We consider a single-server queue with Poisson input operating under first-come--first-served (FCFS) or last-come--first-served (LCFS) disciplines. The service times of the customers are independent and obey a general distribution. The system is subject to costs for holding a customer per unit of time, which can be customer specific or customer class specific. We give general expressions for the corresponding value functions, which have elementary compact forms, similar to the Pollaczek--Khinchine mean value formulae. The results generalize earlier work where similar expressions have been obtained for specific service time distributions. The obtained value functions can be readily applied to develop nearly optimal dispatching policies for a broad range of systems with parallel queues, including multiclass scenarios and the cases where service time estimates are available.
"On the value function of the M/G/1 FCFS and LCFS queues." J. Appl. Probab. 49 (4) 1052 - 1071, December 2012. https://doi.org/10.1239/jap/1354716657