Estimating customer delay and tardiness sensitivity from periodic queue length observations

Liron Ravner, Jiesen Wang

Research output: Contribution to journalArticlepeer-review


A single server commences its service at time zero every day. A random number of customers decide when to arrive to the system so as to minimize the waiting time and tardiness costs. The costs are proportional to the waiting time and the tardiness with rates α and β, respectively. Each customer’s optimal arrival time depends on the others’ decisions; thus, the resulting strategy is a Nash equilibrium. This work considers the estimation of the ratio θ≡ β/ (α+ β) from queue length data observed daily at discrete time points, given that customers use a Nash equilibrium arrival strategy. A method of moments estimator is constructed from the equilibrium conditions. Remarkably, the method does not require estimation of the Nash equilibrium arrival strategy itself, or even an accurate estimate of its support. The estimator is strongly consistent, and the estimation error is asymptotically normal. Moreover, the asymptotic variance of the estimation error as a function of the queue length covariance matrix (at sampling times) is derived. The estimator performance is demonstrated through simulations and is shown to be robust to the number of sampling instants each day.

Original languageEnglish
Pages (from-to)241-274
Number of pages34
JournalQueueing Systems
Issue number3-4
StatePublished - Apr 2023

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.


  • Parameter estimation
  • Strategic arrival times to a queue
  • Tardiness
  • Transient queueing

ASJC Scopus subject areas

  • Statistics and Probability
  • Computer Science Applications
  • Management Science and Operations Research
  • Computational Theory and Mathematics


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