• Kim, Jerim ;
  • Kim, Jeongsim
  • Received : 2011.08.08
  • Published : 2013.04.30


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.


M/PH/1 queue;impatience time;workload;loss probability;waiting time distribution;queue size distribution


  1. 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.
  2. 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.
  3. 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.
  4. D. J. Daley, General customer impatience in the queue GI/G/1, J. Appl. Probability 2 (1965), 186-205.
  5. P. D. Finch, Deterministic customer impatience in the queueing system GI/M/1, Biometrika 47 (1960), 45-52.
  6. B. V. Gnedenko and I. N. Kovalenko, Introduction to Queueing Theory, Israel Program for Scientific Translations, Jerusalem, 1968.
  7. 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.
  8. 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.
  9. A. G. de Kok and H. C. Tijms, A queueing system with impatient customers, J. Appl. Probab. 22 (1985), no. 3, 688-696.
  10. A. Movaghar, On queueing with customer impatience until the beginning of service, Queueing Syst. 29 (1998), no. 2-4, 337-350.
  11. W. Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill, 1987.
  12. R. E. Stanford, Reneging phenomena in single channel queues, Math. Oper. Res. 4 (1979), no. 2, 162-178.
  13. R. E. Stanford, On queues with impatience, Adv. in Appl. Probab. 22 (1990), no. 3, 768-769.
  14. R. W. Wolff, Stochastic Modeling and the Theory of Queues, Prentice Hall, Englewood Cliffs, NJ, 1989.
  15. W. Xiong, D. Jagerman, and T. Altiok, M/G/1 queue with deterministic reneging times, Performance Evaluation 65 (2008), no. 3-4, 308-316.
  16. S. Asmussen, Applied Probability and Queues, Second Edition, Springer, 2003.
  17. 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.
  18. F. Baccelli, P. Boyer, and G. Hebuterne, Single server queues with impatient customers, Adv. in Appl. Probab. 16 (1984), no. 4, 887-905.
  19. F. Baccelli and G. Hebuterne, On queues with impatient customers, Performance '81 (Amsterdam, 1981), 159-179, North-Holland, Amsterdam-New York, 1981.
  20. D. Y. Barrer, Queueing with impatient customers and indifferent clerks, Oper. Res. 5 (1957), 644-649.
  21. D. Y. Barrer, Queueing with impatient customers and ordered service, Oper. Res. 5 (1957), 650-656.

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Supported by : Chungbuk National University