Load Balancing in Quorum Systems

Abstract

This paper introduces and studies the question of balancing the load on processors participating in a given quorum system. Our proposed measure for the degree of balancing is the ratio between the load on the least frequently referenced element and on the most frequently used one. We give some simple sufficient and necessary conditions for perfect balancing. We then look at the balancing properties of the common class of voting systems and prove that every voting system with odd total weight is perfectly balanced. (This holds, in fact, for the more general class of ordered systems.) We also give some characterizations for the balancing ratio in the worst case. It is shown that for any quorum system with a universe of size n, the balancing ratio is no smaller than $1/(n-1)$, and this bound is the best possible. When restricting attention to nondominated coteries (NDCs), the bound becomes $2/\bigl(n-\log_2 n+o(\log n)\bigr)$, and there exists an NDC with ratio $2/\bigl(n-\log_2 n-o(\log n)\bigr)$. Next, we study the interrelations between the two basic parameters of load balancing and quorum size. It turns out that the two size parameters suitable for our investigation are the size of the largest quorum and the optimally weighted average quorum size(OWAQS) of the system. For the class of ordered NDCs (for which perfect balancing is guaranteed), it is shown that over a universe of size n, some quorums of size $\lceil(n+1)/2\rceil$ or more must exist (and this bound is the best possible). A similar lower bound holds for the OWAQS measure if we restrict attention to voting systems. For nonordered systems, perfect balancing can sometimes be achieved with much smaller quorums. A lower bound of $\Omega(\sqrt{n})$ is established for the maximal quorum size and the OWAQS of any perfectly balanced quorum system over n elements, and this bound is the best possible. Finally, we turn to quorum systems that cannot be perfectly balanced, but have some balancing ratio $0 < \rho < 1$. For such systems we study the trade-offs between the required balancing ratio $\rho$ and the quorum size it admits in the best case. It is easy to get an analogue of the result for perfect balancing, yielding a lower bound of $\sqrt{n\rho}$. We actually get a better estimate by a refinement of the argument.