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A fluid approximation for a matching model with general reneging distributions

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Abstract

Motivated by a service platform, we study a two-sided network where heterogeneous demand (customers) and heterogeneous supply (workers) arrive randomly over time to get matched. Customers and workers arrive with a randomly sampled patience time (also known as reneging time in the literature) and are lost if forced to wait longer than that time to be matched. The system dynamics depend on the matching policy, which determines when to match a particular customer class with a particular worker class. Matches between classes use the head-of-line customer and worker from each class. Since customer and worker arrival processes can be very general counting processes, and the reneging times can be sampled from any finite mean distribution that is absolutely continuous, the state descriptor must track the age-in-system for every customer and worker waiting in order to be Markovian, as well as the time elapsed since the last arrival for every class. We develop a measure-valued fluid model that approximates the evolution of the discrete-event stochastic matching model and prove its solution is unique under a fixed matching policy. For a sequence of matching models, we establish a tightness result for the associated sequence of fluid-scaled state descriptors and show that any distributional limit point is a fluid model solution almost surely. When arrival rates are constant, we characterize the invariant states of the fluid model solution and show convergence to these invariant states as time becomes large. Finally, again when arrival rates are constant, we establish another tightness result for the sequence of fluid-scaled state descriptors distributed according to a stationary distribution and show that any subsequence converges to an invariant state. As a consequence, the fluid and time limits can be interchanged, which justifies regarding invariant states as first order approximations to stationary distributions.

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Acknowledgements

We would like to thank Levi DeValve for helpful discussion related to the use of the fluid model invariant states to formulate and solve a matching policy optimization problem, which resulted in the paper [6]. Financial support from the University of Chicago Booth School of Business is gratefully acknowledged.

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Authors and Affiliations

  1. The University of Chicago Booth School of Business, Chicago, USA

    Angelos Aveklouris & Amy R. Ward

  2. California State University San Marcos, San Marcos, USA

    Amber L. Puha

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  1. Angelos Aveklouris
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  2. Amber L. Puha
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  3. Amy R. Ward
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Correspondence to Amy R. Ward.

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Aveklouris, A., Puha, A.L. & Ward, A.R. A fluid approximation for a matching model with general reneging distributions. Queueing Syst 106, 199–238 (2024). https://doi.org/10.1007/s11134-023-09892-w

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