Production fluid-type inventory model associated with a single vacation queue arising from additive manufacturing in the food sector

Three-dimensional (3D) food printing is an emerging trend in additive manufacturing and rapid prototyping. Based on a real-world scenario originating from 3D-printed food manufacturing, this paper investigates a single vacation queue with a connected fluid-type inventory. In such an integrated system, customers' orders for personalized products arrive following a Poisson process, and the first-in-first-out principle governs the service discipline at the service facility. Since some raw materials used in food manufacturing are stored in fluid form, the quantity required for each customer is no longer unit-sized. We suppose that each customer consumes an exponential amount of items upon service completion. When the stock becomes empty, the server leaves the system on vacation and hires specialists to adjust and maintain the processing equipment during this time. Adopting the well-known (������, ������) control policy, internal production with an exponential lead time is used to replenish the inventory. Under the assumption that customers who arrive during the stock -out period and the remaining vacation time are lost, we derive the explicit product form solution for the system state joint distribution by converting integral equations to equivalent ordinary differential equations. Furthermore, with the help of these joint probability distributions, mathematical formulas are also developed to estimate system performance metrics. Finally, incorporating these metrics into a cost function, an optimal (������, ������) inventory policy and its corresponding minimum cost can be numerically determined.