Matrix-geometric method for the analysis of a queuing system with perishable inventory

Markov models of queuing systems with perishable stocks and an infinite buffer are studied using two replenishment policies. In one of them, the volume of orders is constant, while the other depends on the current level of stocks. Customers can join the queue even when the inventory level is zero. After the service is completed, customers either receive supplies or leave the system without receiving them, while the duration of their service depends on whether the customer has received supplies or not. The conditions for the ergodicity of the constructed two-dimensional Markov chains are obtained, and the matrix-geometric method is used to calculate their steady-state distributions. Formulas are found for finding the characteristics of the system using the indicated replenishment policies, and the results of numerical experiments are given.