Skip to main content
SpringerLink
Log in

Asymptotic independence of regenerative processes with a special dependence structure

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

We identify some conditions under which regenerative processes with a certain dependence structure among them are asymptotically independent. The result is applied to various models, in particular independent Lévy processes with dependent secondary jumps at the origin (for example, workloads of parallel M/G/1 queues with server vacations), the asymptotic performance of systems with multiple correlated sources that generate real-time status updates measured by the limiting probability of an updated system, and asymptotic results for clearing processes with dependent arrivals of clearings. Finally, the asymptotic distribution of the classic Jackson network is discussed as yet another example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, Berlin (2003)

    Google Scholar 

  2. Balsamo, S.: Product form queueing networks. In: Haring, G., Lindemann, C., Reiser, M. (eds.) Performance Evaluation: Origins and Directions, pp. 377–401. Springer, Berlin (2000)

    Chapter  Google Scholar 

  3. Chao, X., Pinedo, M., Miyazawa, M.: Queueing Networks: Customers, Signals, and Product Form Solutions. Wiley, New York (1999)

    Google Scholar 

  4. Chen, H., Whitt, W.: Diffusion approximations for open queueing networks with service interruptions. Queueing Syst. 13, 335–359 (1993)

    Article  Google Scholar 

  5. Cohen, J.W.: On regenerative processes in queueing theory, vol. 121. Springer, Berlin (2012)

    Google Scholar 

  6. Doney, R.A.: An analogue of the renewal theorem in higher dimensions. Proc. Lond. Math. Soc. 3(1), 669–684 (1966)

    Article  Google Scholar 

  7. Glynn, P.W.: Some new results in regenerative process theory. Technical report 60, Department of operation research, Stanford university, Stanford, California (1982)

  8. Huang, L., Modiano, E.: Optimizing age-of-information in a multi-class queueing system. In: Proceeding of IEEE International Symposium on Information Theory (ISIT), 1681–1685 (2015)

  9. Hunter, J.: Renewal theory in two dimensions: basic results. Adv. Appl. Probab. 6(2), 376–391 (1974). https://doi.org/10.2307/1426299

    Article  Google Scholar 

  10. Hunter, J.: Renewal theory in two dimensions: asymptotic results. Adv. Appl. Probab. 6(3), 546–562 (1974). https://doi.org/10.2307/1426233

    Article  Google Scholar 

  11. Hunter, J.: Renewal theory in two dimensions: bounds on the renewal function. Adv. Appl. Probab. 9(3), 527–541 (1977). https://doi.org/10.2307/1426112

    Article  Google Scholar 

  12. Kaul, S., Yates, R.D., Gruteser, M.: Status updates through queues. In: Conference on Information Sciences and Systems (CISS), (2012)

  13. Kella, O.: An exhaustive Lévy storage process with intermittent output. Stoch. Models 14, 979–992 (1998)

    Article  Google Scholar 

  14. Kella, O., Stadje, W.: A clearing system with impatient passengers: asymptotics and estimation in a bus stop model. Queueing Syst. 80, 1–14 (2015)

    Article  Google Scholar 

  15. Kella, O., Whitt, W.: Diffusion approximations for queues with server vacations. Adv. Appl. Probab. 22, 706–729 (1990)

    Article  Google Scholar 

  16. Kella, O., Whitt, W.: Queues with server vacations and Lévy processes with secondary jump input. Ann. Appl. Prob. 1, 104–117 (1991)

    Article  Google Scholar 

  17. Kella, O., Whitt, W.: Useful martingales for stochastic storage processes with Lévy input. J. Appl. Probab. 29, 396–403 (1992)

    Article  Google Scholar 

  18. Miyazawa, M., Kella, O.: Parallel fluid queues with constant inflows and simultaneous random reductions. J. Appl. Probab. 38, 609–620 (2001)

    Article  Google Scholar 

  19. Nagaev, A.V.: Renewal theorems in \(R^d\). Theory Probab. Appl. 24(3), 572–581 (1980)

    Google Scholar 

  20. Najm, E., Nasser, R.: The age of information: the gamma awakening. In: Proceeding of IEEE International Symposium on Infomation Theory (ISIT), 2574–2578 (2016)

  21. Najm, E., Yates, R., Soljanin, E.: Status updates through M/G/1/1 queues with HARQ. arXiv preprint arXiv:1704.03937 (2017)

  22. Serfozo, R.F., Stidham Jr., S.: Semistationary clearing processes. Stoch. Proc. Appl. 6, 165–178 (1978)

    Article  Google Scholar 

  23. Sigman, K., Wolff, R.W.: A review of regenerative processes. SIAM Rev. 35(2), 269–288 (1993)

    Article  Google Scholar 

  24. Stam, A.J.: Renewal theory in \(r\) dimensions I. Compos. Math. 21, 383–399 (1969)

    Google Scholar 

  25. Stam, A.J.: Renewal theory in \(r\) dimensions. II. Compos. Math. 23, 1–13 (1971)

    Google Scholar 

  26. Stidham Jr., S.: Stochastic clearing systems. Stoch. Proc. Appl. 2, 85–113 (1974)

    Article  Google Scholar 

  27. Stidham Jr., S.: Cost models for stochastic clearing systems. Oper. Res. 25, 100–127 (1977)

    Article  Google Scholar 

  28. Thorisson, H.: Coupling, Stationarity and Regeneration. Springer, Berlin (2000)

    Book  Google Scholar 

  29. Thorisson, H.: Construction of stationary regenerative process. Stoch. Proc. Appl. 42, 237–253 (1992)

    Article  Google Scholar 

  30. Whitt, W.: The stationary distribution of a stochastic clearing process. Oper. Res. 29, 294–308 (1981)

    Article  Google Scholar 

  31. Yates, R.D., Kaul, S.: Real-time status updating: multiple sources. In: Proceeding of IEEE International Symposium on Information Theory (ISIT), 2666–2670 (2012)

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, The Hebrew University of Jerusalem, 9190501, Jerusalem, Israel

    Royi Jacobovic & Offer Kella

Authors
  1. Royi Jacobovic
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Offer Kella
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Offer Kella.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Offer Kella: Supported by Grant No. 1647/17 from the Israel Science Foundation and the Vigevani Chair in Statistics.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jacobovic, R., Kella, O. Asymptotic independence of regenerative processes with a special dependence structure. Queueing Syst 93, 139–152 (2019). https://doi.org/10.1007/s11134-019-09606-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11134-019-09606-1

Keywords

Mathematics Subject Classification

Search

Navigation