Stochastic decomposition in a queueing-inventory system with batch demands, randomized order policy and multiple vacations

This article studied a queueing-inventory system with batch demands, randomized order policy, and multiple vacations. Customers arrive in this system according to a Poisson process and take away batch items after a service time. The system sends a replenishment order and starts multiple vacations simultaneously once the on-hand inventory level drops to zero. The replenishment is based on a random order size policy. Customers arriving during this period are assumed to be lost. Under the assumption that all the underlying random variables are exponential, the stationary joint probability is obtained in product form. Based on the stationary distribution, the expected sojourn time of a permitted entering customer is derived in recursive form. Several numerical examples are processed to analyze the effect of variant parameters on the performance measures. The numerical results show that the expected total cost function is a convex function with respect to the maximum allowable inventory. The influence of various parameters on the optimal value is also displayed visually.