The M/M/1 queue with inventory, lost sale, and general lead times

We consider an M/M/1 queueing system with inventory under the (r,Q) policy and with lost sales, in which demands occur according to a Poisson process and service times are exponentially distributed. All arriving customers during stockout are lost. We derive the stationary distributions of the joint queue length (number of customers in the system) and on-hand inventory when lead times are random variables and can take various distributions. The derived stationary distributions are used to formulate long-run average performance measures and cost functions in some numerical examples.