A four-input three-stage queuing network approach to model an industrial system

An industrial system is represented as a four-input, three-stage queuing network in this paper. The four-input queuing network receives orders from clients, and the orders are waiting to be served. Each order comprises (i) time of occurrence of the orders, and (ii) quantity of items to be delivered in each order. The objective of this paper is to compute the optimal path which produces the least response time for the delivery of items to the final destination along the three stages of the network. The average number of items that can be delivered with this minimum response time constitute the optimum capacity of the queuing network. After getting serviced by the last node (a queue and its server) in each stage of the queuing network, a decision is made to route the items to the appropriate node in the next stage which can produce the least response time. Performance measures such as average queue lengths, average response times, average waiting times of the jobs in the four-input network are derived and plotted. Closed-form expressions for the equivalent service rate, equivalent average queue lengths, equivalent response and waiting times of a single equivalent queue with a server representing the entire four-input queuing network are also derived and plotted.