Joint Channel Probing and Proportional Fair Scheduling in Wireless Networks

The design of a scheduling scheme is crucial for the efficiency and user-fairness of wireless networks. Assuming that the quality of all user channels is available to a central controller, a simple scheme which maximizes the utility function defined as the sum logarithm throughput of all users has been shown to guarantee proportional fairness. However, to acquire the channel quality information may consume substantial amount of resources. In this work, it is assumed that probing the quality of each user's channel takes a fraction of the coherence time, so that the amount of time for data transmission is reduced. The multiuser diversity gain does not always increase as the number of users increases. In case the statistics of the channel quality is available to the controller, the problem of sequential channel probing for user scheduling is formulated as an optimal stopping time problem. A joint channel probing and proportional fair scheduling scheme is developed. This scheme is extended to the case where the channel statistics are not available to the controller, in which case a joint learning, probing and scheduling scheme is designed by studying a generalized bandit problem. Numerical results demonstrate that the proposed scheduling schemes can provide significant gain over existing schemes.Comment: 26 pages, 8 figure

Out-of-Time-Order Correlation at a Quantum Phase Transition

In this paper we numerically calculate the out-of-time-order correlation functions in the one-dimensional Bose-Hubbard model. Our study is motivated by the conjecture that a system with Lyapunov exponent saturating the upper bound $2\pi/\beta$ will have a holographic dual to a black hole at finite temperature. We further conjecture that for a many-body quantum system with a quantum phase transition, the Lyapunov exponent will have a peak in the quantum critical region where there exists an emergent conformal symmetry and is absent of well-defined quasi-particles. With the help of a relation between the R\'enyi entropy and the out-of-time-order correlation function, we argue that the out-of-time-order correlation function of the Bose-Hubbard model will also exhibit an exponential behavior at the scrambling time. By fitting the numerical results with an exponential function, we extract the Lyapunov exponents in the one-dimensional Bose-Hubbard model across the quantum critical regime at finite temperature. Our results on the Bose-Hubbard model support the conjecture. We also compute the butterfly velocity and propose how the echo type measurement of this correlator in the cold atom realizations of the Bose-Hubbard model without inverting the Hamiltonian.Comment: 7 pages, 6 figures, published versio