arXiv subjects

Frank Aurzada

Publications and source records attributed to Frank Aurzada.

At least 55 records · Page 3Linked to original sources

arXiv2016

Random walks and branching processes in correlated Gaussian environment

We study persistence probabilities for random walks in correlated Gaussian random environment first studied by Oshanin, Rosso and Schehr. From the persistence results, we can deduce properties of critical branching processes with offspring sizes geometrically distributed with correlated random parameters. More precisely, we obtain estimates on the tail distribution of its total population size, of its maximum population, and of its extinction time.

math.PR↗

arXiv2016

Small deviations of sums of correlated stationary Gaussian sequences

We consider the small deviation probabilities (SDP) for sums of stationary Gaussian sequences. For the cases of constant boundaries and boundaries tending to zero, we obtain quite general results. For the case of the boundaries tending to infinity, we focus our attention on the discrete analogs of the fractional Brownian motion (FBM). It turns out that the lower bounds for the SDP can be transferred from the well studied FBM caseto the discrete time setting under the usual assumptions that imply weak convergence while the transfer of the corresponding upper bounds necessarily requires a deeper knowledge of the spectral structure of the underlying stationary sequence.

math.PR↗

arXiv2015

Upstream Polling Protocols for Flow Control in PON/xDSL Hybrid Access Networks

In a hybrid PON/xDSL access network, multiple Customer Premise Equipment (CPE) nodes connect over individual Digital Subscriber Lines (DSLs) to a drop-point device. The drop-point device, which is typically reverse powered from the customer, is co-located with an Optical Network Unit (ONU) of the Passive Optical Network (PON). We demonstrate that the drop-point experiences very high buffer occupancies when no flow control or standard Ethernet PAUSE frame flow control is employed. In order to reduce the buffer occupancies in the drop-point, we introduce two gated flow control protocols that extend the polling-based PON medium access control to the DSL segments between the CPEs and the ONUs. We analyze the timing of the gated flow control mechanisms to specify the latest possible time instant when CPEs can start the DSL upstream transmissions so that the ONU can forward the upstream transmissions at the full PON upstream transmission bit rate. Through extensive simulations for a wide range of bursty traffic models, we find that the gated flow control mechanisms, specifically, the ONU and CPE grant sizing policies, enable effective control of the maximum drop-point buffer occupancies.

cs.NI↗

arXiv2015

Persistence exponent for discrete-time, time-reversible processes

We study the persistence probability for some discrete-time, time-reversible processes. In particular, we deduce the persistence exponent in a number of examples: first, we deal with random walks in random sceneries (RWRS) in any dimension with Gaussian scenery. Second, we deal with sums of stationary Gaussian sequences with correlations exhibiting long-range dependence. Apart from the persistence probability we deal with the position of the maximum and the time spent on the positive half-axis by the process.

math.PR↗

arXiv2015

The first passage time problem over a moving boundary for asymptotically stable Lévy processes

We study the asymptotic tail behaviour of the first-passage time over a moving boundary for asymptotically $α$-stable Lévy processes with $α<1$. Our main result states that if the left tail of the Lévy measure is regularly varying with index $- α$ and the moving boundary is equal to $1 - t^γ$ for some $γ<1/α$, then the probability that the process stays below the moving boundary has the same asymptotic polynomial order as in the case of a constant boundary. The same is true for the increasing boundary $1 + t^γ$ with $γ<1/α$ under the assumption of a regularly varying right tail with index $- α$.

math.PR↗

arXiv2014

Exponential moments of first passage times and related quantities for Lévy processes

For a Lévy process on the real line, we provide complete criteria for the finiteness of exponential moments of the first passage time into the interval $(r,\infty)$, the sojourn time in the interval $(-\infty,r]$, and the last exit time from $(-\infty,r]$. Moreover, whenever these quantities are finite, we derive their respective asymptotic behavior as $r \to \infty$.

math.PR↗

arXiv2014

DyCaPPON: Dynamic Circuit and Packet Passive Optical Network (Extended Version)

Dynamic circuits are well suited for applications that require predictable service with a constant bit rate for a prescribed period of time, such as cloud computing and e-science applications. Past research on upstream transmission in passive optical networks (PONs) has mainly considered packet-switched traffic and has focused on optimizing packet-level performance metrics, such as reducing mean delay. This study proposes and evaluates a dynamic circuit and packet PON (DyCaPPON) that provides dynamic circuits along with packet-switched service. DyCaPPON provides $(i)$ flexible packet-switched service through dynamic bandwidth allocation in periodic polling cycles, and $(ii)$ consistent circuit service by allocating each active circuit a fixed-duration upstream transmission window during each fixed-duration polling cycle. We analyze circuit-level performance metrics, including the blocking probability of dynamic circuit requests in DyCaPPON through a stochastic knapsack-based analysis. Through this analysis we also determine the bandwidth occupied by admitted circuits. The remaining bandwidth is available for packet traffic and we conduct an approximate analysis of the resulting mean delay of packet traffic. Through extensive numerical evaluations and verifying simulations we demonstrate the circuit blocking and packet delay trade-offs in DyCaPPON.

cs.NI↗

arXiv2013

FiWi Access Networks Based on Next-Generation PON and Gigabit-Class WLAN Technologies: A Capacity and Delay Analysis (Extended Version)

Current Gigabit-class passive optical networks (PONs) evolve into next-generation PONs, whereby high-speed 10+ Gb/s time division multiplexing (TDM) and long-reach wavelength-broadcasting/routing wavelength division multiplexing (WDM) PONs are promising near-term candidates. On the other hand, next-generation wireless local area networks (WLANs) based on frame aggregation techniques will leverage physical layer enhancements, giving rise to Gigabit-class very high throughput (VHT) WLANs. In this paper, we develop an analytical framework for evaluating the capacity and delay performance of a wide range of routing algorithms in converged fiber-wireless (FiWi) broadband access networks based on different next-generation PONs and a Gigabit-class multi-radio multi-channel WLAN-mesh front-end. Our framework is very flexible and incorporates arbitrary frame size distributions, traffic matrices, optical/wireless propagation delays, data rates, and fiber faults. We verify the accuracy of our probabilistic analysis by means of simulation for the wireless and wireless-optical-wireless operation modes of various FiWi network architectures under peer-to-peer, upstream, uniform, and nonuniform traffic scenarios. The results indicate that our proposed optimized FiWi routing algorithm (OFRA) outperforms minimum (wireless) hop and delay routing in terms of throughput for balanced and unbalanced traffic loads, at the expense of a slightly increased mean delay at small to medium traffic loads.

cs.IT↗

arXiv2013

Capacity and Delay Analysis of Next-Generation Passive Optical Networks (NG-PONs) - Extended Version

Building on the Ethernet Passive Optical Network (EPON) and Gigabit PON (GPON) standards, Next-Generation (NG) PONs (i) provide increased data rates, split ratios, wavelengths counts, and fiber lengths, as well as (ii) allow for all-optical integration of access and metro networks. In this paper we provide a comprehensive probabilistic analysis of the capacity (maximum mean packet throughput) and packet delay of subnetworks that can be used to form NG-PONs. Our analysis can cover a wide range of NG-PONs through taking the minimum capacity of the subnetworks making up the NG-PON and weighing the packet delays of the subnetworks. Our numerical and simulation results indicate that our analysis quite accurately characterizes the throughput-delay performance of EPON/GPON tree networks, including networks upgraded with higher data rates and wavelength counts. Our analysis also characterizes the trade-offs and bottlenecks when integrating EPON/GPON tree networks across a metro area with a ring, a Passive Star Coupler (PSC), or an Arrayed Waveguide Grating (AWG) for uniform and non-uniform traffic. To the best of our knowledge, the presented analysis is the first to consider multiple PONs interconnected via a metro network.

cs.IT↗

arXiv2013

Small time Chung-type LIL for Lévy processes

We prove Chung-type laws of the iterated logarithm for general Lévy processes at zero. In particular, we provide tools to translate small deviation estimates directly into laws of the iterated logarithm. This reveals laws of the iterated logarithm for Lévy processes at small times in many concrete examples. In some cases, exotic norming functions are derived.

math.PR↗

arXiv2013

Persistence of fractional Brownian motion with moving boundaries and applications

We consider various problems related to the persistence probability of fractional Brownian motion (FBM), which is the probability that the FBM $X$ stays below a certain level until time $T$. Recently, Oshanin et al. study a physical model where persistence properties of FBM are shown to be related to scaling properties of a quantity $J_N$, called steady-state current. It turns out that for this analysis it is important to determine persistence probabilities of FBM with a moving boundary. We show that one can add a boundary of logarithmic order to a FBM without changing the polynomial rate of decay of the corresponding persistence probability which proves a result needed in Oshanin et al. Moreover, we complement their findings by considering the continuous-time version of $J_N$. Finally, we use the results for moving boundaries in order to improve estimates by Molchan concerning the persistence properties of other quantities of interest, such as the time when a FBM reaches its maximum on the time interval $(0,1)$ or the last zero in the interval $(0,1)$.

math.PR↗

arXiv2012

First passage times of Lévy processes over a one-sided moving boundary

We study the asymptotic behaviour of the tail of the distribution of the first passage time of a Lévy process over a one-sided moving boundary. Our main result states that if the boundary behaves as $t^γ$ for large $t$ for some $γ<1/2$ then the probability that the process stays below the boundary behaves asymptotically as in the case of a constant boundary. We do not have to assume Spitzer's condition in contrast to all previously known results. Both positive ($+t^γ$) and negative ($-t^γ$) boundaries are considered. These results extend the findings of Greenwood and Novikov (1986) and are also motivated by results in the case of Brownian motion, for which the above result was proved in Uchiyama (1980).

math.PR↗

arXiv2012

Persistence probabilities for an integrated random walk bridge

We prove that an integrated simple random walk, where random walk and integrated random walk are conditioned to return to zero, has asymptotic probability $n^{-1/2}$ to stay positive. This question is motivated by so-called random polymer models and proves a conjecture by Caravenna and Deuschel.

math.PR↗

arXiv2012

Persistence probabilities \& exponents

This article deals with the asymptotic behaviour as $t\to +\infty$ of the survival function $P[T > t],$ where $T$ is the first passage time above a non negative level of a random process starting from zero. In many cases of physical significance, the behaviour is of the type $P[T > t]=t^{-θ+ o(1)}$ for a known or unknown positive parameter $θ$ which is called a persistence exponent. The problem is well understood for random walks or Lévy processes but becomes more difficult for integrals of such processes, which are more related to physics. We survey recent results and open problems in this field.

math.PR↗

arXiv2012

First exit of Brownian motion from a one-sided moving boundary

We revisit a result of Uchiyama (1980): given that a certain integral test is satisfied, the rate of the probability that Brownian motion remains below the moving boundary $f$ is asymptotically the same as for the constant boundary. The integral test for $f$ is also necessary in some sense. After Uchiyama's result, a number of different proofs appeared simplifying the original arguments, which strongly rely on some known identities for Brownian motion. In particular, Novikov (1996) gives an elementary proof in the case of an increasing boundary. Here, we provide an elementary, half-page proof for the case of a decreasing boundary. Further, we identify that the integral test is related to a repulsion effect of the three-dimensional Bessel process. Our proof gives some hope to be generalized to other processes such as FBM.

math.PR↗

arXiv2011

Small deviations for a family of smooth Gaussian processes

We study the small deviation probabilities of a family of very smooth self-similar Gaussian processes. The canonical process from the family has the same scaling property as standard Brownian motion and plays an important role in the study of zeros of random polynomials. Our estimates are based on the entropy method, discovered in Kuelbs and Li (1992) and developed further in Li and Linde (1999), Gao (2004), and Aurzada et al. (2009). While there are several ways to obtain the result w.r.t. the $L_2$ norm, the main contribution of this paper concerns the result w.r.t. the supremum norm. In this connection, we develop a tool that allows to translate upper estimates for the entropy of an operator mapping into $L_2[0,1]$ by those of the operator mapping into $C[0,1]$, if the image of the operator is in fact a Hölder space. The results are further applied to the entropy of function classes, generalizing results of Gao et al. (2010).

math.PR↗

arXiv2011

Survival probabilities of weighted random walks

We study the asymptotic behaviour of the probability that a weighted sum of centered i.i.d. random variables X_k does not exceed a constant barrier. For regular random walks, the results follow easily from classical fluctuation theory, while this theory does not carry over to weighted random walks, where essentially nothing seems to be known. First we discuss the case of a polynomial weight function and determine the rate of decay of the above probability for Gaussian X_k. This rate is shown to be universal over a larger class of distributions that obey suitable moment conditions. Finally we discuss the case of an exponential weight function. The mentioned universality does not hold in this setup anymore so that the rate of decay has to be determined separately for different distributions of the X_k. We present some results in the Gaussian framework.

math.PR↗

arXiv2011

Moments of recurrence times for Markov chains

We consider moments of the return times (or first hitting times) in a discrete time discrete space Markov chain. It is classical that the finiteness of the first moment of a return time of one state implies the finiteness of the first moment of the first return time of any other state. We extend this statement to moments with respect to a function $f$, where $f$ satisfies a certain, best possible condition. This generalizes results of K. L. Chung (1954) who considered the functions $f(n)=n^p$ and wondered "[...] what property of the power $n^p$ lies behind this theorem [...]" (see Chung (1967), p. 70). We exhibit that exactly the functions that do not increase exponentially -- neither globally nor locally -- fulfill the above statement.

math.PR↗