Optimal Partition for a Multi-Type Queueing System

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We study an optimal server partition and customer assignment problem for an uncapacitated first-come-first-served queueing system with heterogeneous types of customers. Each type of customer is associated with a Poisson arrival, a certain service time distribution, and a unit waiting cost. The goal is to minimize the expected total waiting cost by partitioning the server into subqueues, each with a smaller service capacity, and routing customer types probabilistically. First, we show that by properly partitioning the queue, it is possible to reduce the expected waiting costs by an arbitrarily large ratio. Then, we show that for any given server partition, the optimal customer assignment admits a certain geometric structure, enabling an efficient algorithm to find the optimal assignment. Such an optimal structure also applies when minimizing the expected sojourn time. Finally, we consider the joint partition-assignment optimization problem. The customer assignment under the optimal server partition admits a stronger structure. Specifically, if the first two moments of the service time distributions satisfy certain properties, it is optimal to deterministically assign customer types with consecutive service rates to the same subqueue. This structure allows for more efficient algorithms. Overall, the common rule of thumb to partition customers into continuous segments ranked by service rates could be suboptimal, and our work is the first to comprehensively study the queue partition problem based on customer types. Funding: Financial support from the National Natural Science Foundation of China [Grants 72192830, 72192832, 71825003, 72394361, and 72425013] and the Guangdong Provincial Key Laboratory on Mathematical Foundations for Artificial Intelligence [Grant 2023B1212010001] is gratefully acknowledged. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.0035 .