Article

Free access

Infinite parallel job allocation (extended abstract)

Authors:

Petra Berenbrink,

Tom Friedetzky,

Nikita D. VvedenskayaAuthors Info & Claims

SPAA '00: Proceedings of the twelfth annual ACM symposium on Parallel algorithms and architectures

Pages 99 - 108

https://doi.org/10.1145/341800.341813

Published: 09 July 2000 Publication History

Abstract

In recent years, the task of allocating jobs to servers has been studied with the “balls and bins” abstraction. Results in this area exploit the large decrease in maximum load that can be achieved by allowing each job (ball) a little freedom in choosing its destination server (bin).

In this paper we examine an infinite and parallel allocation process (see [ABS98]) which is related to the “balls and bins” abstraction. The simple process can be used to model many problems arising in applications like load balancing, data accesses for parallel data servers, hashing, and PRAM simulations.

Unfortunately, the parallel allocation process behaves in a highly non-uniform manner which makes its analysis challenging. Even the typically simple question of for which arrival rates the process is stable, is highly non-trivial. In order to cope with this non-uniform behavior we introduce a new sequential process and show (via simulations) that the sequential process models the behavior of the parallel one very accurately. We develop a system of ordinary differential equations in order to describe the behavior of our sequential process and present a thorough analysis of the performance this process. For example, we show that the queue length distribution decreases double-exponentially. Finally, we present simulation results indicating that the solutions to the differential equations very well predict the queue length distribution of our sequential process and the largest injection rate for which it is stable.

Summarizing, we can conclude that in all the performance characteristics we have measured experimentally, the parallel and the sequential process are closely related. This indicates that the obtained solution of the differential equations and the results presented above are applicable to the parallel process, too.

References

[1]

A.O. Allen. Probability, Statistics, and QueueinE Theory with Computer Science Applications. Academic Press, 1990.]]

Digital Library

[2]

Y. Azar, A. Z. Broder, A. R. Karlin, and E. Upfal. Balanced allocations. SIAM Journal on ComputinE, 29(1):180-200, September 1999. A preliminary version appeared in ProceedinEs of the 26th Annual ACM Symposium on Theory of Computing, pages 593-602, Montreal, Quebec, Canada, May 23-25, 1994. ACM Press, New York, NY.]]

Digital Library

[3]

M. Adler, P. Berenbrink, and K. SchrSder. Analyzing an infinite parallel job allocation process. In Proceedings of the 6th Annual European Symposium on Algorithms, pages 417-428, 1998.]]

Digital Library

[4]

M. Adler, S. Chakrabarti, M. Mitzenmacher, and L. Rasmussen. Parallel randomized load balancing. Random Structures and Algorithms, 13(2):159-188, 1998. A preliminary version appeared in Proceedings of the 27th Annual ACM Symposium on Theory of Computing, pages 238-247, 1995.]]

Digital Library

[5]

P. Berenbrink, A. Czumaj, A. Steger, and B. VScking. Balanced allocations: The heavily loaded case. To appear in Proceedings of the 32th Annual ACM Symposium on Theory of Computing, 2000.]]

Digital Library

[6]

P. Berenbrink, F. Meyer auf der Heide, and K. Schr6der. Allocating weighted jobs in parallel. Theory of Computing Systems, 32(3):281-300, 1999.]]

[7]

R. Cole, A. Frieze, B. M. Maggs, M. Mitzenmacher, A. W. Richa, R. K. Sitaraman, and E. Upfal. On balls and bins with deletions. In Proceedings of the 2nd International Workshop on Randomization and Approximation Techniques in Computer Science, pages 145- 158, 1998.]]

Digital Library

[8]

R. Cole, B. M. Maggs, F. Meyer auf der Heide, M. Mitzenmacher, A. W. Richa, K. Schr6der, R. K. Sitaraman, and B. V6cking. Randomized protocols for low-congestion circuit routing in multistage interconnection networks. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 378- 388, 1998.]]

Digital Library

[9]

A. Czumaj and V. Stemann. Randomized allocation processes. In Proceedings of the 38th IEEE Symposium on Foundations of Computer Science, pages 194-203, 1997.]]

Digital Library

[10]

A. Czumaj. Recovery time of dynamic allocation processes. In Proceedings of the lOth Annual ACM Symposium on Parallel Algorithms and Architectures, pages 202-211, 1998.]]

Digital Library

[11]

S.N. Ethier and T. G. Kurtz. Markov Processes: Characterization and Convergence. John Wiley & Sons, New York, NY, 1986.]]

[12]

P. Jacquet and N. Vvedenskaya. On/off sources in an interconnection network: Performance analysis when packets are routed to the shortest queue of two randomly selected nodes. Technical Report N~ 3570, IN- RIA Rocquencourt, France, 1998.]]

[13]

L. Kleinrock. Queueing Systems, Volume I: Theory. John Wiley & Sons, New York, NY, 1976.]]

[14]

R.M. Karp, M. Luby, and F. Meyer auf der Heide. Efficient PRAM simulation on a distributed memory machine. Algorithmica, 16(4/5):517-542, 1996.]]

[15]

V. F. Kolchin, B. A. Sevast'yanov, and V. P. Chistyakov. Random Allocations. V. H. Winston & Sons, Washington, D.C., 1978.]]

[16]

T.G. Kurtz. Solutions of ordinary differential equations as limits of pure jump Markov processes. Journal of Applied Probability, 7:49-58, 1970.]]

[17]

T.G. Kurtz. Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. Journal of Applied Probability, 8:344-356, 1971.]]

[18]

T.G. Kurtz. Approximation of Population Processes. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), 1981.]]

[19]

M. Luczak, E. Upfal. Reducing Network Congestion and Blocking Probability Through Balanced Allocation, In Proceedings of the 29th IEEE Symposium on Foundations of Computer Science, pages 587-595, 1999.]]

Digital Library

[20]

M.D. Mitzenmacher. The Power of Two Choices in Randomized Load Balancing. PhD thesis, Computer Science Department, University of California at Berkeley, CA, USA, September 1996.]]

Digital Library

[21]

M. Mitzenmacher. Load balancing and density dependent jump Markov processes (Extended abstract). In Proceedings of the 37th IEEE Symposium on Foundations of Computer Science, pages 213-222, 1996.]]

Digital Library

[22]

M. Mitzenmacher. Analyses of load stealing models based on differential equations. In Proceedings of the lOth Annual ACM Symposium on Parallel Algorithms and Architectures, pages 212-221, 1998.]]

Digital Library

[23]

M. Mitzenmacher. On the analysis of randomized load balancing schemes. Theory of Computing Systems, 32(3):292-301, 1999.]]

[24]

D.R. McDonald and S. R. E. Turner. Comparing load balancing algorithms for distributed queueing networks. To appear in Fields Institute Communication Series volume on Analysis of Communication Networks: Call Centres, Traffic and Performance.]]

[25]

S.S. Sheu. The Poisson approximation to the binomial distribution. The American 5tatistican, 38(3), 206- 207, 1984.]]

[26]

N.D. Vvedenskaya, R. L. Dobrushin, and F. I. Karpelevich. Queueing system with selection of the shortest of two queues: An assymptotic approach. Problems of Information Transmission, 32(1):15-27, January-March 1996.]]

[27]

B. V6cking. How asymetry helps load balancing. In Proceedings of the 40th IEEE Symposium on Foundations of Computer Science, pages 131-141, 1999.]]

Digital Library

[28]

N.D. Vvedenskaya and Y. M. Suhov. Dobrushin's mean-field approximation for queue with dynamic routing. Technical Report N~ 3328, INRIA, France, December 1997.]]

[29]

N.D. Vvedenskaya. The delay in a network with many multiple routes and feedback. In Proceedings of the 7th Joint Swedish-Russian International Workshop on Information Theory, pages 243-245, St. Peterburg, Russia, 1995, and in Proceedings of the 16th Symposium on Information Theory in Benelux, pages 49-51, Enschede, the Netherlands, 1995.]]

[30]

N.D. Vvedenskaya. Large queueing systems where selection of the shortest of several queues is allowed. In 5dminaire sur la mdcanique statistique des grannds rdseaux, pages 19-21, INRIA Rocquencourt, France, 1996.]]

[31]

N.D. Vvedenskaya. Large queuing system where messages are transmitted via several routers. Problems of Information Transmission, 34(2):180-189, 1998.]]

[32]

N.C. Wormald. Differential equations for random processes and random graphs. The Annals of Applied Probability, 5(4):1217-1235, 1995.]]

Cited By

Dalal KDevroye LMalalla E(2023)Two-Way Linear Probing RevisitedAlgorithms10.3390/a1611050016:11(500)Online publication date: 28-Oct-2023
https://doi.org/10.3390/a16110500
Berenbrink PFriedetzky THahn CHintze LKaaser DKling PNagel L(2021)Infinite Balanced Allocation via Finite Capacities2021 IEEE 41st International Conference on Distributed Computing Systems (ICDCS)10.1109/ICDCS51616.2021.00096(965-975)Online publication date: Jul-2021
https://doi.org/10.1109/ICDCS51616.2021.00096
Becchetti LClementi ANatale EPasquale FPosta G(2019)Self-stabilizing repeated balls-into-binsDistributed Computing10.1007/s00446-017-0320-432:1(59-68)Online publication date: 1-Feb-2019
https://dl.acm.org/doi/10.1007/s00446-017-0320-4
Show More Cited By

Index Terms

Infinite parallel job allocation (extended abstract)

Recommendations

Analyzing an Infinite Parallel Job Allocation Process
ESA '98: Proceedings of the 6th Annual European Symposium on Algorithms

In recent years the task of allocating jobs to servers has been studied with the "balls and bins" abstraction. Results in this area exploit the large decrease in maximum load that can be achieved by allowing each job (ball) a very small amount of choice ...
The NYU Ultracomputer—designing a MIMD, shared-memory parallel machine (Extended Abstract)
ISCA '82: Proceedings of the 9th annual symposium on Computer Architecture

We present the design for the NYU Ultracomputer, a shared-memory MIMD parallel machine composed of thousands of autonomous processing elements. This machine uses an enhanced message switching network with the geometry of an Omega-network to approximate ...
Stochastic performance models of parallel task systems (extended abstract)

This paper considers the class of parallel computations represented by directed, acyclic task graphs. These include parallel loops, multiphase algorithms, partitioning and merging algorithms, as well as any arbitrary parallel computation that can be ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

SPAA '00: Proceedings of the twelfth annual ACM symposium on Parallel algorithms and architectures

July 2000

224 pages

ISBN:1581131852

DOI:10.1145/341800

Chairmen:
Gary Miller
Carnegie Mellon Univ. Pittsburgh PA; and Akamai Technologies, Inc.
,
Shang-Hua Teng
Univ. of Illinois at Urbana-Champaign, Urbana

Copyright © 2000 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 09 July 2000

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Conference

SPAA00

Sponsor:

SPAA00: ACM symposium on Parallel Algorithms and Architectures

July 9 - 13, 2000

Maine, Bar Harbor, USA

Acceptance Rates

SPAA '00 Paper Acceptance Rate 24 of 45 submissions, 53%;

Overall Acceptance Rate 447 of 1,461 submissions, 31%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

9
Total Citations
View Citations
346
Total Downloads

Downloads (Last 12 months)55
Downloads (Last 6 weeks)11

Reflects downloads up to 01 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Dalal KDevroye LMalalla E(2023)Two-Way Linear Probing RevisitedAlgorithms10.3390/a1611050016:11(500)Online publication date: 28-Oct-2023
https://doi.org/10.3390/a16110500
Berenbrink PFriedetzky THahn CHintze LKaaser DKling PNagel L(2021)Infinite Balanced Allocation via Finite Capacities2021 IEEE 41st International Conference on Distributed Computing Systems (ICDCS)10.1109/ICDCS51616.2021.00096(965-975)Online publication date: Jul-2021
https://doi.org/10.1109/ICDCS51616.2021.00096
Becchetti LClementi ANatale EPasquale FPosta G(2019)Self-stabilizing repeated balls-into-binsDistributed Computing10.1007/s00446-017-0320-432:1(59-68)Online publication date: 1-Feb-2019
https://dl.acm.org/doi/10.1007/s00446-017-0320-4
Berenbrink PFriedetzky TKling PMallmann-Trenn FNagel LWastell C(2018)Self-Stabilizing Balls and Bins in BatchesAlgorithmica10.5555/3288645.328867880:12(3673-3703)Online publication date: 1-Dec-2018
https://dl.acm.org/doi/10.5555/3288645.3288678
Berenbrink PFriedetzky TKling PMallmann-Trenn FNagel LWastell C(2018)Self-Stabilizing Balls and Bins in BatchesAlgorithmica10.1007/s00453-018-0411-z80:12(3673-3703)Online publication date: 15-Feb-2018
https://doi.org/10.1007/s00453-018-0411-z
Mocquard YSericola BAnceaume E(2018)Balanced Allocations and Global Clock in Population Protocols: An Accurate AnalysisStructural Information and Communication Complexity10.1007/978-3-030-01325-7_26(296-311)Online publication date: 31-Oct-2018
https://doi.org/10.1007/978-3-030-01325-7_26
Berenbrink PFriedetzky TKling PMallmann-Trenn FNagel LWastell CGiakkoupis G(2016)Self-stabilizing Balls & Bins in BatchesProceedings of the 2016 ACM Symposium on Principles of Distributed Computing10.1145/2933057.2933092(83-92)Online publication date: 25-Jul-2016
https://dl.acm.org/doi/10.1145/2933057.2933092
Sanders P(2003)Asynchronous Scheduling of Redundant Disk ArraysIEEE Transactions on Computers10.1109/TC.2003.122851252:9(1170-1184)Online publication date: 1-Sep-2003
https://dl.acm.org/doi/10.1109/TC.2003.1228512
Sanders PMiller GTeng S(2000)Asynchronous scheduling of redundant disk arraysProceedings of the twelfth annual ACM symposium on Parallel algorithms and architectures10.1145/341800.341812(89-98)Online publication date: 9-Jul-2000
https://dl.acm.org/doi/10.1145/341800.341812

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Table of Contents