DOI QR코드

DOI QR Code

M/PH/1 QUEUE WITH DETERMINISTIC IMPATIENCE TIME

  • Kim, Jerim (Department of Business Administration Yongin University) ;
  • Kim, Jeongsim (Department of Mathematics Education Chungbuk National University)
  • Received : 2011.08.08
  • Published : 2013.04.30

Abstract

We consider an M/PH/1 queue with deterministic impatience time. An exact analytical expression for the stationary distribution of the workload is derived. By modifying the workload process and using Markovian structure of the phase-type distribution for service times, we are able to construct a new Markov process. The stationary distribution of the new Markov process allows us to find the stationary distribution of the workload. By using the stationary distribution of the workload, we obtain performance measures such as the loss probability, the waiting time distribution and the queue size distribution.

Keywords

Acknowledgement

Supported by : Chungbuk National University

References

  1. S. Asmussen, Applied Probability and Queues, Second Edition, Springer, 2003.
  2. J. Bae and S. Kim, The stationary workload of the G/M/1 queue with impatient customers, Queueing Syst. 64 (2010), no. 3, 253-265. https://doi.org/10.1007/s11134-009-9159-0
  3. F. Baccelli, P. Boyer, and G. Hebuterne, Single server queues with impatient customers, Adv. in Appl. Probab. 16 (1984), no. 4, 887-905. https://doi.org/10.2307/1427345
  4. F. Baccelli and G. Hebuterne, On queues with impatient customers, Performance '81 (Amsterdam, 1981), 159-179, North-Holland, Amsterdam-New York, 1981.
  5. D. Y. Barrer, Queueing with impatient customers and indifferent clerks, Oper. Res. 5 (1957), 644-649. https://doi.org/10.1287/opre.5.5.644
  6. D. Y. Barrer, Queueing with impatient customers and ordered service, Oper. Res. 5 (1957), 650-656. https://doi.org/10.1287/opre.5.5.650
  7. O. J. Boxma and P. R. de Waal, Multiserver queues with impatient customers, In: The Fundamental Role of Teletraffic in the Evolution of Telecommunications Networks (Proc. ITC 14), 743-756, North-Holland, Amsterdam, 1994.
  8. A. Brandt and M. Brandt, On the M(n)/M(n)/s queues with impatient calls, Performance Evaluation 35 (1999), no. 1-2, 1-18. https://doi.org/10.1016/S0166-5316(98)00042-X
  9. A. Brandt and M. Brandt, Asymptotic results and a Markovian approximation for the M(n)/M(n)/s+GI system, Queueing Syst. 41 (2002), no. 1-2, 73-94. https://doi.org/10.1023/A:1015781818360
  10. D. J. Daley, General customer impatience in the queue GI/G/1, J. Appl. Probability 2 (1965), 186-205. https://doi.org/10.2307/3211884
  11. P. D. Finch, Deterministic customer impatience in the queueing system GI/M/1, Biometrika 47 (1960), 45-52. https://doi.org/10.1093/biomet/47.1-2.45
  12. B. V. Gnedenko and I. N. Kovalenko, Introduction to Queueing Theory, Israel Program for Scientific Translations, Jerusalem, 1968.
  13. O. M. Jurkevic, On the investigation of many-server queueing systems with bounded waiting time (in Russian), Izv. Akad. Nauk SSSR Techniceskaja Kibernetika 5 (1970), 50-58.
  14. O. M. Jurkevic, On many-server systems with stochastic bounds for the waiting time (in Russian), Izv. Akad. Nauk SSSR Techniceskaja Kibernetika 4 (1971), 39-46.
  15. A. G. de Kok and H. C. Tijms, A queueing system with impatient customers, J. Appl. Probab. 22 (1985), no. 3, 688-696. https://doi.org/10.2307/3213871
  16. A. Movaghar, On queueing with customer impatience until the beginning of service, Queueing Syst. 29 (1998), no. 2-4, 337-350. https://doi.org/10.1023/A:1019196416987
  17. W. Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill, 1987.
  18. R. E. Stanford, Reneging phenomena in single channel queues, Math. Oper. Res. 4 (1979), no. 2, 162-178. https://doi.org/10.1287/moor.4.2.162
  19. R. E. Stanford, On queues with impatience, Adv. in Appl. Probab. 22 (1990), no. 3, 768-769. https://doi.org/10.2307/1427473
  20. R. W. Wolff, Stochastic Modeling and the Theory of Queues, Prentice Hall, Englewood Cliffs, NJ, 1989.
  21. W. Xiong, D. Jagerman, and T. Altiok, M/G/1 queue with deterministic reneging times, Performance Evaluation 65 (2008), no. 3-4, 308-316. https://doi.org/10.1016/j.peva.2007.07.003

Cited by

  1. MAP/M/c and M/PH/c queues with constant impatience times vol.82, pp.3-4, 2016, https://doi.org/10.1007/s11134-015-9455-9
  2. Discrete-time renewal input queue with balking and multiple working vacations vol.10, pp.3, 2015, https://doi.org/10.1080/17509653.2014.954651
  3. Analysis of the loss probability in the M/G/1+G queue vol.80, pp.4, 2015, https://doi.org/10.1007/s11134-015-9449-7
  4. Multi-class M/PH/1 queues with deterministic impatience times vol.33, pp.1, 2017, https://doi.org/10.1080/15326349.2016.1197778