Parametric Analysis of Queuing Networks

We consider a queuing network with M exponential service stations and with N customers. We study the behavior of a subsystem u, which has a single node as input and a single node as output, when the subsystem parameters are varied. An “equivalent” network is constructed in which all queues except those in subsystem u are replaced by a single composite queue. We show that for certain classes of system parameters, the behavior of subsystem u in the equivalent network is the same as in the given network. The analogy to Norton’s theorem in electrical circuit theory is demonstrated. In addition, the equivalent network analysis can be applied to open exponential networks. Introduction Queuing models are widely used to analyze and design a variety of systems. These models are generally used to study the variation of certain system parameters such as response time as a function of the network structure and service times. In this paper, the authors determine a relationship that exists between some queuing networks and electrical networks, with customers and throughput (rate of flow of customers) being analogous to electrical charge and electrical current, respectively. The authors show how to directly apply some methods from electrical circuit theory, in particular Norton’s theorem [ 11, to queuing networks. Consider an electrical circuit consisting of batteries and resistors, Fig. 1 . To study the behavior of a subsystem u between terminals 1 and 2 of Fig. 1 as the subsystem parameters are varied, construct an “equivalent” circuit in which all the components of the given circuit, except those of subsystem u, are replaced by a single current source and a parallel internal resistance, Fig. 2. The value of the current source is set equal to the current flowing between terminals 1 and 2 when subsystem u Figure 1 Electrical network.