The Regulation of Queue Size by Levying Tolls
Abstract
This paper considers an M/M/1 queueing system where arriving customers observe the number in the system and decide whether to join. When the reward from service V exceeds the expected waiting cost, they join; otherwise they balk. The individual equilibrium strategy and the socially optimal strategy are derived, showing they diverge.
★ Curator Summary
Naor's 1969 paper is the genesis of everything we now call 'strategic queueing.' The core insight — customers observe queue state and make rational join/balk decisions — is obvious in retrospect but was revolutionary for queueing theory, which had always treated arrivals as mindless Poisson processes. The formula n_e = floor(V/Cmu) gives the threshold queue length below which customers join. When automation increases mu or reduces C, n_e increases — more customers join.
Why It Matters
This paper explains at a mathematical level why AI deployment increases contact volume. When you speed up service (higher mu) or reduce effort (lower C), the joining threshold relaxes. Customers who previously would have balked — tolerated their problem, found a workaround — now find it worthwhile to contact. This is the micro-foundation of the Jevons Paradox applied to service operations, expressed in the language of queueing theory.
Caveats
The model is M/M/1 (single server) with observable queue — real contact centers are multi-server with imperfect information. The assumption of fully rational customers with known V and C is strong. Modern extensions (Hassin 2016) relax many of these assumptions. The paper is 55+ years old and predates digital channels entirely.
Discussion
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