Advertisement

Queueing Systems

, Volume 79, Issue 1, pp 87–115 | Cite as

A model for deceased-donor transplant queue waiting times

  • Steve Drekic
  • David A. Stanford
  • Douglas G. Woolford
  • Vivian C. McAlister
Article

Abstract

In many jurisdictions, organ allocation is done on the basis of the health status of the patient, either explicitly or implicitly. This paper presents a self-promoting priority queueing model for patient waiting times which takes into account changes in health status over time. In this model, most patients arrive as “regular” customers to the queue, but as the health of a patient degrades, their status is promoted to “priority” to reflect the increased urgency of the transplant. We model the queueing system as a level-dependent quasi-birth-and-death process, and the steady-state joint queue length distribution as well as the marginal delay distributions for each queue are computed via the use of matrix analytic techniques. The model is calibrated using liver transplantation wait-list data, provided by a regional health centre in Canada, which tracked approximately 1,100 patients over nearly 13 years. Blood-type-specific models are fit and performance measures, such as the mean and distribution of the time until transplant, are obtained and compared to empirical estimates calculated using the raw data.

Keywords

Abandonments Competing risks Estimation Liver transplantation Phase-type distribution Quasi-birth-and-death process Reneging 

Mathematics Subject Classification

60K25 90B22 

Notes

Acknowledgments

The authors thank the Associate Editor and the referees for their useful comments and valuable suggestions that helped improve this paper. The authors also wish to thank D. Bellhouse who suggested we consider a competing risks framework for calibrating our model. This research was supported by the Natural Sciences and Engineering Research Council of Canada through its Discovery Grants program.

References

  1. 1.
    Abellán, J.J., Armero, C., Conesa, D., Pérez-Panadés, J., Martínez-Beneito, M.A., Zurriaga, O., García-Blasco, M.J., Vanaclocha, H.: Analysis of the renal transplant waiting list in the País Valencià (Spain). Stat. Med. 25, 345–358 (2006)CrossRefGoogle Scholar
  2. 2.
    Barone, M., Avolio, A.W., Di Leo, A., Burra, P., Francavilla, A.: ABO blood group-related waiting list disparities in liver transplant candidates: effect of the MELD adoption. Transplantation 85, 844–849 (2008)CrossRefGoogle Scholar
  3. 3.
    Bazarah, S.M., Peltekian, K.M., McAlister, V.C., Bitter-Suermann, H., MacDonald, A.S.: Utility of MELD and Child–Turcotte–Pugh scores and the Canadian waitlisting algorithm in predicting short-term survival after liver transplant. Clin. Investig. Med. 27, 162–167 (2004)Google Scholar
  4. 4.
    Bright, L., Taylor, P.G.: Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes. Stoch. Models 11, 497–525 (1995)CrossRefGoogle Scholar
  5. 5.
  6. 6.
    Gaver, D.P., Jacobs, P.A., Latouche, G.: Finite birth-and-death models in randomly changing environments. Adv. Appl. Probab. 16, 715–731 (1984)CrossRefGoogle Scholar
  7. 7.
    Glander, P., Budde, K., Schmidt, D., Fuller, T.F., Giessing, M., Neumayer, H.-H., Liefeldt, L.: The ‘blood group O problem’ in kidney transplantation: time to change? Nephrol. Dial. Transplant. 25, 1998–2004 (2010)CrossRefGoogle Scholar
  8. 8.
    Gómez-Corral, A., Krishnamoorthy, A., Narayanan, V.C.: The impact of self-generation of priorities on multi-server queues with finite capacity. Stoch. Models 21, 427–447 (2005)CrossRefGoogle Scholar
  9. 9.
    Hussey, J.C., Parameshwar, J., Banner, N.R.: Influence of blood group on mortality and waiting time before heart transplantation in the United Kingdom: implications for equity of access. J. Heart Lung Transplant. 26, 30–33 (2007)CrossRefGoogle Scholar
  10. 10.
    Kalbfleisch, J.D., Prentice, R.L.: The Statistical Analysis of Failure Time Data, 2nd edn. Wiley, Hoboken (2002)CrossRefGoogle Scholar
  11. 11.
    Krishnamoorthy, A., Narayanan, V.C.: On a queueing system with self generation of priorities. In: Srinivasan, S.K., Vijayakumar, A. (eds.) Stochastic Point Processes, pp. 212–217. Narosa Publishing House, Chennai (2003)Google Scholar
  12. 12.
    Krishnamoorthy, A., Narayanan, V.C., Deepak, T.G.: On a queueing system with self generation of priorities. Neural Parallel Sci. Comput. 13, 119–130 (2005)Google Scholar
  13. 13.
    Krishnamoorthy, A., Babu, S., Narayanan, V.C.: MAP/(PH/PH)/\(c\) queue with self-generation of priorities and non-preemptive service. Stoch. Anal. Appl. 26, 1250–1266 (2008)CrossRefGoogle Scholar
  14. 14.
    Krishnamoorthy, A., Babu, S., Narayanan, V.C.: The MAP/(PH/PH)/1 queue with self-generation of priorities and non-preemptive service. Eur. J. Oper. Res. 195, 174–185 (2009)CrossRefGoogle Scholar
  15. 15.
    Krishnamoorthy, A., Narayanan, V. C., Chakravarthy, S. R.: The impact of priority generations in a multi-priority queueing system: a simulation approach. In: Proceedings of the 2009 Winter Simulation Conference (Eds. M.D. Rossetti, R.R. Hill, B. Johansson, A. Dunkin & R.G. Ingalls), IEEE Computer Society, New York, USA, pp. 1622-1633 (2009)Google Scholar
  16. 16.
    Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  17. 17.
    Lawless, J.F.: Statistical Models and Methods for Lifetime Data, 2nd edn. Wiley, Hoboken (2003)Google Scholar
  18. 18.
    Liefeldt, L., Budde, K., Glander, P.: Accumulation of elderly ESRD patients with blood group O on the waiting list (Letter to editors). Transpl. Int. 24, e83–e84 (2011)CrossRefGoogle Scholar
  19. 19.
    Maertens, T., Walraevens, J., Bruneel, H.: Performance comparison of several priority schemes with priority jumps. Ann. Oper. Res. 162, 109–125 (2008)CrossRefGoogle Scholar
  20. 20.
    Pinsky, M.A., Karlin, S.: An Introduction to Stochastic Modeling, 4th edn. Academic Press, Boston (2011)Google Scholar
  21. 21.
    Stanford, D.A., Renouf, E.M., McAlister, V.C.: Waiting for liver transplantation in Canada: Waitlist history 2000–04 and sensitivity analysis for the future. Health Care Manag. Sci. 11, 184–195 (2008)CrossRefGoogle Scholar
  22. 22.
    Stanford, D.A., Lee, J.M., Chandok, N., McAlister, V.C.: A queueing model to address wait time inconsistency in solid-organ transplantation. Oper. Res. Health Care 3, 40–45 (2014)Google Scholar
  23. 23.
    Su, X., Zenios, S.: Patient choice in kidney allocation: the role of the queueing discipline. Manuf. Serv. Oper. Manag. 6, 280–301 (2004)Google Scholar
  24. 24.
    Syski, R.: Passage Times for Markov Chains. IOS Press, Amsterdam (1992)Google Scholar
  25. 25.
    Tijms, H.C.: A First Course in Stochastic Models. Wiley, Chichester (2003)CrossRefGoogle Scholar
  26. 26.
    Wang, Q.: Modeling and analysis of high risk patient queues. Eur. J. Oper. Res. 155, 502–515 (2004)CrossRefGoogle Scholar
  27. 27.
    Wiesner, R., Edwards, E., Freeman, R., Harper, A., Kim, R., Kamath, P., et al.: Model for end-stage liver disease (MELD) and allocation of donor livers. Gastroenterology 124, 91–96 (2003)CrossRefGoogle Scholar
  28. 28.
    Zenios, S.A.: Modeling the transplant waiting list: a queueing model with reneging. Queueing Syst. 31, 239–251 (1999)CrossRefGoogle Scholar
  29. 29.
    Zenios, S.A., Chertow, G.M., Wein, L.M.: Dynamic allocation of kidneys to candidates on the transplant waiting list. Oper. Res. 48, 549–569 (2000)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Steve Drekic
    • 1
  • David A. Stanford
    • 2
  • Douglas G. Woolford
    • 3
  • Vivian C. McAlister
    • 4
  1. 1.Department of Statistics and Actuarial ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Department of Statistical & Actuarial SciencesWestern UniversityLondonCanada
  3. 3.Department of MathematicsWilfrid Laurier UniversityWaterlooCanada
  4. 4.Department of SurgeryWestern UniversityLondonCanada

Personalised recommendations