Behavior-Aware Queueing: The Finite-Buffer Setting with Many Strategic Servers

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+∪{∞}) 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 due to 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.

Keywords: service systems; strategic servers; finite buffer; queueing game; equilibrium