Core Concepts

The optimal policy partitions the state-space into a well-defined no-transfer region and its complement, such that transferring is optimal if and only if the system is sufficiently imbalanced. In the absence of fixed transfer costs, the optimal policy moves the state to the boundary of the no-transfer region, while in the presence of fixed costs, it moves the state to the relative interior.

Abstract

The paper considers the problem of load balancing in parallel queues by transferring customers between them at discrete points in time. Holding costs accrue as customers wait in the queues, while transfer decisions incur both fixed (setup) and variable costs proportional to the number and direction of transfers.
The key insights are:
Structural results on the optimal fluid policy:
Under fairly general assumptions, the optimal policy partitions the state-space into a well-defined no-transfer region and its complement.
Transferring is optimal if and only if the system is sufficiently imbalanced.
In the absence of fixed transfer costs, the optimal policy moves the state to the boundary of the no-transfer region.
In the presence of fixed costs, the optimal policy moves the state to the relative interior of the no-transfer region.
Role of idleness:
When holding costs are linear and equal across queues, non-idleness at certain queues serves as a sufficient condition for when transferring to them is suboptimal.
In the absence of transfer and setup costs, non-idleness becomes a sufficient and necessary condition for when transferring is suboptimal.
Numerical results and case study:
Simulation experiments confirm the structure of the optimal policy for the stochastic control problem.
A case study using real data from a network of intensive care units during the COVID-19 pandemic demonstrates that the fluid policy can improve the total expected system cost by up to 27.7% over a one-week horizon.

Stats

Customers incur holding costs in queues according to queue-dependent convex non-decreasing functions.
Transferring U customers in period m incurs a setup cost of ̃K(U) and a variable transfer cost of rij per transferred customer from queue i to j.

Quotes

"The optimal policy partitions the state-space into a well-defined no-transfer region and its complement, such that transferring is optimal if and only if the system is sufficiently imbalanced."
"In the absence of fixed transfer costs, the optimal policy moves the state to the boundary of the no-transfer region, while in the presence of fixed costs, it moves the state to the relative interior."

Key Insights Distilled From

by Timothy C. Y... at **arxiv.org** 04-02-2024

Deeper Inquiries

If the holding cost functions were not convex, the structure of the optimal policy would likely change. Convexity of the holding cost functions plays a crucial role in determining the optimal transfer decisions in the model. Non-convex holding costs could lead to more complex and potentially discontinuous optimal policies. The no-transfer region may not be as well-defined, and the trade-off between holding costs and transfer costs could be more intricate. The optimal policy might involve more frequent and smaller transfers to balance the costs effectively. Additionally, the characterization of the target states and the boundary of the no-transfer region may vary, leading to different transfer patterns based on the state of the system.

Relaxing the assumption of exponential service times in the model would have several implications. Exponential service times are commonly used in queueing models due to their memoryless property, which simplifies the analysis. By relaxing this assumption, the model could better capture real-world scenarios where service times are not exponentially distributed. Introducing more realistic service time distributions could impact the system's dynamics, affecting the optimal transfer decisions and the overall performance of the system. The model would need to be adjusted to accommodate the new service time distributions, potentially leading to more complex analytical and computational challenges.

Extending the insights from this work to settings with multiple decision makers, such as independent hospitals, would require adapting the model to account for decentralized decision-making. Each decision maker would need to optimize their transfer policies independently, considering their local constraints and objectives. The coordination and information sharing between decision makers would be crucial to ensure efficient load balancing across the network of facilities. Insights from this work could be applied to develop collaborative transfer strategies, considering the interplay between holding costs, transfer costs, and system imbalances. Game theory and cooperative game theory frameworks could be employed to analyze the interactions between multiple decision makers and derive optimal transfer policies that benefit the entire network while respecting individual goals and constraints.

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