Doubly Exponential Solution for Randomized Load Balancing Models with General Service Times

In this paper, we provide a novel and simple approach to study the supermarket model with general service times. This approach is based on the supplementary variable method used in analyzing stochastic models extensively. We organize an infinite-size system of integral-differential equations by means of the density dependent jump Markov process, and obtain a close-form solution: doubly exponential structure, for the fixed point satisfying the system of nonlinear equations, which is always a key in the study of supermarket models. The fixed point is decomposited into two groups of information under a product form: the arrival information and the service information. based on this, we indicate two important observations: the fixed point for the supermarket model is different from the tail of stationary queue length distribution for the ordinary M/G/1 queue, and the doubly exponential solution to the fixed point can extensively exist even if the service time distribution is heavy-tailed. Furthermore, we analyze the exponential convergence of the current location of the supermarket model to its fixed point, and study the Lipschitz condition in the Kurtz Theorem under general service times. Based on these analysis, one can gain a new understanding how workload probing can help in load balancing jobs with general service times such as heavy-tailed service.Comment: 40 pages, 4 figure

Nonlinear Markov Processes in Big Networks

Big networks express various large-scale networks in many practical areas such as computer networks, internet of things, cloud computation, manufacturing systems, transportation networks, and healthcare systems. This paper analyzes such big networks, and applies the mean-field theory and the nonlinear Markov processes to set up a broad class of nonlinear continuous-time block-structured Markov processes, which can be applied to deal with many practical stochastic systems. Firstly, a nonlinear Markov process is derived from a large number of interacting big networks with symmetric interactions, each of which is described as a continuous-time block-structured Markov process. Secondly, some effective algorithms are given for computing the fixed points of the nonlinear Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff center, the Lyapunov functions and the relative entropy are used to analyze stability or metastability of the big network, and several interesting open problems are proposed with detailed interpretation. We believe that the results given in this paper can be useful and effective in the study of big networks.Comment: 28 pages in Special Matrices; 201

Commutation Relations in Mesoscopic Electric Circuits

In the talk, I briefly demonstrate the quantum theory for mesoscopic electric circuits and its applications. In the theory, the importance of the charge discreteness in a mesoscopic electric circuit is addressed. As a result, a new kind of commutation relation for electric charge and current occurred inevitably. The charge representation, canonical current representation and pseudo-current representation are discussed extensively. It not only provides a concrete realization of mathematical models which discuss the space quantization in high energy physics and quantum gravity but also presents a sequence of applications in condensed matter physics from a different point of view. A possible generalization to coupled circuits is also proposed.Comment: 7 pages, talk at ``Spin-Statistics Connection and Commutation Relations'' Capri 200

Super-Exponential Solution in Markovian Supermarket Models: Framework and Challenge

Marcel F. Neuts opened a key door in numerical computation of stochastic models by means of phase-type (PH) distributions and Markovian arrival processes (MAPs). To celebrate his 75th birthday, this paper reports a more general framework of Markovian supermarket models, including a system of differential equations for the fraction measure and a system of nonlinear equations for the fixed point. To understand this framework heuristically, this paper gives a detailed analysis for three important supermarket examples: M/G/1 type, GI/M/1 type and multiple choices, explains how to derive the system of differential equations by means of density-dependent jump Markov processes, and shows that the fixed point may be simply super-exponential through solving the system of nonlinear equations. Note that supermarket models are a class of complicated queueing systems and their analysis can not apply popular queueing theory, it is necessary in the study of supermarket models to summarize such a more general framework which enables us to focus on important research issues. On this line, this paper develops matrix-analytical methods of Markovian supermarket models. We hope this will be able to open a new avenue in performance evaluation of supermarket models by means of matrix-analytical methods.Comment: Randomized load balancing, supermarket model, matrix-analytic method, super-exponential solution, density-dependent jump Markov process, Batch Markovian Arrival Process (BMAP), phase-type (PH) distribution, fixed poin

The effects of optically induced non-Abelian gauge field in cold atoms

We show that $N-1$ degenerate dark states can be generated by coupling $N$-fold degenerate ground states and a common excited state with $N$ laser fields. Interferences between light waves with different frequencies can produce laser fields with time-dependent amplitudes, which can induce not only U(N) non-Abelian vector fields but also the scalar ones for the adiabatic motion of atoms in such laser fields. As an example, a time-periodic gauge potential is produced by applying specific laser fields to a tripod system. Some features of the Landau levels and the ground-state phase diagram of a rotating Bose-Einstein condensate for a concrete gauge field are also discussed.Comment: Revtex 6 pages, 2 figures, version to be published in PR

Einstein-Podolsky-Rosen correlations and Bell correlations in the simplest scenario

Einstein-Podolsky-Rosen (EPR) steering is an intermediate type of quantum nonlocality which sits between entanglement and Bell nonlocality. A set of correlations is Bell nonlocal if it does not admit a local hidden variable (LHV) model, while it is EPR nonlocal if it does not admit a local hidden variable-local hidden state (LHV-LHS) model. It is interesting to know what states can generate EPR-nonlocal correlations in the simplest nontrivial scenario, that is, two projective measurements for each party sharing a two-qubit state. Here we show that a two-qubit state can generate EPR-nonlocal full correlations (excluding marginal statistics) in this scenario if and only if it can generate Bell-nonlocal correlations. If full statistics (including marginal statistics) is taken into account, surprisingly, the same scenario can manifest the simplest one-way steering and the strongest hierarchy between steering and Bell nonlocality. To illustrate these intriguing phenomena in simple setups, several concrete examples are discussed in detail, which facilitates experimental demonstration. In the course of study, we introduce the concept of restricted LHS models and thereby derive a necessary and sufficient semidefinite-programming criterion to determine the steerability of any bipartite state under given measurements. Analytical criteria are further derived in several scenarios of strong theoretical and experimental interest.Comment: New results added, 13 pages, 3 figures; published in Phys. Rev.