Abstract
Motivated by distributed schedulers that combine the power-of-d-choices with late binding and systems that use replication with cancellation-on-start, we study the performance of the LL(d) policy which assigns a job to a server that currently has the least workload among d randomly selected servers in large-scale homogeneous clusters. We consider general job size distributions and propose a partial integro-differential equation to describe the evolution of the system. This equation relies on the earlier proven ansatz for LL(d) which asserts that the workload distribution of any finite set of queues becomes independent of one another as the number of servers tends to infinity. Based on this equation we propose a fixed point iteration for the limiting workload distribution and study its convergence. For exponential job sizes we present a simple closed form expression for the limiting workload distribution that is valid for any work-conserving service discipline as well as for the limiting response time distribution in case of first-come-first-served scheduling. We further show that for phase-type distributed job sizes the limiting workload and response time distribution can be expressed via the unique solution of a simple set of ordinary differential equations. Numerical and analytical results that compare response time of the classic power-of-d-choices algorithm and the LL(d) policy are also presented and the accuracy of the limiting response time distribution for finite systems is illustrated using simulations.
- M. Abramowitz and I. A. Stegun. 1964. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (ninth dover printing, tenth gpo printing ed.). Dover, New York.Google Scholar
- R. Aghajani, X. Li, and K. Ramanan. 2017. The PDE Method for the Analysis of Randomized Load Balancing Networks. Proc. ACM Meas. Anal. Comput. Syst. 1, 2, Article 38 (Dec. 2017), 28 pages. Google Scholar
Digital Library
- Y. Azar, A.Z. Broder, A.R. Karlin, and E. Upfal. 1999. Balanced Allocations. SIAM J. Comput. 29, 1 (1999), 180--200. Google Scholar
Digital Library
- R. Bekker, S.C. Borst, O.J. Boxma, and O. Kella. 2004. Queues with Workload-Dependent Arrival and Service Rates. Queueing Systems 46, 3 (01 Mar 2004), 537--556. Google Scholar
Digital Library
- M. Bramson. 2011. Stability of join the shortest queue networks. Ann. Appl. Probab. 21, 4 (2011), 1568--1625.Google Scholar
- M. Bramson, Y. Lu, and B. Prabhakar. 2010. Randomized load balancing with general service time distributions. In ACM SIGMETRICS 2010. 275--286. Google Scholar
Digital Library
- M. Bramson, Y. Lu, and B. Prabhakar. 2012. Asymptotic independence of queues under randomized load balancing. Queueing Syst. 71, 3 (2012), 247--292. Google Scholar
Digital Library
- M. Bramson, Y. Lu, and B. Prabhakar. 2013. Decay of tails at equilibrium for FIFO join the shortest queue networks. Ann. Appl. Probab. 23, 5 (10 2013), 1841--1878.Google Scholar
- R. D. Driver. 1977. Ordinary and Delay Differential Equations. Springer-Verlag, Berlin-Heidelberg-New York.Google Scholar
- S. Foss and N. Chernova. 1998. On the Stability of a Partially Accessible Multi-station Queue with State-dependent Routing. Queueing Syst. Theory Appl. 29, 1 (May 1998), 55--73. Google Scholar
Digital Library
- K. Gardner, M. Harchol-Balter, A. Scheller-Wolf, M. Velednitsky, and S. Zbarsky. 2017. Redundancy-d: The Power of d Choices for Redundancy. Operations Research 65, 4 (2017), 1078--1094.Google Scholar
Digital Library
- K. Gardner, S. Zbarsky, M. Harchol-Balter, and A. Scheller-Wolf. 2016. The Power of d Choices for Redundancy. In Proceedings of the 2016 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Science, Antibes Juan-Les-Pins, France, June 14--18, 2016. 409--410. Google Scholar
Digital Library
- L. M. Graves. 1946. The theory of functions of real variables. McGraw-Hill book company, inc.Google Scholar
- J. Kriege and P. Buchholz. 2014. PH and MAP Fitting with Aggregated Traffic Traces. Springer International Publishing, Cham, 1--15. Google Scholar
Digital Library
- G. Latouche and V. Ramaswami. 1999. Introduction to Matrix Analytic Methods and stochastic modeling. SIAM, Philadelphia.Google Scholar
- M. Mitzenmacher. 2000. How Useful Is Old Information? IEEE Trans. Parallel Distrib. Syst. 11, 1 (Jan. 2000), 6--20. Google Scholar
Digital Library
- M. Mitzenmacher. 2001. The Power of Two Choices in Randomized Load Balancing. IEEE Trans. Parallel Distrib. Syst. 12 (October 2001), 1094--1104. Issue 10. Google Scholar
Digital Library
- D. Mukherjee, S. Borst, J. van Leeuwaarden, and P. Whiting. 2016. Universality of Power-of-d Load Balancing Schemes. SIGMETRICS Perform. Eval. Rev. 44, 2 (Sept. 2016), 36--38. Google Scholar
Digital Library
- K. Ousterhout, P. Wendell, M. Zaharia, and I. Stoica. 2013. Sparrow: Distributed, Low Latency Scheduling. In Proceedings of the Twenty-Fourth ACM Symposium on Operating Systems Principles (SOSP '13). ACM, New York, NY, USA, 69--84. Google Scholar
Digital Library
- A. Panchenko and A. Thümmler. 2007. Efficient Phase-type Fitting with Aggregated Traffic Traces. Perform. Eval. 64, 7--8 (Aug. 2007), 629--645. Google Scholar
Digital Library
- A. Tarski. 1955. A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math. 5, 2 (1955), 285--309. https://projecteuclid.org:443/euclid.pjm/1103044538Google Scholar
Cross Ref
- N.D. Vvedenskaya, R.L. Dobrushin, and F.I. Karpelevich. 1996. Queueing System with Selection of the Shortest of Two Queues: an Asymptotic Approach. Problemy Peredachi Informatsii 32 (1996), 15--27.Google Scholar
Index Terms
On the Power-of-d-choices with Least Loaded Server Selection
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