Performance analysis and optimization of state‑dependent replenishment policy in queuing‑inventory system

Summary: This study proposes a new queue-dependent (s, S)-type replenishment policy in the queueing-inventory systems. If the inventory level is greater than s, no restocking order is sent; otherwise, the replenishment is performed to reach the maximum value S, regardless of the stock level at the moment. Orders are replenished in two ways: the regular order and the urgent order. Lead times of the two-type orders are exponentially distributed with different parameters. The urgent orders require a shorter delivery time than the regular orders. The queue-dependent replenishment policy is defined as follows: when the inventory level drops to s, if the number of the customers is less than predefined threshold value r, the regular order is made; if the number is more or equal r, the urgent order is sent. When the inventory level drops to zero, one customer becomes impatient, regardless of the customer’s number in the queue. Arrival of customers is according to a Markovian arrival process and the service times are adapted by a phase-type distribution. The mathematical model of the system is developed using a continuous-time Markov chain with an infinite state space. Stability condition and then the steady-state distribution are derived by using the matrix-geometric method. The influences of the parameters on the performance measures are discussed with numerical examples. An optimization problem is solved, where the criterion is the expected total cost, and the controlled parameters are the reorder point s and the threshold parameter r.