Abstract
The paper deals with the workload and busy period for the \(M/GI/1\) system with impatience under FCFS discipline. The customers may become impatient during their waiting for service with generally distributed maximal waiting times and also during their service with generally distributed maximal service times depending on the time waited for service. This general impatience mechanism was originally introduced by Kovalenko (1961) and considered by Daley (1965), too. It covers the special cases of impatience on waiting times as well as impatience on sojourn times, for which Boxma et al. (2010, 2011) gave new results and outlined special cases recently. Our unified approach bases on the vector process of workload and busy time. Explicit representations for the LSTs of workload and busy period are given in case of phase-type distributed impatience.
Similar content being viewed by others
References
Asmussen, S.: Applied Probability and Queues. Wiley, Chichester (1987)
Baccelli, F., Boyer, P., Hebuterne, G.: Single-server queues with impatient customers. Adv. Appl. Probab. 16, 887–905 (1984)
Bekker, R., Koole, G.M., Nielsen, B.F., Nielsen, T.B.: Queues with waiting time dependent service. Queueing Syst. 68, 61–78 (2011)
Boxma, O.J., Perry, D., Stadje, W., Zacks, S.: The busy period of an \(M/G/1\) queue with customer impatience. J. Appl. Probab. 47, 130–145 (2010)
Boxma, O.J., Perry, D., Stadje, W.: The \(M/G/1+G\) queue revisited. Queueing Syst. 67, 207–220 (2011)
Daley, D.J.: General customer impatience in the queue GI/G/1. J. Appl. Probab. 2, 186–205 (1965)
Kim, S., Bae, J., Lee, E.Y.: Busy periods of Poisson arrival queues with loss. Queueing Syst. 39, 201–212 (2001)
Kovalenko, I.N.: Some queuing problems with restrictions. Theory Probab. Appl. 6, 204–208 (1961)
O’Cinneide, C.A.: Characterization of phase-type distributions. Commun. Stat. Stoch. Models 6, 1–57 (1990)
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)
Stanford, R.E.: Reneging phenomena in single channel queues. Math. Oper. Res. 4, 162–178 (1979)
Stanford, R.E.: On queues with impatience. Adv. Appl. Probab. 22, 768–769 (1990)
Subba Rao, S.: Queuing with balking and reneging in \(M/G/1\) systems. Metrika 12, 173–188 (1968)
Xiong, W., Jagerman, D., Altiok, T.: \(M/G/1\) queue with deterministic reneging times. Perform. Eval. 65, 308–316 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brandt, A., Brandt, M. Workload and busy period for \(M/GI/1\) with a general impatience mechanism. Queueing Syst 75, 189–209 (2013). https://doi.org/10.1007/s11134-013-9373-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-013-9373-7