Behavior-Aware Queueing: The Finite-Buffer Setting with Many Strategic Servers
Yueyang Zhong, Ragavendran Gopalakrishnan, Amy R. Ward
Operations Research · 2025 · Volume 73 · Issue 1
Abstract
Service system design is often informed by queueing theory. Traditional queueing theory assumes that servers work at constant speeds. That is reasonable in computer science and manufacturing contexts. However, servers in service systems are people, and in contrast to machines, the incentives created by design decisions influence their work speeds. We study how server work speed is affected by managerial decisions concerning (i) how many servers to staff and how much to pay them and (ii) whether and when to turn away customers in the context of many-server queues with finite or infinite buffers ( M / M / N / k with 𝑘 ∈ ℤ + ∪ { ∞ } k ∈ Z + ∪ { ∞ } ) in which the work speeds emerge as the solution to a noncooperative game. We show that a symmetric equilibrium always exists in a loss system ( N = k ) and provide conditions for equilibrium existence in a single-server system ( N = 1). For the general M / M / N / k system, we provide a sufficient condition for the existence of a solution to the first-order condition and bounds on such a solution; however, showing that it is an equilibrium is challenging because of the existence of multiple local maxima in the utility function. Nevertheless, in an asymptotic regime in which demand becomes large, the utility function becomes concave, allowing us to characterize underloaded, critically loaded, and overloaded equilibria. Funding: This work was supported in part by funding from the Social Sciences and Humanities Research Council of Canada [Grant 430-2020-00334] and the Charles M. Harper Faculty Fellowship at the University of Chicago Booth School of Business. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2023.2487 .
Keywords
Stochastic Models, service systems, strategic servers, finite buffer, queueing game, equilibrium, asymptotic analysis
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