Systems weakend by failures

AbstractThe ideas of a dynamic approach to the analysis of multivariate life length distributions, introduced in Arjas (1981a) and Arjas and Norros (1984), are developed further. Basic definitions are given in terms of prediction processes. Properties of martingales jumping downwards at failure times are studied. Finally, the spaecial case of a general multivariate exponential distribution is considered

Regular decomposition of large graphs and other structures: scalability and robustness towards missing data

A method for compression of large graphs and matrices to a block structure is further developed. Szemer\'edi's regularity lemma is used as a generic motivation of the significance of stochastic block models. Another ingredient of the method is Rissanen's minimum description length principle (MDL). We continue our previous work on the subject, considering cases of missing data and scaling of algorithms to extremely large size of graphs. In this way it would be possible to find out a large scale structure of a huge graphs of certain type using only a tiny part of graph information and obtaining a compact representation of such graphs useful in computations and visualization.Comment: Accepted for publication in: Fourth International Workshop on High Performance Big Graph Data Management, Analysis, and Mining, December 11, 2017, Bosto U.S.

Large cliques in a power-law random graph

We study the size of the largest clique $\omega(G(n,\alpha))$ in a random graph $G(n,\alpha)$ on $n$ vertices which has power-law degree distribution with exponent $\alpha$. We show that for `flat' degree sequences with $\alpha>2$ whp the largest clique in $G(n,\alpha)$ is of a constant size, while for the heavy tail distribution, when $0<\alpha<2$, $\omega(G(n,\alpha))$ grows as a power of $n$. Moreover, we show that a natural simple algorithm whp finds in $G(n,\alpha)$ a large clique of size $(1+o(1))\omega(G(n,\alpha))$ in polynomial time.Comment: 13 page

On Spatial Point Processes with Uniform Births and Deaths by Random Connection

This paper is focused on a class of spatial birth and death process of the Euclidean space where the birth rate is constant and the death rate of a given point is the shot noise created at its location by the other points of the current configuration for some response function $f$. An equivalent view point is that each pair of points of the configuration establishes a random connection at an exponential time determined by $f$, which results in the death of one of the two points. We concentrate on space-motion invariant processes of this type. Under some natural conditions on $f$, we construct the unique time-stationary regime of this class of point processes by a coupling argument. We then use the birth and death structure to establish a hierarchy of balance integral relations between the factorial moment measures. Finally, we show that the time-stationary point process exhibits a certain kind of repulsion between its points that we call $f$-repulsion

On convergence to stationarity of fractional Brownian storage

With $M(t):=\sup_{s\in[0,t]}A(s)-s$ denoting the running maximum of a fractional Brownian motion $A(\cdot)$ with negative drift, this paper studies the rate of convergence of $\mathbb {P}(M(t)>x)$ to $\mathbb{P}(M>x)$. We define two metrics that measure the distance between the (complementary) distribution functions $\mathbb{P}(M(t)>\cdot)$ and $\mathbb{P}(M>\cdot)$. Our main result states that both metrics roughly decay as $\exp(-\vartheta t^{2-2H})$, where $\vartheta$ is the decay rate corresponding to the tail distribution of the busy period in an fBm-driven queue, which was computed recently [Stochastic Process. Appl. (2006) 116 1269--1293]. The proofs extensively rely on application of the well-known large deviations theorem for Gaussian processes. We also show that the identified relation between the decay of the convergence metrics and busy-period asymptotics holds in other settings as well, most notably when G\"artner--Ellis-type conditions are fulfilled.Comment: Published in at http://dx.doi.org/10.1214/08-AAP578 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spatial Interactions of Peers and Performance of File Sharing Systems

We propose a new model for peer-to-peer networking which takes the network bottlenecks into account beyond the access. This model allows one to cope with key features of P2P networking like degree or locality constraints or the fact that distant peers often have a smaller rate than nearby peers. We show that the spatial point process describing peers in their steady state then exhibits an interesting repulsion phenomenon. We analyze two asymptotic regimes of the peer-to-peer network: the fluid regime and the hard--core regime. We get closed form expressions for the mean (and in some cases the law) of the peer latency and the download rate obtained by a peer as well as for the spatial density of peers in the steady state of each regime, as well as an accurate approximation that holds for all regimes. The analytical results are based on a mix of mathematical analysis and dimensional analysis and have important design implications. The first of them is the existence of a setting where the equilibrium mean latency is a decreasing function of the load, a phenomenon that we call super-scalability.Comment: No. RR-7713 (2012

Origin-destination matrix estimation with a conditionally binomial model

A doubly stochastic, conditionally binomial model is proposed to describe volumes of vehicular origin-destination flows in regular vehicular traffic, such as morning rush hours. The statistical properties of this model are motivated by the data obtained from inductive loop traffic counts. The model parameters can be expressed as rational functions of the first and second order moments of the observed link counts. Challenges arising from the inaccuracy of moment estimates are studied. A real origin-destination traffic problem of Tampere city is solved by optimisation methods and the accuracy of the solution is examined.Peer reviewe