17,465 research outputs found
Mathematical Models of Gene Expression
In this paper we analyze the equilibrium properties of a large class of
stochastic processes describing the fundamental biological process within
bacterial cells, {\em the production process of proteins}. Stochastic models
classically used in this context to describe the time evolution of the numbers
of mRNAs and proteins are presented and discussed. An extension of these
models, which includes elongation phases of mRNAs and proteins, is introduced.
A convergence result to equilibrium for the process associated to the number of
proteins and mRNAs is proved and a representation of this equilibrium as a
functional of a Poisson process in an extended state space is obtained.
Explicit expressions for the first two moments of the number of mRNAs and
proteins at equilibrium are derived, generalizing some classical formulas.
Approximations used in the biological literature for the equilibrium
distribution of the number of proteins are discussed and investigated in the
light of these results. Several convergence results for the distribution of the
number of proteins at equilibrium are in particular obtained under different
scaling assumptions
Interacting multi-class transmissions in large stochastic networks
The mean-field limit of a Markovian model describing the interaction of
several classes of permanent connections in a network is analyzed. Each of the
connections has a self-adaptive behavior in that its transmission rate along
its route depends on the level of congestion of the nodes of the route. Since
several classes of connections going through the nodes of the network are
considered, an original mean-field result in a multi-class context is
established. It is shown that, as the number of connections goes to infinity,
the behavior of the different classes of connections can be represented by the
solution of an unusual nonlinear stochastic differential equation depending not
only on the sample paths of the process, but also on its distribution.
Existence and uniqueness results for the solutions of these equations are
derived. Properties of their invariant distributions are investigated and it is
shown that, under some natural assumptions, they are determined by the
solutions of a fixed-point equation in a finite-dimensional space.Comment: Published in at http://dx.doi.org/10.1214/09-AAP614 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Flow-aware MAC Protocol for a Passive Optical Metropolitan Area Network
The paper introduces an original MAC protocol for a passive optical
metropolitan area network using time-domain wavelength interleaved networking
(TWIN)% as proposed recently by Bell Labs . Optical channels are shared under
the distributed control of destinations using a packet-based polling algorithm.
This MAC is inspired more by EPON dynamic bandwidth allocation than the
slotted, GPON-like access control generally envisaged for TWIN. Management of
source-destination traffic streams is flow-aware with the size of allocated
time slices being proportional to the number of active flows. This emulates a
network-wide, distributed fair queuing scheduler, bringing the well-known
implicit service differentiation and robustness advantages of this mechanism to
the metro area network. The paper presents a comprehensive performance
evaluation based on analytical modelling supported by simulations. The proposed
MAC is shown to have excellent performance in terms of both traffic capacity
and packet latency
A stochastic analysis of resource sharing with logarithmic weights
The paper investigates the properties of a class of resource allocation
algorithms for communication networks: if a node of this network has
requests to transmit, then it receives a fraction of the capacity proportional
to , the logarithm of its current load. A detailed fluid scaling
analysis of such a network with two nodes is presented. It is shown that the
interaction of several time scales plays an important role in the evolution of
such a system, in particular its coordinates may live on very different time
and space scales. As a consequence, the associated stochastic processes turn
out to have unusual scaling behaviors. A heavy traffic limit theorem for the
invariant distribution is also proved. Finally, we present a generalization to
the resource sharing algorithm for which the function is replaced by an
increasing function. Possible generalizations of these results with nodes
or with the function replaced by another slowly increasing function are
discussed.Comment: Published at http://dx.doi.org/10.1214/14-AAP1057 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A scaling analysis of a cat and mouse Markov chain
Motivated by an original on-line page-ranking algorithm, starting from an arbitrary Markov chain on a discrete state space , a Markov chain on the product space , the cat and mouse Markov chain, is constructed. The first coordinate of this Markov chain behaves like the original Markov chain and the second component changes only when both coordinates are equal. The asymptotic properties of this Markov chain are investigated. A representation of its invariant measure is in particular obtained. When the state space is infinite it is shown that this Markov chain is in fact null recurrent if the initial Markov chain is positive recurrent and reversible. In this context, the scaling properties of the location of the second component, the mouse, are investigated in various situations: simple random walks in and , reflected simple random walk in and also in a continuous time setting. For several of these processes, a time scaling with rapid growth gives an interesting asymptotic behavior related to limit results for occupation times and rare events of Markov processes.\u
Self-adaptive congestion control for multi-class intermittent connections in a communication network
A Markovian model of the evolution of intermittent connections of various
classes in a communication network is established and investigated. Any
connection evolves in a way which depends only on its class and the state of
the network, in particular as to the route it uses among a subset of the
network nodes. It can be either active (ON) when it is transmitting data along
its route, or idle (OFF). The congestion of a given node is defined as a
functional of the transmission rates of all ON connections going through it,
and causes losses and delays to these connections. In order to control this,
the ON connections self-adaptively vary their transmission rate in TCP-like
fashion. The connections interact through this feedback loop. A Markovian model
is provided by the states (OFF, or ON with some transmission rate) of the
connections. The number of connections in each class being potentially huge, a
mean-field limit result is proved with an appropriate scaling so as to reduce
the dimensionality. In the limit, the evolution of the states of the
connections can be represented by a non-linear system of stochastic
differential equations, of dimension the number of classes. Additionally, it is
shown that the corresponding stationary distribution can be expressed by the
solution of a fixed-point equation of finite dimension
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