Scheduling policies for real-time systems exhibit threshold behavior that is related to the utilization of the task set they schedule, and in some cases this threshold is sharp. For the rate monotonic scheduling policy, we show that periodic workload with utilization less than a threshold @math can be scheduled almost surely and that all workload with utilization greater than @math is almost surely not schedulable. We study such sharp threshold behavior in the context of processor scheduling using static task priorities, not only for periodic real-time tasks but for aperiodic real-time tasks as well. The notion of a utilization threshold provides a simple schedulability test for most real-time applications. These results improve our understanding of scheduling policies and provide an interesting characterization of the typical behavior of policies. The threshold is sharp (small deviations around the threshold cause schedulability, as a property, to appear or disappear) for most policies; this is a happy consequence that can be used to address the limitations of existing utilization-based tests for schedulability. We demonstrate the use of such an approach for balancing power consumption with the need to meet deadlines in web servers.
Determining the satisfiability of randomly generated Boolean expressions with k variables per clause is a popular test for the performance of search algorithms in artificial intelligence and computer science. It is known that for k = 2, formulas are almost always satisfiable when the ratio of clauses to variables is less than 1; for ratios larger than 1, the formulas are almost never satisfiable. Similar sharp threshold behavior is observed for higher values of k. Finite-size scaling, a method from statistical physics, can be used to characterize size-dependent effects near the threshold. A relationship can be drawn between thresholds and computational complexity.  We report results from large-scale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible to generate random formulas that are hard, that is, for which satisfiability testing is quite difficult. Our results provide a benchmark for the evaluation of satisfiability-testing procedures.  In their seminal work which initiated random graph theory Erdos and Renyi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem. Let Vn(p) = 0, 1 n denote the Hamming space endowed with the probability measure μp defined by μp( 1, 2, . . . , n) = pk · (1 − p)n−k, where k = 1 + 2 + · · · + n. Let A be a monotone subset of Vn. We say that A is symmetric if there is a transitive permutation group Γ on 1, 2, . . . , n such that A is invariant under Γ. Theorem. For every symmetric monotone A, if μp(A) > then μq(A) > 1− for q = p+ c1 log(1 2 ) logn. (c1 is an absolute constant.) It is well known that for many NP-complete problems, such as K-Sat, etc., typical cases are easy to solve; so that computationally hard cases must be rare (assuming P = NP). This paper shows that NP-complete problems can be summarized by at least one "order parameter", and that the hard problems occur at a critical value of such a parameter. This critical value separates two regions of characteristically different properties. For example, for K-colorability, the critical value separates overconstrained from underconstrained random graphs, and it marks the value at which the probability of a solution changes abruptly from near 0 to near 1. It is the high density of well-separated almost solutions (local minima) at this boundary that cause search algorithms to "thrash". This boundary is a type of phase transition and we show that it is preserved under mappings between problems. We show that for some P problems either there is no phase transition or it occurs for bounded N (and so bounds the cost). These results suggest a way of deciding if a problem is in P or NP and why they are different.