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.
Sharp thresholds are indicators of phase transitions. Phase transitions are common in physical systems. Freezing of ice and superconductivity are phenomena that have temperature as the critical parameter. Phase transitions have been identified in many combinatorial optimization problems, especially constraint satisfaction problems @cite_12 @cite_28 @cite_4 . Phase transitions provide very interesting insight into the behavior of combinatorial optimization problems, of which scheduling is an instance, and mayhold the key to faster, near-optimal solutions. Sharp thresholds for properties of random graphs were identified initially by Erd "o s and R 'e nyi @cite_22 and these results have been generalized by many mathematicians including Friedgut and Kalai @cite_13 @cite_27 .
@cite_4 @cite_22 @cite_28 @cite_27 @cite_13 @cite_12
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For many random constraint satisfaction problems such as random satisfiability or random graph or hypergraph coloring, the best current estimates of the threshold for the existence of solutions are based on the first and the second moment method. However, in most cases these techniques do not yield matching upper and lower bounds. Sophisticated but non-rigorous arguments from statistical mechanics have ascribed this discrepancy to the existence of a phase transition called condensation that occurs shortly before the actual threshold for the existence of solutions and that affects the combinatorial nature of the problem (Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova: PNAS 2007). In this paper we prove for the first time that a condensation transition exists in a natural random CSP, namely in random hypergraph 2-coloring. Perhaps surprisingly, we find that the second moment method applied to the number of 2-colorings breaks down strictly before the condensation transition. Our proof also yields slightly improved bounds on the threshold for random hypergraph 2-colorability. Many NP-complete constraint satisfaction problems appear to undergo a “phase transition” from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that the threshold for both random hypergraph 2-colorability (Property B) and random Not-All-Equal @math -SAT is @math . As a corollary, we establish that the threshold for random @math -SAT is of order @math , resolving a long-standing open problem. We prove the existence of an asymptotic phase transition threshold on community detectability for the spectral modularity method [M. E. J. Newman, Phys. Rev. E 74, 036104 (2006) and Proc. National Academy of Sciences. 103, 8577 (2006)] under a stochastic block model. The phase transition on community detectability occurs as the inter-community edge connection probability @math grows. This phase transition separates a sub-critical regime of small @math , where modularity-based community detection successfully identifies the communities, from a super-critical regime of large @math where successful community detection is impossible. We show that, as the community sizes become large, the asymptotic phase transition threshold @math is equal to @math , where @math is the within-community edge connection probability. Thus the phase transition threshold is universal in the sense that it does not depend on the ratio of community sizes. The universal phase transition phenomenon is validated by simulations for moderately sized communities. Using the derived expression for the phase transition threshold we propose an empirical method for estimating this threshold from real-world data. We study a spectral initialization method that serves a key role in recent work on estimating signals in nonconvex settings. Previous analysis of this method focuses on the phase retrieval problem and provides only performance bounds. In this paper, we consider arbitrary generalized linear sensing models and present a precise asymptotic characterization of the performance of the method in the high-dimensional limit. Our analysis also reveals a phase transition phenomenon that depends on the ratio between the number of samples and the signal dimension. When the ratio is below a minimum threshold, the estimates given by the spectral method are no better than random guesses drawn from a uniform distribution on the hypersphere, thus carrying no information; above a maximum threshold, the estimates become increasingly aligned with the target signal. The computational complexity of the method, as measured by the spectral gap, is also markedly different in the two phases. Worked examples and numerical results are provided to illustrate and verify the analytical predictions. In particular, simulations show that our asymptotic formulas provide accurate predictions for the actual performance of the spectral method even at moderate signal dimensions.