Closed-Form Approximations for Optimal (&ITr&IT, &ITq&IT) and (&ITS&IT, &ITT&IT) Policies in a Parallel Processing Environment

We consider a single-item continuous-review (r, q) inventory system with a renewal demand process and Independent, identically distributed stochastic lead times. Using a stationary marked-point process technique and a heavy-traffic limit, we prove a previous conjecture that inventory position and inventory on-order are asymptotically independent. We also establish closed-form expressions for the optimal policy parameters and system cost in heavy-traffic limit, the first of their kind, to our knowledge. These expressions sharpen our understanding of the key determinants of the optimal policy and their quantitative and qualitative impacts. For example, the results demonstrate that the well-known square-root relationship between the optimal order quantity and demand rate under a sequential processing environment is replaced by the cube root under a stochastic parallel processing environment. We further extend the study to periodic-review (S, T) systems with constant lead times.