Heavy traffic limit for a processor sharing queue with soft deadlines

Annals of Applied Probability

Volumn 17, Issue 3, 2007, Pages 1049-1101

  Gromoll, H Christian ^a   Kruk, Łukasz ^a,b  

^a STANFORD UNIVERSITY   (United States)

^b MARIA CURIE SK ODOWSKA UNIVERSITY   (Poland)

Author keywords

Deadlines; Empirical process; Heavy traffic; Measure valued; Process; Processor sharing; Real time queue

Indexed keywords

EID: 52349118603     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/105051607000000014     Document Type: Article

References (26)

1. BILLINGSLEY, P. (1986). Probability and Measure, 2nd ed. Wiley, New York. MR0830424
2. BILLINGSLEY, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York. MR1700749
3. BRAMSON, M. (1998). State space collapse with applications to heavy traffic limits for multi-class queueing networks. Queueing Syst. 30 89-148. MR1663763
4. BRAMSON, M. (2001). Stability of earliest-due-date, first-served queueing networks. Queueing Syst. 39 79-102. MR1865459
5. BRAMSON, M. and DAI, J. (2001). Heavy traffic limits for some queueing networks. Ann. Appl. Probab. 11 49-90. MR1825460
6. DOYTCHINOV, B., LEHOCZKY, J. and SHREVE, S. (2001). Real-time queues in heavy traffic with earliest-deadline-first queue discipline. Ann. Appl. Probab. 11 332-378. MR1843049
7. ETHIER, S. N. and KURTZ, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. MR0838085
8. GRISHECHKIN, S. (1994). GI/G/1 processor sharing queue in heavy traffic. Adv. in Appl. Probab. 26 539-555. MR1272726
9. GROMOLL, H. C. (2004). Diffusion approximation for a processor sharing queue in heavy traffic. Ann. Appl. Probab. 14 555-611. MR2052895
10. GROMOLL, H. C., PUHA, A. L. and WILLIAMS, R. J. (2002). The fluid limit of a heavily loaded processor sharing queue. Ann. Appl. Probab. 12 797-859. MR1925442
11. IGLEHART, D. L. and WHITT, W. (1970). Multiple channel queues in heavy traffic I. Adv. in Appl. Probab. 2 150-177. MR0266331
12. KALLENBERG, O. (1986). Random Measures. Academic Press, New York. MR0854102
13. KRUK, Ł., LEHOCZKY, J. and SHREVE, S. (2003). Second order approximation for the customer time in queue distribution under the FIFO service discipline. Ann. Univ. Mariae Curie-Skłodowska Sect. AI Inform. 1 37-48. MR2254131
14. KRUK, Ł., LEHOCZKY, J. and SHREVE, S. (2006). Accuracy of state space collapse for earliest-deadline-first queues. Ann Appl. Probab. 16 516-561. MR2244424
15. KRUK, Ł., LEHOCZKY, J., SHREVE, S. and YEUNG, S.-N. (2003). Multiple-input heavy-traffic real-time queues. Ann. Appl. Probab. 13 54-99. MR1951994
16. KRUK, Ł., LEHOCZKY, J., SHREVE, S. and YEUNG, S.-N. (2004). Earliest-deadline-first service in heavy-traffic acyclic networks. Ann. Appl. Probab. 14 1306-1352. MR2071425
17. KRUK, Ł. and ZIEBA, W. (1994). On tightness of randomly indexed sequences of random variables. Bull. Pol. Acad. Sci. Math. 42 237-241. MR1811853
18. LEHOCZKY, J. P. (1996). Real-time queueing theory. In Proceedings of the IEEE Real-Time Systems Symposium. IEEE, New York.
19. PROHOROV, Y. V. (1956). Convergence of random processes and limit theorems in probability theory. Theory Probab. Appl. 1 157-214. MR0084896
20. PUHA, A. L., STOLYAR, A. L. and WILLIAMS, R. J. (2006). The fluid limit of an overloaded processor sharing queue. Math. Oper. Res. 31 316-350. MR2234000
21. PUHA, A. L. and WILLIAMS, R. J. (2004). Invariant states and rates of convergence for a critical fluid model of a processor sharing queue. Ann. Appl. Probab. 14 517-554. MR2052894
22. REIMAN, M. I. (1984). Open queueing networks in heavy traffic. Math. Oper. Res. 9 441-458. MR0757317
23. VAN DER VAART, A. and WELLNER, J. A. (1996). Weak convergence and empirical processes. Springer, New York. MR1385671
24. WILLIAMS, R. J. (1998). Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Syst. 30 27-88. MR1663759
25. YASHKOV, S. F. (1987). Processor-sharing queues: Some progress in analysis. Queueing Syst. 2 1-17. MR0903424
26. YEUNG, S.-N. and LEHOCZKY, J. P. (2004). Real-time queueing networks in heavy traffic with EDF and FIFO queue discipline. Preprint.