Efficient allocation of load-balancing and differentiation tasks in tandem queue services

Abstract

Within service systems, tasks can encompass diverse functionalities. In a two-phase queuing model featuring two customer priority classes, our study discerns two distinct task functionalities executed by the first-phase server (referred to as the auxiliary server). These tasks aim to facilitate priority-based service by the second-phase server (referred to as the expert). Load-balancing tasks aim to alleviate the expert’s workload, while differentiation tasks seek to enhance accurate customer prioritization in the second phase by reducing misclassifications. With customers queuing for both the auxiliary server and the expert in tandem, our investigation focuses on determining the optimal allocation of the auxiliary server’s time between these load-balancing and differentiation tasks. Through queuing optimization, we aim to minimize customers’ expected total delay cost. In scenarios where the auxiliary server is allowed to perform only one task type (either load-balancing or differentiation), we delineate the optimal solutions based on specific functional forms dictating the server’s efficiency in executing each task type. These solutions strike a balance between excess phase capacities and the square root of marginal cost-to-saving ratios arising from each task type. Additionally, we partially characterize the optimal solution in scenarios permitting both load-balancing and differentiation tasks. Notably, under high system loads, executing load-balancing tasks proves more efficient than differentiation tasks. However, the relationship between the optimal task durations and system load showcases a non-monotonic pattern. As AI decision support products increasingly enable expert providers to delegate “routine” tasks to mid-level providers, our study sheds light on the efficient allocation of different tasks to different provider types to minimize delay costs in service systems.