66,586 research outputs found
Extinction probabilities of branching processes with countably infinitely many types
We present two iterative methods for computing the global and partial
extinction probability vectors for Galton-Watson processes with countably
infinitely many types. The probabilistic interpretation of these methods
involves truncated Galton-Watson processes with finite sets of types and
modified progeny generating functions. In addition, we discuss the connection
of the convergence norm of the mean progeny matrix with extinction criteria.
Finally, we give a sufficient condition for a population to become extinct
almost surely even though its population size explodes on the average, which is
impossible in a branching process with finitely many types. We conclude with
some numerical illustrations for our algorithmic methods
Ergodic SDEs on submanifolds and related numerical sampling schemes
In many applications, it is often necessary to sample the mean value of
certain quantity with respect to a probability measure {\mu} on the level set
of a smooth function , .
A specially interesting case is the so-called conditional probability measure,
which is useful in the study of free energy calculation and model reduction of
diffusion processes. By Birkhoff's ergodic theorem, one approach to estimate
the mean value is to compute the time average along an infinitely long
trajectory of an ergodic diffusion process on the level set whose invariant
measure is {\mu}. Motivated by the previous work of Ciccotti, Leli\`evre, and
Vanden-Eijnden [11], as well as the work of Leli\`evre, Rousset, and Stoltz
[33], in this paper we construct a family of ergodic diffusion processes on the
level set of whose invariant measures coincide with the given one. For
the conditional measure, in particular, we show that the corresponding SDEs of
the constructed ergodic processes have relatively simple forms, and, moreover,
we propose a consistent numerical scheme which samples the conditional measure
asymptotically. The numerical scheme doesn't require computing the second
derivatives of and the error estimates of its long time sampling
efficiency are obtained.Comment: 45 pages. Accepted versio
Heavy-traffic limits for waiting times in many-server queues with abandonment
We establish heavy-traffic stochastic-process limits for waiting times in
many-server queues with customer abandonment. If the system is asymptotically
critically loaded, as in the quality-and-efficiency-driven (QED) regime, then a
bounding argument shows that the abandonment does not affect waiting-time
processes. If instead the system is overloaded, as in the efficiency-driven
(ED) regime, following Mandelbaum et al. [Proceedings of the Thirty-Seventh
Annual Allerton Conference on Communication, Control and Computing (1999)
1095--1104], we treat customer abandonment by studying the limiting behavior of
the queueing models with arrivals turned off at some time . Then, the
waiting time of an infinitely patient customer arriving at time is the
additional time it takes for the queue to empty. To prove stochastic-process
limits for virtual waiting times, we establish a two-parameter version of
Puhalskii's invariance principle for first passage times. That, in turn,
involves proving that two-parameter versions of the composition and inverse
mappings appropriately preserve convergence.Comment: Published in at http://dx.doi.org/10.1214/09-AAP606 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Avoiding Kernel Fixed Points: Computing with ELU and GELU Infinite Networks
Analysing and computing with Gaussian processes arising from infinitely wide
neural networks has recently seen a resurgence in popularity. Despite this,
many explicit covariance functions of networks with activation functions used
in modern networks remain unknown. Furthermore, while the kernels of deep
networks can be computed iteratively, theoretical understanding of deep kernels
is lacking, particularly with respect to fixed-point dynamics. Firstly, we
derive the covariance functions of MLPs with exponential linear units and
Gaussian error linear units and evaluate the performance of the limiting
Gaussian processes on some benchmarks. Secondly, and more generally, we
introduce a framework for analysing the fixed-point dynamics of iterated
kernels corresponding to a broad range of activation functions. We find that
unlike some previously studied neural network kernels, these new kernels
exhibit non-trivial fixed-point dynamics which are mirrored in finite-width
neural networks.Comment: 18 pages, 9 figures, 2 tables. Corrected name particle capitalisation
and formattin
Wait-Freedom with Advice
We motivate and propose a new way of thinking about failure detectors which
allows us to define, quite surprisingly, what it means to solve a distributed
task \emph{wait-free} \emph{using a failure detector}. In our model, the system
is composed of \emph{computation} processes that obtain inputs and are supposed
to output in a finite number of steps and \emph{synchronization} processes that
are subject to failures and can query a failure detector. We assume that, under
the condition that \emph{correct} synchronization processes take sufficiently
many steps, they provide the computation processes with enough \emph{advice} to
solve the given task wait-free: every computation process outputs in a finite
number of its own steps, regardless of the behavior of other computation
processes. Every task can thus be characterized by the \emph{weakest} failure
detector that allows for solving it, and we show that every such failure
detector captures a form of set agreement. We then obtain a complete
classification of tasks, including ones that evaded comprehensible
characterization so far, such as renaming or weak symmetry breaking
Strong Equivalence Relations for Iterated Models
The Iterated Immediate Snapshot model (IIS), due to its elegant geometrical
representation, has become standard for applying topological reasoning to
distributed computing. Its modular structure makes it easier to analyze than
the more realistic (non-iterated) read-write Atomic-Snapshot memory model (AS).
It is known that AS and IIS are equivalent with respect to \emph{wait-free
task} computability: a distributed task is solvable in AS if and only if it
solvable in IIS. We observe, however, that this equivalence is not sufficient
in order to explore solvability of tasks in \emph{sub-models} of AS (i.e.
proper subsets of its runs) or computability of \emph{long-lived} objects, and
a stronger equivalence relation is needed. In this paper, we consider
\emph{adversarial} sub-models of AS and IIS specified by the sets of processes
that can be \emph{correct} in a model run. We show that AS and IIS are
equivalent in a strong way: a (possibly long-lived) object is implementable in
AS under a given adversary if and only if it is implementable in IIS under the
same adversary. %This holds whether the object is one-shot or long-lived.
Therefore, the computability of any object in shared memory under an
adversarial AS scheduler can be equivalently investigated in IIS
A group membership algorithm with a practical specification
Presents a solvable specification and gives an algorithm for the group membership problem in asynchronous systems with crash failures. Our specification requires processes to maintain a consistent history in their sequences of views. This allows processes to order failures and recoveries in time and simplifies the programming of high level applications. Previous work has proven that the group membership problem cannot be solved in asynchronous systems with crash failures. We circumvent this impossibility result building a weaker, yet nontrivial specification. We show that our solution is an improvement upon previous attempts to solve this problem using a weaker specification. We also relate our solution to other methods and give a classification of progress properties that can be achieved under different models
On the Space Complexity of Set Agreement
The -set agreement problem is a generalization of the classical consensus
problem in which processes are permitted to output up to different input
values. In a system of processes, an -obstruction-free solution to the
problem requires termination only in executions where the number of processes
taking steps is eventually bounded by . This family of progress conditions
generalizes wait-freedom () and obstruction-freedom (). In this
paper, we prove upper and lower bounds on the number of registers required to
solve -obstruction-free -set agreement, considering both one-shot and
repeated formulations. In particular, we show that repeated set agreement
can be solved using registers and establish a nearly matching lower
bound of
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