Time Dependent Behaviour of Waiting Line Models in Discrete Time

Abstract

The use of traditional closed form solutions for waiting line networks is sometimes an oversimplification, particularly where the flow of customers fluctuate with time.   Two approximation methods are used to overcome this simplification.  Two approximation models are developed which differ in the manner in which they model the transfer process between service modes.  Approximation I uses aggregate departure probabilities to model the transfer process.  Approximation II on the other hand, takes into account the inter arrival time process to model the transfer process.  Test case examples show that when the distribution of service time at node one is of negative  exponential in shape, the error of the mean system size at node two for Approximation I is within 3%, whereas in approximation II is within 7%.  For other types of services time distribution at node two is lower for approximation II than Approximation I at traffic intensities which cause frequent fluctuate on the number of customers at the first time.