Continuous-Time Markov Chain Modeling of Divergent Multi-Echelon Inventory Systems with Erlang Replenishment Processes

Background: Divergent inventory systems must balance product availability and operating costs under uncertain demand and replenishment. This study examines how replenishment-time variability and retailer-specific inventory decisions affect a two-echelon network. Methods: A continuous-time Markov chain model is developed for one saturated upstream supplier and multiple independent retailers with heterogeneous continuous-review policies, Poisson demand, Erlang-distributed replenishment lead times, and complete lost sales. A recursive matrix-analytic formulation constructs the generator matrix and yields stationary probabilities for calculating operational and economic performance measures. Analytical results are checked against simulation, and numerical experiments examine replenishment variability, rates, and order-up-to levels. Results: Within the settings investigated, reducing replenishment-time variability across retailers while preserving mean lead times increases throughput and reduces lost-sales and total operating costs. Reducing variability only at one retailer yields smaller savings, mainly through lower holding costs. Faster replenishment improves fill rates and reduces total operating costs, with diminishing marginal savings and benefits that depend on the order-up-to level. Conclusions: The framework supports exact steady-state evaluation under the stated assumptions. The findings support jointly assessing replenishment responsiveness and inventory policies using both cost and service measures, while targeting variability reductions according to retailer-specific performance effects.