Queueing Systems

, Volume 80, Issue 1–2, pp 1–14 | Cite as

A clearing system with impatient passengers: asymptotics and estimation in a bus stop model

  • Offer Kella
  • Wolfgang Stadje


At some location buses stop to pick up the passengers waiting there and leave immediately, empty or occupied. Passenger arrival times as well as bus arrival times form independent renewal processes. Every passenger is willing to wait for a random amount of time before leaving, and every bus takes away all waiting passengers. For this pickup model, we study the distributions of the number of waiting passengers and of the individual sojourn times. The sojourn times lead to a Markov chain embedded in the superposition of the two underlying renewal arrival processes, for which we study its convergence toward stationarity.


Clearing system Impatience Rate of convergence Strongly mixing \(M/G/\infty \) Pickup problem 

Mathematics Subject Classification

Primary: 60K25 90B22 Secondary: 60K05 60J05 



We are very grateful to the reviewer and the associate editor for the careful reading and for important observations and corrections that improved the presentation of the paper. In particular, (12) was suggested by the associate editor and replaces a considerably more cumbersome derivation. Offer Kella was supported in part by Grant 1462/13 from the Israel Science Foundation and the Vigevani Chair in Statistics. Wolfgang Stadje was supported by Grant No. 306/13-2 of the Deutsche Forschungsgemeinschaft.


  1. 1.
    Carlstein, E.: The use of subseries values for estimating the variance of a general statistic from a stationary sequence. Ann. Stat. 14, 1171–1179 (1986)CrossRefGoogle Scholar
  2. 2.
    Dehling, H., Fried, R., Sharipov, OSh, Vogel, D., Wornowizk, M.: Estimation of the variance of partial sums of dependent processes. Stat. Probab. Lett. 83, 141–147 (2012)CrossRefGoogle Scholar
  3. 3.
    Ibragimov, I.A., Linnik, Y.V.: Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen (1971)Google Scholar
  4. 4.
    Jones, G.L.: On the Markov chain central limit theorem. Prob. Surv. 1, 299–320 (2004)CrossRefGoogle Scholar
  5. 5.
    Ivanovs, J., Kella, O.: Another look into decomposition results. Queueing Syst. 75, 19–28 (2013)CrossRefGoogle Scholar
  6. 6.
    Linton, D., Purdue, P.: An \(M/G/\infty \) queue with \(m\) customer types subject to periodic clearing. Opsearch 16, 80–88 (1979)Google Scholar
  7. 7.
    Stone, C.: On absolutely continuous components and renewal theory. Ann. Math. Stat. 37, 271–275 (1966)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of StatisticsThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Institute of MathematicsUniversity of OsnabrückOsnabrückGermany

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