266 research outputs found
The Critical Exponent is Computable for Automatic Sequences
The critical exponent of an infinite word is defined to be the supremum of
the exponent of each of its factors. For k-automatic sequences, we show that
this critical exponent is always either a rational number or infinite, and its
value is computable. Our results also apply to variants of the critical
exponent, such as the initial critical exponent of Berthe, Holton, and Zamboni
and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes
or recovers previous results of Krieger and others, and is applicable to other
situations; e.g., the computation of the optimal recurrence constant for a
linearly recurrent k-automatic sequence.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Waiting time dynamics of priority-queue networks
We study the dynamics of priority-queue networks, generalizations of the
binary interacting priority queue model introduced by Oliveira and Vazquez
[Physica A {\bf 388}, 187 (2009)]. We found that the original AND-type protocol
for interacting tasks is not scalable for the queue networks with loops because
the dynamics becomes frozen due to the priority conflicts. We then consider a
scalable interaction protocol, an OR-type one, and examine the effects of the
network topology and the number of queues on the waiting time distributions of
the priority-queue networks, finding that they exhibit power-law tails in all
cases considered, yet with model-dependent power-law exponents. We also show
that the synchronicity in task executions, giving rise to priority conflicts in
the priority-queue networks, is a relevant factor in the queue dynamics that
can change the power-law exponent of the waiting time distribution.Comment: 5 pages, 3 figures, minor changes, final published versio
Optimization in task--completion networks
We discuss the collective behavior of a network of individuals that receive,
process and forward to each other tasks. Given costs they store those tasks in
buffers, choosing optimally the frequency at which to check and process the
buffer. The individual optimizing strategy of each node determines the
aggregate behavior of the network. We find that, under general assumptions, the
whole system exhibits coexistence of equilibria and hysteresis.Comment: 18 pages, 3 figures, submitted to JSTA
On the Number of Unbordered Factors
We illustrate a general technique for enumerating factors of k-automatic
sequences by proving a conjecture on the number f(n) of unbordered factors of
the Thue-Morse sequence. We show that f(n) = 4 and that f(n) = n
infinitely often. We also give examples of automatic sequences having exactly 2
unbordered factors of every length
On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases
This article studies the expressive power of finite automata recognizing sets
of real numbers encoded in positional notation. We consider Muller automata as
well as the restricted class of weak deterministic automata, used as symbolic
set representations in actual applications. In previous work, it has been
established that the sets of numbers that are recognizable by weak
deterministic automata in two bases that do not share the same set of prime
factors are exactly those that are definable in the first order additive theory
of real and integer numbers. This result extends Cobham's theorem, which
characterizes the sets of integer numbers that are recognizable by finite
automata in multiple bases.
In this article, we first generalize this result to multiplicatively
independent bases, which brings it closer to the original statement of Cobham's
theorem. Then, we study the sets of reals recognizable by Muller automata in
two bases. We show with a counterexample that, in this setting, Cobham's
theorem does not generalize to multiplicatively independent bases. Finally, we
prove that the sets of reals that are recognizable by Muller automata in two
bases that do not share the same set of prime factors are exactly those
definable in the first order additive theory of real and integer numbers. These
sets are thus also recognizable by weak deterministic automata. This result
leads to a precise characterization of the sets of real numbers that are
recognizable in multiple bases, and provides a theoretical justification to the
use of weak automata as symbolic representations of sets.Comment: 17 page
Invasion percolation on a tree and queueing models
We study the properties of the Barabási model of queuing [ A.-L. Barabási Nature (London) 435 207 (2005); J. G. Oliveira and A.-L. Barabási Nature (London) 437 1251 (2005)] in the hypothesis that the number of tasks grows with time steadily. Our analytical approach is based on two ingredients. First we map exactly this model into an invasion percolation dynamics on a Cayley tree. Second we use the theory of biased random walks. In this way we obtain the following results: the stationary-state dynamics is a sequence of causally and geometrically connected bursts of execution activities with scale-invariant size distribution. We recover the correct waiting-time distribution PW(τ)∼τ−3/2 at the stationary state (as observed in different realistic data). Finally we describe quantitatively the dynamics out of the stationary state quantifying the power-law slow approach to stationarity both in single dynamical realization and in average. These results can be generalized to the case of a stochastic increase in the queue length in time with limited fluctuations. As a limit case we recover the situation in which the queue length fluctuates around a constant average value
Enumeration and Decidable Properties of Automatic Sequences
We show that various aspects of k-automatic sequences -- such as having an
unbordered factor of length n -- are both decidable and effectively enumerable.
As a consequence it follows that many related sequences are either k-automatic
or k-regular. These include many sequences previously studied in the
literature, such as the recurrence function, the appearance function, and the
repetitivity index. We also give some new characterizations of the class of
k-regular sequences. Many results extend to other sequences defined in terms of
Pisot numeration systems
Priority queues with bursty arrivals of incoming tasks
Recently increased accessibility of large-scale digital records enables one
to monitor human activities such as the interevent time distributions between
two consecutive visits to a web portal by a single user, two consecutive emails
sent out by a user, two consecutive library loans made by a single individual,
etc. Interestingly, those distributions exhibit a universal behavior,
, where is the interevent time, and or 3/2. The universal behaviors have been modeled via the
waiting-time distribution of a task in the queue operating based on priority;
the waiting time follows a power law distribution with either or 3/2 depending on the detail of
queuing dynamics. In these models, the number of incoming tasks in a unit time
interval has been assumed to follow a Poisson-type distribution. For an email
system, however, the number of emails delivered to a mail box in a unit time we
measured follows a powerlaw distribution with general exponent . For
this case, we obtain analytically the exponent , which is not
necessarily 1 or 3/2 and takes nonuniversal values depending on . We
develop the generating function formalism to obtain the exponent ,
which is distinct from the continuous time approximation used in the previous
studies.Comment: 19 pages, 5 figure
Modeling bursts and heavy tails in human dynamics
Current models of human dynamics, used from risk assessment to
communications, assume that human actions are randomly distributed in time and
thus well approximated by Poisson processes. We provide direct evidence that
for five human activity patterns the timing of individual human actions follow
non-Poisson statistics, characterized by bursts of rapidly occurring events
separated by long periods of inactivity. We show that the bursty nature of
human behavior is a consequence of a decision based queuing process: when
individuals execute tasks based on some perceived priority, the timing of the
tasks will be heavy tailed, most tasks being rapidly executed, while a few
experiencing very long waiting times. We discuss two queueing models that
capture human activity. The first model assumes that there are no limitations
on the number of tasks an individual can hadle at any time, predicting that the
waiting time of the individual tasks follow a heavy tailed distribution with
exponent alpha=3/2. The second model imposes limitations on the queue length,
resulting in alpha=1. We provide empirical evidence supporting the relevance of
these two models to human activity patterns. Finally, we discuss possible
extension of the proposed queueing models and outline some future challenges in
exploring the statistical mechanisms of human dynamics.Comment: RevTex, 19 pages, 8 figure
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