network, from the source to the sink node. Hence, the method leads to determining the distribution function of the time required to complete a product in this system (called the manufacturing lead time). This is done through solving a system of linear differential equations with non-constant coefficients, which is obtained from a related continuous-time Markov process. The results are verified by simulation.