Distributed selfish load balancing with weights and speeds

In this paper we consider neighborhood load balancing in the context of selfish clients. We assume that a network of n processors is given, with m tasks assigned to the processors. The processors may have different speeds and the tasks may have different weights. Every task is controlled by a selfish user. The objective of the user is to allocate his/her task to a processor with minimum load, where the load of a processor is defined as the weight of its tasks divided by its speed. We investigate a concurrent probabilistic protocol which works in sequential rounds. In each round every task is allowed to query the load of one randomly chosen neighboring processor. If that load is smaller than the load of the task's current processor, the task will migrate to that processor with a suitably chosen probability. Using techniques from spectral graph theory we obtain upper bounds on the expected convergence time towards approximate and exact Nash equilibria that are significantly better than previous results for this protocol. We show results for uniform tasks on non-uniform processors and the general case where the tasks have different weights and the machines have speeds. To the best of our knowledge, these are the first results for this general setting.

[1]  H. Weyl.  Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung) , 1912 .

[2]  Robert Elsässer,et al.  Diffusion Schemes for Load Balancing on Heterogeneous Networks , 2002, Theory of Computing Systems.

[3]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[4]  Rajendra Bhatia,et al.  Linear Algebra to Quantum Cohomology: The Story of Alfred Horn's Inequalities , 2001, Am. Math. Mon..

[5]  P. Mahonen,et al.  Load Balancing for Dynamic Spectrum Assignment with Local Information for Secondary Users , 2008, 2008 3rd IEEE Symposium on New Frontiers in Dynamic Spectrum Access Networks.

[6]  Yishay Mansour,et al.  Convergence time to Nash equilibrium in load balancing , 2007, TALG.

[7]  Bojan Mohar,et al.  Isoperimetric numbers of graphs , 1989, J. Comb. Theory, Ser. B.

[8]  S. Muthukrishnan,et al.  First- and Second-Order Diffusive Methods for Rapid, Coarse, Distributed Load Balancing , 1996, Theory of Computing Systems.

[9]  Petra Berenbrink,et al.  Improved Bounds for Discrete Diffusive Load Balancing , 2012, 2012 IEEE 26th International Parallel and Distributed Processing Symposium.

[10]  Iman Hajirasouliha,et al.  Convergence to Equilibria in Distributed, Selfish Reallocation Processes with Weighted Tasks , 2007, ESA.

[11]  Yuval Rabani,et al.  Local divergence of Markov chains and the analysis of iterative load-balancing schemes , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[12]  Robert Elsässer,et al.  Distributing Unit Size Workload Packages in Heterogeneous Networks , 2006, J. Graph Algorithms Appl..

[13]  Martin Hoefer,et al.  Distributed Selfish Load Balancing on Networks , 2014, TALG.

[14]  Paul W. Goldberg,et al.  Distributed selfish load balancing , 2005, SODA '06.

[15]  Thomas Sauerwald,et al.  Near-perfect load balancing by randomized rounding , 2009, STOC '09.

[16]  Berthold Vöcking,et al.  Adaptive routing with stale information , 2009, Theor. Comput. Sci..

[17]  Martin Gairing,et al.  Nashification and the Coordination Ratio for a Selfish Routing Game , 2003, ICALP.

[18]  Paul G. Spirakis,et al.  Atomic Congestion Games: Fast, Myopic and Concurrent , 2009, Theory of Computing Systems.

[19]  Yishay Mansour,et al.  Fast convergence of selfish rerouting , 2005, SODA '05.

[20]  George Cybenko,et al.  Dynamic Load Balancing for Distributed Memory Multiprocessors , 1989, J. Parallel Distributed Comput..

[21]  Jacques E. Boillat,et al.  Load Balancing and Poisson Equation in a Graph , 1990, Concurr. Pract. Exp..

[22]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[23]  B. Mohar.  THE LAPLACIAN SPECTRUM OF GRAPHS y , 1991 .

[24]  Berthold Vöcking.  Algorithmic Game Theory: Selfish Load Balancing , 2007 .

[25]  Yossi Azar,et al.  Fast load balancing via bounded best response , 2008, SODA '08.

[26]  Martin Hoefer,et al.  Distributed algorithms for QoS load balancing , 2011, Distributed Computing.

[27]  A. Klyachko.  Random walks on symmetric spaces and inequalities for matrix spectra , 2000 .

[28]  Bojan Mohar,et al.  Eigenvalues, diameter, and mean distance in graphs , 1991, Graphs Comb..