Ingemar Kaj

Why Time Matters: Codon Evolution and the Temporal Dynamics of dN/dS

The ratio of divergence at nonsynonymous and synonymous sites, dN/dS, is a widely used measure in evolutionary genetic studies to investigate the extent to which selection modulates gene sequence evolution. Originally tailored to codon sequences of distantly related lineages, dN/dS represents the ratio of fixed nonsynonymous to synonymous differences. The impact of ancestral and lineage-specific polymorphisms on dN/dS, which we here show to be substantial for closely related lineages, is generally neglected in estimation techniques of dN/dS. To address this issue, we formulate a codon model that is firmly anchored in population genetic theory, derive analytical expressions for the dN/dS measure by Poisson random field approximation in a Markovian framework and validate the derivations by simulations. In good agreement, simulations and analytical derivations demonstrate that dN/dS is biased by polymorphisms at short time scales and that it can take substantial time for the expected value to settle at its time limit where only fixed differences are considered. We further show that in any attempt to estimate the dN/dS ratio from empirical data the effect of the intrinsic fluctuations of a ratio of stochastic variables, can even under neutrality yield extreme values of dN/dS at short time scales or in regions of low mutation rate. Taken together, our results have significant implications for the interpretation of dN/dS estimates, the McDonald–Kreitman test and other related statistics, in particular for closely related lineages.

Convergence to Fractional Brownian Motion and to the Telecom Process: the Integral Representation Approach

It has become common practice to use heavy-tailed distributions in order to describe the variations in time and space of network traffic workloads. The asymptotic behavior of these workloads is complex; different limit processes emerge depending on the specifics of the work arrival structure and the nature of the asymptotic scaling. We focus on two variants of the infinite source Poisson model and provide a coherent and unified presentation of the scaling theory by using integral representations. This allows us to understand physically why the various limit processes arise.

Limiting Fractal Random Processes in Heavy-Tailed Systems

We give an overview of limit results for a selection of stochastic models that involve heavy-tailed distributions, exhibit long-range dependence and are naturally parametrized by a tail-index. Under aggregation of independent subsystems and simultaneous time or space rescaling, the asymptotic behavior of such systems varies considerably. It is the relative speed of aggregation degree and rescaling that determines the nature of the limit process, ranging from fractional Brownian motion to stable Levy processes. The limits obtained for a generalized example based on a spatial Poisson grains model include a fractional Brownian field. We are particularly interested in the intermediate scaling regime, bridging Gaussian and stable asymptotics. One feature shared by all limit processes is identified as an aggregatesimilarity property.

Scaling limits for random fields with long range dependence

This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density ? of the sets grows to infinity and the mean volume ? of the sets tends to zero. Assuming that the volume distribution has a regularly varying tail with infinite variance, we show that the centered and renormalized random field can have three different limits, depending on the relative speed at which ? and ? are scaled. If ? grows much faster than ? shrinks, the limit is Gaussian with long-range dependence, while in the opposite case, the limit is independently scattered with infinite second moments. In a special intermediate scaling regime, there exists a nontrivial limiting random field that is not stable.

Stochastic Modeling in Broadband Communications Systems

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Limit Processes for Age-Dependent Branching Particle Systems

We consider systems of spatially distributed branching particles in Rd. The particle lifelengths are of general form, hence the time propagation of the system is typically not Markov. A natural time-space-mass scaling is applied to a sequence of particle systems and we derive limit results for the corresponding sequence of measure-valued processes. The limit is identified as the projection on Rd of a superprocess in R+×Rd. The additive functional characterizing the superprocess is the scaling limit of certain point processes, which count generations along a line of descent for the branching particles.

Throughput Modeling and Simulation for Single Connection TCP-Tahoe

Abstract We present a stochastic model for the window dynamics in TCP and investigate the throughput performance of TCP-Tahoe during persistent transmission of a bulk data flow. Our overall purpose is to investigate the impact of packet loss probability on the resulting TCP throughput. We analyze several distinct aspects of the TCP flow control, such as maximal advertised window, large initial window, fast retransmit, time-out delays, and duplicate ACK threshold, as well as combinations of these. In this manner we gain insight into each features effect on throughput. Under the further assumption that packets are lost independently of each other, we obtain numerically explicit formulas for the throughput. The main tools are the application of renewal-reward arguments and a new technique to approximate window size and cycle lengths by continuous quantities, which greatly simplifies calculations. Validation of the model is done by simulations using the ns simulator.

Modelling the Arrival Process for Packet Audio

Packets in an audio stream can be distorted relative to one another during the traversal of a packet switched network. This distortion can be mainly attributed to queues in routers between the source and the destination. The queues can consist of packets either from our own flow, or from other flows. The contribution of this work is a Markov model for the time delay variation of packet audio in this scenario. Our model is extensible, and show this by including sender silence suppression and packet loss into the model. By comparing the model to wide area traffic traces we show the possibility to generate an audio arrival process similar to those created by real conditions. This is done by comparing the probability density functions of our model to the real captured data.

Propagation properties for a message in a Brownian sensor network

A wireless multi-hop sensor network, in which node positions are fixed, may fail to transmit a message over longer distances. This could occur, for example, due to low node density or small node transmission range. In mobile systems where nodes are allowed to move, it is natural to expect a better reachability, with the condition that messages are not time-critical and longer propagation delays are permitted. In order to understand the relation of mobility to node density and node transmission range, we study a simple network model where active sensors move according to independent Brownian motions. In the one-dimensional case, the propagation of a message can be viewed as a Brownian growth process among Poisson points on the real line. We investigate the distributional properties of the mobile nodes and show that the system grows linearly at a remarkably uniform rate. For the spatial model where planar Brownian motions transport and transfer the message to those nodes which eventually come within transmission range of active messenger nodes, we provide a discussion and some insight based primarily on simulations.

Probabilistic Analysis of Hierarchical Cluster Protocols for Wireless Sensor Networks

Wireless sensor networks are designed to extract data from the deployment environment and combine sensing, data processing and wireless communication to provide useful information for the network users. Hundreds or thousands of small embedded units, which operate under low-energy supply and with limited access to central network control, rely on interconnecting protocols to coordinate data aggregation and transmission. Energy efficiency is crucial and it has been proposed that cluster based and distributed architectures such as LEACH are particularly suitable. We analyse the random cluster hierarchy in this protocol and provide a solution for low-energy and limited-loss optimization. Moreover, we extend these results to a multi-level version of LEACH, where clusters of nodes again self-organize to form clusters of clusters, and so on.