Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T06:39:45.334Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic Scheduling in in-Forest Networks

Part of: Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Zhen Liu  and
Don Towsley
Zhen Liu*
Affiliation:
INRIA, Centre Sophia Antipolis
Don Towsley*
Affiliation:
University of Massachusetts
*
* Postal address: INRIA, Centre Sophia Antipolis, 2004 Route des Lucioles, 06560 Valbonne, France.
** Postal address: Department of Computer and Information Science, University of Massachusetts, Amherst, MA 01003, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the extremal properties of several scheduling policies in an in-forest network consisting of multiserver queues. Each customer has a due date, and we assume that service times at the different queues form mutually independent sequences of independent and identically distributed random variables independent of the arrival times and due dates. Furthermore, the network is assumed to consist of a mixture of nodes, some of which permit only non-preemptive service policies whereas the others permit preemptive resume policies. In the case of non-preemptive queues, service times may be generally distributed if there is only one server; otherwise the service times are required to be increasing in likelihood ratio (ILR). In the case of preemptive queues, service times are restricted to exponential distributions. Using stochastic majorizations and partial orders on permutations, we establish that, within the class of work conserving service policies, the stochastically smallest due date (SSDD) and the stochastically largest due date (SLDD) policies minimize and maximize, respectively, the vector of the customer latenesses of the first n customers in the sense of the Schur-convex order and some weaker orders, provided the due dates are comparable in some stochastic sense. It then follows that the first come-first served (FCFS) and the last come-first served (LCFS) policies minimize and maximize, respectively, the vector of the response times of the first n customers in the sense of the Schur-convex order. We also show that the FCFS and LCFS policies minimize and maximize, respectively, the vector of customer end-to-end delays in the sense of the strong stochastic order. Extensions to the class of non-idling policies and to the stationary regime are also given.

Keywords

QUEUEING SYSTEMSERVICE DISCIPLINESAMPLE PATH ANALYSISSTOCHASTIC ORDERINGMAJORIZATIONPERMUTATION ORDERINGLATENESSRESPONSE TIMEEND-TO-END DELAY

MSC classification

Secondary: 90B35: Scheduling theory, deterministic
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 1 , March 1994 , pp. 222 - 241
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work was supported in part by the National Science Foundation under grants ASC 88–8802764, NCR-9116183.

The work of this author was also partially supported by CEC DG-XIII under the ESPRIT-BRA grant QMIPS.

References

[1] Baccelli, F., Gelenbe, E. and Plateau, B. E. (1984) An end-to-end approach to the resequencing problem. J. Assoc. Comput. Mach. 31, 474–485.Google Scholar
[2] Baccelli, F. and Makowski, A. M. (1989) Queueing models for systems with synchronization constraints. Proc. IEEE 77 (Special Issue on Dynamics of Discrete Event Systems) 138–161.Google Scholar
[3] Baccelli, F., Liu, Z. and Towsley, D. (1993) Extremal scheduling of parallel processing with and without real-time constraints. J. Assoc. Comput. Mach. To appear.CrossRefGoogle Scholar
[4] Barlow, R. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
[5] Brandt, A., Franken, P. and Lisek, B. (1989) Stationary Stochastic Models. Akademie-Verlag, New York.Google Scholar
[6] Chang, C. S. and Yao, D. D. (1990) Rearrangement, majorization and stochastic scheduling. IBM Research Report RC 16250.Google Scholar
[7] Daley, D. J. (1987) Certain optimality properties of the first come first served discipline for G/G/s queues. Stoch. Proc. Appl. 25, 301–308.Google Scholar
[8] Day, P. W. (1972) Rearrangement inequalities. Canad. J. Math. 24, 930–943.Google Scholar
[9] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. Wiley, New York.Google Scholar
[10] Foss, S. G. (1980) Approximation of multichannel queueing systems (in Russian). Sibirski Mat. Zh. 21, 132–140. English translation: Siberian Math. J. 21, 851-857.Google Scholar
[11] Foss, S. G. (1981) Comparison of servicing strategies in multichannel queueing systems (in Russian). Sibirski Mat. Zh. 22, 190–197. English translation: Siberian Math. J. 22, 142-147.Google Scholar
[12] Hirayama, T. and Kijima, M. (1989) An extremal property of FIFO discipline in G/IFR/1 queues. Adv. Appl. Prob. 21, 481–484.Google Scholar
[13] Iliadis, I. and Lien, L. (1988) Resequencing delay for a queueing system with two heterogeneous servers under a threshold-type scheduling. IEEE Trans. Commun. 36, 692–702.Google Scholar
[14] Kingman, J. F. C. (1970) Inequalities in the theory of queues. J. R. Statist. Soc. B 32, 102–110.Google Scholar
[15] Kleinrock, L., Kamoun, F. and Muntz, R. (1981) Queueing analysis of the reordering issue in a distributed database concurrency control mechanism. Proc. 2nd Internat. Conf. Distributed Computing Systems, Versailles, France.Google Scholar
[16] Liu, Z. and Nain, P. (1992) Optimal scheduling in some multi-queue single-server systems. IEEE Trans. Autom. Control 37, 247–252.Google Scholar
[17] Liu, Z. and Towsley, D. (1992) Effects of service discipline in G/G/s queueing systems. Ann. Operat. Res. To appear.Google Scholar
[18] Liu, Z. and Towsley, D. (1992) Stochastic scheduling in in-forest networks. COINS Technical Report TR 92-27; INRIA Rapport de Recherche No. 1719.Google Scholar
[19] Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and its Applications. Academic Press, New York.Google Scholar
[20] Pinedo, M. (1983) Stochastic scheduling with release dates and due dates. Operat. Res. 31, 559–572.Google Scholar
[21] Righter, R. and Shanthikumar, J. G. (1992) Extremal properties of the FIFO distribution in queueing networks. J. Appl. Prob. 22, 967–978.Google Scholar
[22] Ross, S. (1983) Stochastic Processes. Wiley, New York.Google Scholar
[23] Shanthikumar, J. G. and Sumita, U. (1987) Convex ordering of sojourn times in single-server queues: extremal properties of FIFO and LIFO service disciplines. J. Appl. Prob. 24, 736–748.Google Scholar
[24] Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. English translation, ed. Daley, D. J.. Wiley, New York.Google Scholar
[25] Towsley, D. and Baccelli, F. (1991) Comparison of service disciplines in a tandem queueing network with delay-dependent customer behavior. Operat. Res. Lett. 10, 49–55. Operat. Res. 31, 559-572.Google Scholar
[26] Towsley, D. and Panwar, S. S. (1991) Optimality of the stochastic earliest deadline policy for the G/M/c queue serving customers with deadlines. COINS Technical Report 91-61.Google Scholar
[27] Vasicek, O. A. (1977) An inequality for the variance of waiting time under a general queueing discipline. Operat. Res. 25, 879–884.Google Scholar
[28] Whitt, W. (1984) The amount of overtaking in a network of queues. Networks 14, 411–426.Google Scholar
[29] Wolff, R. W. (1977) An upper bound for multi-channel queues. J. Appl. Prob. 14, 884–888.Google Scholar
[30] Wolff, R. W. (1987) Upper bounds on work in system for multichannel queues. J. Appl. Prob. 24, 547–551.Google Scholar