Multi-component Matching Queues in Heavy Traffic

Abstract

We consider multi-component matching queue systems in heavy traffic. We assume a product made of K ≥ 2 distinct perishable components is mass produced under the assemble-to-order production strategy. These components arrive randomly over time at high speed at the assembling station, and they wait in their respective queues according to their categories until matched or their "patience" runs out. If all categories are available, matching occurs instantaneously, and thereafter the matched components leave immediately. For a sequence of such matching queue systems parameterized by n, when the arrival rates of all categories tend to infinity in concert as n tends to infinity, we obtain a weak convergence result for the queue length processes in heavy traffic under mild assumptions. We demonstrate that the heavy traffic limit of the appropriately scaled queue length vector is characterized by a coupled stochastic integral equation with a scalar-valued non-linear term. We prove some crucial properties of such a coupling behavior for certain coupled equations. We also exhibit that a generalized coupled stochastic integral equation admits a unique weak solution that has the strong Markov property. Moreover, we establish an asymptotic Little's law for each component queue, which reveals the asymptotic relationship between the queue length and its waiting time in the queue. We introduce an infinite-horizon discounted cost functional associated with the matching queue model with abandonment, where we consider two types of costs: a holding cost generated by storing the components in queues and a penalty cost generated by abandoned components. We show that the expected value of this cost functional for the nth system converges to that of the heavy traffic limiting process as n tends to infinity.