Queueing Systems

, 58:239 | Cite as

On priority queues with impatient customers

  • Foad Iravani
  • Barış Balcıog̃luEmail author


In this paper, we study three different problems where one class of customers is given priority over the other class. In the first problem, a single server receives two classes of customers with general service time requirements and follows a preemptive-resume policy between them. Both classes are impatient and abandon the system if their wait time is longer than their exponentially distributed patience limits. In the second model, the low-priority class is assumed to be patient and the single server chooses the next customer to serve according to a non-preemptive priority policy in favor of the impatient customers. The third problem involves a multi-server system that can be used to analyze a call center offering a call-back option to its impatient customers. Here, customers requesting to be called back are considered to be the low-priority class. We obtain the steady-state performance measures of each class in the first two problems and those of the high-priority class in the third problem by exploiting the level crossing method. We furthermore adapt an algorithm from the literature to obtain the factorial moments of the low-priority queue length of the multi-server system exactly.


Call centers M/GI/1+M queue Priority queues Impatient customers Level-crossing method Call-back 

Mathematics Subject Classification (2000)

60K25 68M20 90B22 


  1. 1.
    Ahghari, M., Balcıog̃lu, B.: Benefits of cross-training in a skill-based routing contact center with priority queues and impatient customers. Technical Report MIE-OR-TR2006-05, D. of Mechanical & Industrial Engineering, University of Toronto (2006) Google Scholar
  2. 2.
    Argon, N.T., Ziya, S., Righter, R.: Scheduling impatient jobs in a clearing system with insights on patient triage in mass casualty incidents. Probab. Eng. Inf. Sci. (2008, forthcoming) Google Scholar
  3. 3.
    Armony, M., Maglaras, C.: On customer contact centers with a call-back option: customer decisions, routing rules, and system design. Oper. Res. 52(2), 271–292 (2004) CrossRefGoogle Scholar
  4. 4.
    Armony, M., Maglaras, C.: Contact centers with a call-back option and real-time delay information. Oper. Res. 52(4), 527–545 (2004) CrossRefGoogle Scholar
  5. 5.
    Baccelli, F., Hebuterne, G.: On queues with impatient customers. In: Kylstra, F.J. (ed.) Performance ’81, vol. 32, pp. 159–179. North-Holland, Amsterdam (1981) Google Scholar
  6. 6.
    Baccelli, F., Boyer, P., Hebuterne, G.: Single-server queues with impatient customers. Adv. Appl. Probab. 16, 887–905 (1984) CrossRefGoogle Scholar
  7. 7.
    Brandt, A., Brandt, M.: On a two-queue priority system with impatience and its applications to a call center. Methodol. Comput. Appl. Probab. 1(2), 191–210 (1999) CrossRefGoogle Scholar
  8. 8.
    Brandt, A., Brandt, M.: On the two-class M/M/1 system under preemptive resume and impatience of the prioritized customers. Queueing Syst. 47, 147–168 (2004) CrossRefGoogle Scholar
  9. 9.
    Brill, P.H.: System point theory in exponential queues. Ph.D. Thesis, Department of Industrial Engineering, University of Toronto (1975) Google Scholar
  10. 10.
    Brill, P.H.: An embedded level crossing technique for dams and queues. J. Appl. Probab. 16(1), 174–186 (1979) CrossRefGoogle Scholar
  11. 11.
    Brill, P.H., Posner, M.J.M.: Level crossings in point processes applied to queues: single-server case. Oper. Res. 25(4), 662–674 (1977) Google Scholar
  12. 12.
    Burton, T.A.: Volterra Integral and Differential Equations, 2nd edn. Elsevier, Amsterdam (2005) Google Scholar
  13. 13.
    Choi, B.D., Kim, B., Chung, J.: M/M/1 queue with impatient customers of higher priority. Queueing Syst. 38, 49–66 (2001) CrossRefGoogle Scholar
  14. 14.
    Daley, D.J.: General customer impatience in the queue GI/G/1. J. Appl. Probab. 2(1), 186–205 (1965) CrossRefGoogle Scholar
  15. 15.
    Gaver, D.P.: A waiting line with interrupted service, including priorities. J. R. Stat. Soc. 24(1), 73–90 (1962) Google Scholar
  16. 16.
    Iravani, F., Balcıog̃lu, B.: Approximations for the M/GI/N+GI type call center. Queueing Syst. 58(2), 137–153 (2008) CrossRefGoogle Scholar
  17. 17.
    Jagerman, D.L.: An inversion technique for the Laplace transform. Bell Syst. Tech. J. 61(8), 1995–2002 (1982) Google Scholar
  18. 18.
    Rao, S.S.: Queueing with balking and reneging in M/G/1 systems. Metrika 12, 173–188 (1967) CrossRefGoogle Scholar
  19. 19.
    Sanajian, N., Balcioglu, B.: The impact of production time variability on make-to-stock queue performance. Eur. J. Oper. Res. (2008). doi: 10.1016/j.ejor.2008.01.020 Google Scholar
  20. 20.
    Sandhu, D., Posner, M.J.M.: A priority M/G/1 queue with application to voice/data communication. Eur. J. Oper. Res. 40, 99–108 (1989) CrossRefGoogle Scholar
  21. 21.
    Shanthikumar, J.G.: On level crossing analysis of queues. Aust. J. Stat. 23, 377–411 (1981) Google Scholar
  22. 22.
    Stanford, R.E.: Reneging phenomenon in single channel queues. Math. Oper. Res. 4(2), 162–178 (1979) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.UCLA Anderson School of ManagementLos AngelesUSA
  2. 2.Department of Mechanical and Industrial EngineeringUniversity of TorontoTorontoCanada

Personalised recommendations