19 research outputs found
Expected Values Estimated via Mean-Field Approximation are 1/N-Accurate
International audienceMean-field approximation is a powerful tool to study large-scale stochastic systems such as data-centers – one example being the famous power of two-choice paradigm. It is shown in the literature that under quite general conditions, the empirical measure of a system of N interacting objects converges at rate O (1/ √ N) to a deterministic dynamical system, called its mean-field approximation. In this paper, we revisit the accuracy of mean-field approximation by focusing on expected values. We show that, under almost the same general conditions, the expectation of any performance functional converges at rate O (1/N) to its mean-field approximation. Our result applies for finite and infinite-dimensional mean-field models. We also develop a new perturbation theory argument that shows that the result holds for the stationary regime if the dynamical system is asymptotically exponentially stable. We provide numerical experiments that demonstrate that this rate of convergence is tight and that illustrate the necessity of our conditions. As an example, we apply our result to the classical two-choice model. By combining our theory with numerical experiments, we claim that, as the load ρ goes to 1, the average queue length of a two-choice system with N servers is log 2 1 1−ρ + 1 2N (1−ρ) + O 1 N 2
Global attraction of ODE-based mean field models with hyperexponential job sizes
Mean field modeling is a popular approach to assess the performance of large
scale computer systems. The evolution of many mean field models is
characterized by a set of ordinary differential equations that have a unique
fixed point. In order to prove that this unique fixed point corresponds to the
limit of the stationary measures of the finite systems, the unique fixed point
must be a global attractor. While global attraction was established for various
systems in case of exponential job sizes, it is often unclear whether these
proof techniques can be generalized to non-exponential job sizes. In this paper
we show how simple monotonicity arguments can be used to prove global
attraction for a broad class of ordinary differential equations that capture
the evolution of mean field models with hyperexponential job sizes. This class
includes both existing as well as previously unstudied load balancing schemes
and can be used for systems with either finite or infinite buffers. The main
novelty of the approach exists in using a Coxian representation for the
hyperexponential job sizes and a partial order that is stronger than the
componentwise partial order used in the exponential case.Comment: This paper was accepted at ACM Sigmetrics 201
Analysis of the Refined Mean-Field Approximation for the 802.11 Protocol Model
Mean-field approximation is a method to investigate the behavior of stochastic models formed by a large number of interacting objects. A new approximation was recently established, i.e., the refined mean-field approximation, and its high accuracy when the number of objects is small has been shown. In this work, we consider the model of the 802.11 protocol, which is a discrete-time model and show how the refined mean-field approximation can be adapted to this model. Our results confirm the accuracy of the refined mean-field approximation when the model with N objects is in discrete time.This research was founded by the Department of Education of the Basque Government, Spain, through the Consolidated Research Group MATHMODE (IT1456-22) and by the Marie Sklodowska-Curie, grant agreement number 777778
Hyper-Scalable JSQ with Sparse Feedback
Load balancing algorithms play a vital role in enhancing performance in data
centers and cloud networks. Due to the massive size of these systems,
scalability challenges, and especially the communication overhead associated
with load balancing mechanisms, have emerged as major concerns. Motivated by
these issues, we introduce and analyze a novel class of load balancing schemes
where the various servers provide occasional queue updates to guide the load
assignment.
We show that the proposed schemes strongly outperform JSQ() strategies
with comparable communication overhead per job, and can achieve a vanishing
waiting time in the many-server limit with just one message per job, just like
the popular JIQ scheme. The proposed schemes are particularly geared however
towards the sparse feedback regime with less than one message per job, where
they outperform corresponding sparsified JIQ versions.
We investigate fluid limits for synchronous updates as well as asynchronous
exponential update intervals. The fixed point of the fluid limit is identified
in the latter case, and used to derive the queue length distribution. We also
demonstrate that in the ultra-low feedback regime the mean stationary waiting
time tends to a constant in the synchronous case, but grows without bound in
the asynchronous case
Age of Information in Ultra-Dense IoT Systems: Performance and Mean-Field Game Analysis
In this paper, a dense Internet of Things (IoT) monitoring system is
considered in which a large number of IoT devices contend for channel access so
as to transmit timely status updates to the corresponding receivers using a
carrier sense multiple access (CSMA) scheme. Under two packet management
schemes with and without preemption in service, the closed-form expressions of
the average age of information (AoI) and the average peak AoI of each device is
characterized. It is shown that the scheme with preemption in service always
leads to a smaller average AoI and a smaller average peak AoI, compared to the
scheme without preemption in service. Then, a distributed noncooperative medium
access control game is formulated in which each device optimizes its waiting
rate so as to minimize its average AoI or average peak AoI under an average
energy cost constraint on channel sensing and packet transmitting. To overcome
the challenges of solving this game for an ultra-dense IoT, a mean-field game
(MFG) approach is proposed to study the asymptotic performance of each device
for the system in the large population regime. The accuracy of the MFG is
analyzed, and the existence, uniqueness, and convergence of the mean-field
equilibrium (MFE) are investigated. Simulation results show that the proposed
MFG is accurate even for a small number of devices; and the proposed CSMA-type
scheme under the MFG analysis outperforms two baseline schemes with fixed and
dynamic waiting rates, with the average AoI reductions reaching up to 22% and
34%, respectively. Moreover, it is observed that the average AoI and the
average peak AoI under the MFE do not necessarily decrease with the arrival
rate.Comment: Fixed typos in Equations (4) and (7). 30 pages, 9 figure
Improved estimations of stochastic chemical kinetics by finite state expansion
Stochastic reaction networks are a fundamental model to describe interactions
between species where random fluctuations are relevant. The master equation
provides the evolution of the probability distribution across the discrete
state space consisting of vectors of population counts for each species.
However, since its exact solution is often elusive, several analytical
approximations have been proposed. The deterministic rate equation (DRE) gives
a macroscopic approximation as a compact system of differential equations that
estimate the average populations for each species, but it may be inaccurate in
the case of nonlinear interaction dynamics. Here we propose finite state
expansion (FSE), an analytical method mediating between the microscopic and the
macroscopic interpretations of a stochastic reaction network by coupling the
master equation dynamics of a chosen subset of the discrete state space with
the mean population dynamics of the DRE. An algorithm translates a network into
an expanded one where each discrete state is represented as a further distinct
species. This translation exactly preserves the stochastic dynamics, but the
DRE of the expanded network can be interpreted as a correction to the original
one. The effectiveness of FSE is demonstrated in models that challenge
state-of-the-art techniques due to intrinsic noise, multi-scale populations,
and multi-stability.Comment: 33 pages, 9 figure
Bias and Refinement of Multiscale Mean Field Models
Mean field approximation is a powerful technique which has been used in many
settings to study large-scale stochastic systems. In the case of two-timescale
systems, the approximation is obtained by a combination of scaling arguments
and the use of the averaging principle. This paper analyzes the approximation
error of this `average' mean field model for a two-timescale model
, where the slow component
describes a population of interacting particles which is fully coupled with a
rapidly changing environment . The model is parametrized by a
scaling factor , e.g. the population size, which as gets large decreases
the jump size of the slow component in contrast to the unchanged dynamics of
the fast component. We show that under relatively mild conditions the `average'
mean field approximation has a bias of order compared to
. This holds true under any continuous performance
metric in the transient regime as well as for the steady-state if the model is
exponentially stable. To go one step further, we derive a bias correction term
for the steady-state from which we define a new approximation called the
refined `average' mean field approximation whose bias is of order .
This refined `average' mean field approximation allows computing an accurate
approximation even for small scaling factors, i.e., . We
illustrate the developed framework and accuracy results through an application
to a random access CSMA model.Comment: 28 page