Queues with State-Dependent Stochastic Service Rates

The standard M/G/1 queuing system is generalized so that the service time parameter becomes a stochastic process, {M n , n = 1, 2, …}, indexed on the length of the queue at the moment service is begun. The service time, T i , of a customer entering into service when a total of i customers are in the system is to be conditioned upon the random variable M i . Some general theory is developed for the model and three specific cases are explored. For each of the examples, both the conditional service-time distributions, {B Tn ∣M n (t∣μ n ), n = 1, 2, …}, and the prior distributions of {M n }, {F M n (μ n ), n = 1, 2, …}, are specified, and results are obtained that characterize queue behavior using the imbedded Markov chain approach. The first case is an illustration of a random, non-state-dependent parameter, while the other two describe different ways a service parameter may be state-dependent. In addition, an industrial example based on the third case is cited.