Markovian queueing models with periodic-review

This article concerns the production/inventory system with periodic-review policy. The system is modelled as an M/M/c-type of queue and is analysed by using the embedded Markov chain technique. The probability generating functions for the queue length are derived, which are used to determine the mean queue length, mean job waiting times, and higher moments of these quantities. Analytic results are obtained for M/M/1 queues and M/M/∞ queues with periodic-review. For M/M/c queues of 1 < c < ∞, analytic solutions do not exist. Thus, numerical methods have to be applied. An optimization model is imposed on the queueing model that balances the inventory (waiting) cost and the cost of reviewing the system. The cost is minimized by determining the length of the review period. Both constant and state-dependent lengths of the review period are considered and compared. The optimal review periods are given for various numerical examples.