Technical Note—Asymptotically Optimal Control of Omnichannel Service Systems with Pick-up Guarantees

Motivated by the recent popularity of omnichannel service systems, we analyze the joint admission and scheduling control of a queueing system with two classes of customers: online and walk-in. Unlike walk-in customers, online customers are given a target time for pick up upon placing an order. Thus, in addition to minimizing the waiting costs of walk-in customers and the rejection cost of both classes, we need to minimize the earliness and tardiness costs of online customers. Such a distinctive objective makes the control problem difficult to analyze. We develop a novel analysis by adopting the idea of proving $H=\lambda G$ to establish an asymptotic relationship between the waiting time cost and the queue length cost under general control policies in the heavy-traffic regime. We also design a policy and show that it is asymptotically optimal.

Funding: X. Gao’s research is supported in part by the Hong Kong Research Grants Council [Grants 14201520, 14201421, and 14212522]. J. Huang’s research is supported in part by the Hong Kong Research Grants Council [Grants 14500819, 14505820, and 14501621] and the National Natural Science Foundation of China [Grant 72222023]. J. Zhang’s research is supported in part by the Hong Kong Research Grants Council [Grants 16208120 and 16214121].