Queuing Models for Analyzing the Steady-State Distribution of Stochastic Inventory Systems with Random Lead Time and Impatient Customers

Abstract

In material management, the inventory systems may have good management aspects in terms of materials; however, this negatively affects the relationship between the facility and customers. In classical inventory models, arriving demands are satisfied immediately if there is enough on-hand inventory. Traditional inventory models consider optimization problems and find the optimal policy of decision variables without computing the stationary distribution of the inventory states for random demand. Hence, a detailed analysis of inventory management systems requires a joint distribution of system stock levels and the number of requests to be investigated thoroughly. This research provides a new stochastic mathematical model for inventory systems with lead times and impatient customers under deterministic and uniform order sizes. The proposed model identifies the performance measures in a stochastic environment, analyzing the properties of the inventory system with stochastic and probabilistic parameters, and finally, validating the model’s accuracy. To analyze the system, balance equations were derived from a mathematical characterization of the underlying queuing model dependent on the Markov chain formalism. The precise performance was achieved by examining the graphical representation of the service process in a steady-state as a function of both arrival distribution and the customer patience coefficient, while it was challenging to derive an optimal curve fit in a three-dimensional space that features two input variables and a single output variable.