Methodology and Computing in Applied Probability

, Volume 21, Issue 4, pp 1023–1044 | Cite as

The Single Server Queue with Mixing Dependencies

  • Youri RaaijmakersEmail author
  • Hansjörg Albrecher
  • Onno Boxma


We study a single server queue, where a certain type of dependence is introduced between the service times, or between the inter-arrival times, or both between the service times and the inter-arrival times. This dependence arises via mixing, i.e., a parameter pertaining to the distribution of the service times, or of the inter-arrival times, is itself considered to be a random variable. We give a duality result between such queueing models and the corresponding insurance risk models, for which the respective dependence structures have been studied before. For a number of examples we provide exact expressions for the waiting time distribution, and compare these to the ones for the standard M/M/1 queue. We also investigate the effect of dependence and derive first order asymptotics for some of the obtained waiting time tails. Finally, we discuss this dependence concept for the waiting time tail of the G/M/1 queue.


Waiting time distribution Duality between risk and queueing models Dependence Mixing 

Mathematics Subject Classification (2010)

60K25 90B20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abate J, Whitt W (1999) Modeling service-time distributions with non-exponential tails: Beta mixtures of exponentials. Stochastic Models 15:517–546MathSciNetCrossRefGoogle Scholar
  2. Abate J, Choudhury GL, Whitt W (1994) Waiting-time tail probabilities in queues with long-tail service-time distributions. Queueing Syst 16:311–338MathSciNetCrossRefGoogle Scholar
  3. Abramowitz M, Stegun I (1965) Handbook of mathematical functions. National Bureau of Standards Applied Mathematics Series 55Google Scholar
  4. Abrarov S, Quine B (2015) A rational approximation for efficient computation of the Voigt function in quantitative spectroscopy. J Mater Res 7:163–174Google Scholar
  5. Albrecher H, Asmussen S (2006) Ruin probabilities and aggregate claims distributions for shot noise Cox processes. Scandinavian Actuarial Journal, pp 86–110CrossRefGoogle Scholar
  6. Albrecher H, Constantinescu C, Loisel S (2011) Explicit ruin formulas for models with dependence among risks. Insur Math Econ 48:265–270MathSciNetCrossRefGoogle Scholar
  7. Albrecher H, Beirlant J, Teugels J (2017) Reinsurance: Actuarial and statistical aspects. Wiley, ChichesterCrossRefGoogle Scholar
  8. Asmussen S (2003) Applied probability and queues. Springer Verlag, New YorkzbMATHGoogle Scholar
  9. Asmussen S, Albrecher H (2010) Ruin probabilities. World Scientific Publ. Cy., SingaporeCrossRefGoogle Scholar
  10. Badila ES, Boxma OJ, Resing JAC (2015) Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times. Insur Math Econ 61:48–61MathSciNetCrossRefGoogle Scholar
  11. Boxma OJ, Cohen JW (1998) The M/G/1 queue with heavy-tailed service time distribution. IEEE J Sel Areas Commun 16:749–763CrossRefGoogle Scholar
  12. Boxma OJ, Cohen JW (1999) Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions. Queueing Syst 33:177–204MathSciNetCrossRefGoogle Scholar
  13. Bühlmann H. (1972) Ruinwahrscheinlichkeit bei erfahrungstarifiertem portefeuille. Bulletin de l’Association des Actuaires Suisses 2:131–140zbMATHGoogle Scholar
  14. Chaudry M, Temme N, Veling E (1996) Asymptotic and closed form of a generalized incomplete gamma function. J Comput Appl Math 67:371–379MathSciNetCrossRefGoogle Scholar
  15. Cohen JW (1982) The single server queue. North-Holland Publ. Cy., AmsterdamzbMATHGoogle Scholar
  16. Cohen JW (1997) On the M/G/1 queue with heavy-tailed service time distributions. CWI Report PNA-R9702Google Scholar
  17. Constantinescu C, Hashorva E, Ji L (2011) Archimedean copulas in finite and infinite dimensions – with application to ruin problems. Insur Math Econ 49:487–495MathSciNetCrossRefGoogle Scholar
  18. Dutang C, Lefèvre C, Loisel S (2013) On an asymptotic rule a + b/u for ultimate ruin probabilities under dependence by mixing. Insur Math Econ 53:774–785MathSciNetCrossRefGoogle Scholar
  19. Eick S, Massey W, Whitt W (1993) The physics of the \(M_{t}/G/\infty \) queue. Manag Sci 39:241–252CrossRefGoogle Scholar
  20. Gautschi W (1998) The incomplete gamma functions since Tricomi. In Tricomi’s Ideas Contemp Appl Math 147:203–237MathSciNetzbMATHGoogle Scholar
  21. Jongbloed G, Koole G (2001) Managing uncertainty in call centers using Poisson mixtures. Appl Stoch Model Bus Ind 17:307–318CrossRefGoogle Scholar
  22. Kim S-H, Whitt W (2014) Are call center and hospital arrivals well modeled by nonhomogeneous Poisson processes? Manuf Serv Oper Manag 16:464–480CrossRefGoogle Scholar
  23. Koops D, Boxma OJ, Mandjes MRH (2017) Networks of ⋅/G/∞ queues with shot-noise driven arrival intensities. Queueing Syst 86:301–325MathSciNetCrossRefGoogle Scholar
  24. Nelsen R (1999) An introduction to copulas. Springer, New YorkCrossRefGoogle Scholar
  25. Ross SM (1996) Stochastic processes. Wiley, New YorkzbMATHGoogle Scholar
  26. Temme NM (1979) The asymptotic expansion of the incomplete gamma functions. SIAM J Math Anal 10:757–766MathSciNetCrossRefGoogle Scholar
  27. Zan J, Hasenbein J, Morton DP (2014) Asymptotically optimal staffing of service systems with joint QoS constraints. Queueing Syst 78:359–386MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Youri Raaijmakers
    • 1
    Email author
  • Hansjörg Albrecher
    • 2
    • 3
  • Onno Boxma
    • 1
  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Department of Actuarial Science, Faculty of Business and EconomicsUniversity of LausanneLausanneSwitzerland
  3. 3.Swiss Finance InstituteLausanneSwitzerland

Personalised recommendations