25,122 research outputs found
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 degenerate dark states can be generated by coupling
-fold degenerate ground states and a common excited state with 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.
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