Analysis of a single-server queueing system with state-dependent service time

Abstract

This thesis discusses a class of queueing systems in which the service time of a customer at a single-server facility is dependent on the queue state at the onset of his service: Customers are composed of m distinct types. m distinct types of customers arrive randomly and independently at a service facility all processed with service times that are randomly and independently distributed. Customer arrival is Poisson. Customers are serviced according to arrival order. If a customer arrives when the server is idle, his service time has a distribution function $G^e$(x), while if he arrives when the server is busy, his service time has a different distribution function according to queue size at the onset of his service, i.e., if no newly arrived customers are behind him at the onset of his service, his service time has a distribution function $G^o$(x) and if at least one customer is behind him at the onset of his service, his service time has a distribution function G(x). The Laplace-Stieltjes transform for the waiting time distribution function is derived, and the mean waiting time is obtained. So, the probability that a customer does not wait for service in the queue in steady state is obtained.