Abstract
We consider a G/M/1 queue in which the patience time of the customers is constant. The stationary distribution of the workload of the server, or the virtual waiting time, is derived by the level crossing argument. To this end, we obtain the expected downcrossings of a level in the workload process during a busy cycle and then the expected length of a busy cycle. For both the expectations, we use the dual property between the M/G/1 and G/M/1 queue.
Similar content being viewed by others
References
Adan, I., Boxma, O., Perry, D.: The G/M/1 queue revisited. Math. Methods Oper. Res. 62, 437–452 (2005)
Asmussen, S.: Applied Probability and Queues. Wiley, New York (1987)
Baccelli, F., Boyer, P., Hebuterne, G.: Single-server queues with impatient customers. Adv. Appl. Probab. 16, 887–905 (1984)
Bae, J., Kim, S., Lee, E.Y.: The virtual waiting time of the M/G/1 queue with impatient customers. Queueing Syst. 38, 485–494 (2001)
Bae, J., Kim, S., Lee, E.Y.: A \(P^{M}_{\lambda}\) policy for an M/G/1 queueing system. Appl. Math. Model. 26, 929–939 (2002)
Barrer, D.Y.: Queuing with impatient customers and ordered service. Oper. Res. 5, 650–656 (1957)
Brandt, A., Brandt, M.: On the M(n)/M(n)/s queue with impatient calls. Perform. Eval. 35, 1–18 (1999)
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)
Brill, P.H.: Level Crossings Methods in Stochastic Models. Springer, Berlin (2008)
Brill, P.H., Posner, M.J.M.: Level crossings in point processes applied to queues: single-server case. Oper. Res. 25, 662–674 (1977)
Choi, B.D., Kim, B., Chung, J.: M/M/1 queue with impatient customers of higher priority. Queueing Syst. 38, 49–66 (2001)
Choi, B.D., Kim, B., Zhu, D.: MAP/M/c queue with constant impatience time. Math. Oper. Res. 29, 309–325 (2004)
Cohen, J.W.: Extreme value distribution for the M/G/1 and the G/M/1 queueing systems. Ann. l’Inst. Henri Poincaré, Sect. B, Calc. Probab. Stat. 4, 83–98 (1968)
Cohen, J.W.: On up- and downcrossings. J. Appl. Probab. 14, 405–410 (1977)
Cohen, J.W.: Properties of the process of level crossings during a busy cycle of the M/G/1 queueing system. Math. Oper. Res. 3, 133–144 (1978)
Cohen, J.W.: The Single Server Queue, 2nd edn. North-Holland, Amsterdam (1982)
Daley, D.J.: General customer impatience in the queue GI/G/1. J. Appl. Probab. 2, 186–205 (1965)
De Kok, A.G., Tijms, H.C.: A queueing system with impatient customers. J. Appl. Probab. 22, 688–696 (1985)
Finch, P.D.: Deterministic customer impatience in the queueing system GI/M/1. Biometrika 47, 45–52 (1960)
Kim, S., Bae, J., Lee, E.Y.: Busy periods of Poisson arrival queues with loss. Queueing Syst. 39, 201–212 (2001)
Lillo, R.E., Martin, M.: Stability in queues with impatient customers. Stoch. Models 17, 375–389 (2001)
Martin, M., Artalejo, J.R.: Analysis of an M/G/1 queue with two types of impatient units. Adv. Appl. Probab. 27, 840–861 (1995)
Movaghar, A.: On queueing with customer impatience until the beginning of service. Queueing Syst. 29, 337–350 (1998)
Perry, D., Asmussen, S.: Rejection rules in the M/G/1 queue. Queueing Syst. 19, 105–130 (1995)
Perry, D., Stadje, W., Zacks, S.: Busy period analysis for M/G/1 and G/M/1 type queues with restricted accessibility. Oper. Res. Lett. 27, 163–174 (2000)
Swensen, A.R.: On a GI/M/c queue with bounded waiting times. Oper. Res. 34, 895–908 (1986)
Teghem, J., Jr.: Use of discrete transforms for the study of a GI/M/S queue with impatient customer phenomena. Math. Methods Oper. Res. 23, 95–106 (1979)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bae, J., Kim, S. The stationary workload of the G/M/1 queue with impatient customers. Queueing Syst 64, 253–265 (2010). https://doi.org/10.1007/s11134-009-9159-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-009-9159-0