Abstract
We consider a GI/GI/1 queue with the shortest remaining processing time discipline (SRPT) and light-tailed service times. Our interest is focused on the tail behavior of the sojourn-time distribution. We obtain a general expression for its large-deviations decay rate. The value of this decay rate critically depends on whether there is mass in the endpoint of the service-time distribution or not. An auxiliary priority queue, for which we obtain some new results, plays an important role in our analysis. We apply our SRPT results to compare SRPT with FIFO from a large-deviations point of view.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Abate and W. Whitt, Asymptotics for M/G/1 low-priority waiting-time tail probabilities. Queueing Systems 25 (1997) 173–233.
S. Asmussen, Applied Probability and Queues, 2nd edition (Springer, 2003).
F. Baccelli and P. Brémaud, Elements of Queueing Theory 3rd edition (Springer, 2003).
N. Bansal and M. Harchol-Balter, Analysis of SRPT scheduling: investigating unfairness. In Proceedings of ACM Sigmetrics (2001) pp. 279–290.
N. Bansal, On the average sojourn time under M/M/1 SRPT. Operations Research Letters 33 (2004) 195–200.
N. Bansal and D. Gamarnik, Handling load with less stress (Submitted for publication, 2005).
S.C. Borst, O.J. Boxma, R. Nunez-Queija, and A.P. Zwart, The impact of the service discipline on delay asymptotics. Performance Evaluation 54 (2003) 177–206.
D. Cox and W. Smith, Queues. Methuen (1961).
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications (Springer, 1998).
R. Egorova, B. Zwart, and O.J. Boxma, Sojourn time tails in the M/D/1 Processor Sharing queue. Probability in the Engineering and Informational Sciences 20 (2006) 429–446.
A. Ganesh, N. O’Connell, and D. Wischik, Big Queues (Springer, 2003).
P. Glynn and W. Whitt, Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. Journal of Applied Probability 31A (1994) 131–156.
M. Harchol-Balter, B. Schroeder, N. Bansal, and M. Agrawal, Sizebased scheduling to improve web performance. ACM Transactions on Computer Systems 21 (2003) 207–233.
J.F.C. Kingman, A martingale inequality in the theory of queues. In Proceedings of the Cambridge Philosophical Society, vol. 59 (1964) pp. 359–361.
M. Mandjes and M. Nuyens, Sojourn times in the M/G/1 FB queue with light-tailed service times. Probability in the Engineering and Informational Sciences 19 (2005) 351–361.
M. Mandjes and M. Van Uitert, Sample path large deviations for tandem and priority queues with Gaussian input. Annals of Applied Probability 15 (2005) 1193–1226.
M. Mandjes and B. Zwart, Large deviations for sojourn times in processor sharing queues. Queueing Systems 52 (2006) 237–250.
R. Núñez-Queija, Processor-Sharing Models for Integrated-Service Networks. PhD thesis, Eindhoven University of Technology (2000).
Z. Palmowski and T. Rolski, On busy period asymptotics in the GI/GI/1 queue. To appear in Advances in Applied Probability (2006), available at http://www.math.uni.wroc.pl/~zpalma/publication.html
K. Ramanan and A. Stolyar, Largest weighted delay first scheduling: large deviations and optimality. Annals of Applied Probability 11 (2001) 1–48.
W. Rosenkrantz, Calculation of the Laplace transform of the length of the busy period for the M/G/1 queue via martingales. Annals of Probability 11 (1983) 817–818.
L. Schrage, A proof of the optimality of the shortest remaining service time discipline. Operations Research 16 (1968) 670–690.
Author information
Authors and Affiliations
Corresponding author
Additional information
2000 Mathematics Subject Classification: Primary—60K25; Secondary—60F10; 90B22
Rights and permissions
About this article
Cite this article
Nuyens, M., Zwart, B. A large-deviations analysis of the GI/GI/1 SRPT queue. Queueing Syst 54, 85–97 (2006). https://doi.org/10.1007/s11134-006-8767-1
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11134-006-8767-1