Abstracts - math.uni.wroc.pllorek/bedlewo2009/bedl09_abstracts.pdf · ASYMPTOTICS OF SUPREMUM DISTRIBUTION OF A GAUSSIAN PROCESS OVER A RANDOM TIME INTERALV MAREK ARENDARCZYK Institute

Stochastic networks and related topics II

Abstracts

Banach Center, B¦dlewo, Poland,17 May � 22 May, 2009http://www.math.uni.wroc.pl/∼lorek/bedlewo2009/index.php

1

Organizing committee

Krzysztof D¦bicki (head) [email protected] Palmowski [email protected] Rolski [email protected] Szekli [email protected]

The organization of this conference is supported by

• Institute of Mathematics, University of Wrocªaw

• Stefan Banach International Mathematical Center

• HANAP Transfer Of Knowlegde EU program

2

Contents

Abstracts 3

Anantharam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Arendarczyk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Asmussen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Baccelli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Baurdoux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Bªaszczyszyn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Bordenave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Daduna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Daley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Damek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Foss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Ivanovs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Klusik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Konstantopoulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Kosi«ski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Kruk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Last . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Lelarge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Löpker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Mandjes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Michna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Mikosch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Miyazawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Norros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Pistorius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Robert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Samorodnitsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Schreiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Shneer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Sierpi«ska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Szczotka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Tabi± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Vlasiou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Weiss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Wichelhaus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Zwart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

List of participants 39

3

ANONYMITY VIA NETWORKS OF MIXES

VENKAT ANANTHARAM EECS Department, University of California, Berkeley,[email protected]

Mixes are relay nodes that accept packets arriving from multiple sources and release them aftervariable delays to prevent an eavesdropper from associating outgoing packets to their sources. Weassume that each mix has a hard latency constraint. Using an entropy-based measure to quantifyanonymity, we analyze the anonymity provided by networks of such latency-constrained mixes, fo-cusing on the single destination case. Our results are of most interest under light tra�c conditions.A general upper bound is presented that bounds the anonymity of a single-destination mix net-work in terms of a linear combination of the anonymity of two-stage networks. By using a speci�cmixing strategy, a lower bound is provided on the light tra�c derivative of the anonymity of single-destination mix networks. The light tra�c derivative of the upper bound coincides with the lowerbound for the case of mix-cascades (linear single-destination mix networks). Thus, the optimal lighttra�c derivative of the anonymity is characterized for mix cascades.

(Joint work with Parv Venkitasubramaniam.)

4

ASYMPTOTICS OF SUPREMUM DISTRIBUTION OFA GAUSSIAN PROCESS OVER A RANDOM TIMEINTERVAL

MAREK ARENDARCZYK Institute of Mathematics, University of Wrocªaw,[email protected]

Let {X(t) : t ∈ [0,∞)} be a centered Gaussian process with stationary increments and variancefunction σ2

X(t). The classical result of Berman shows that under some conditions on the variancefunction σ2

X(·), providing it is convex, the following asymptotics holds for deterministic T :

P( supt∈[0,T ]

X(t) > u) = P(X(T ) > u)(1 + o(1)),

as u→∞. In the �rst part of the presentation we extend this result to the case of random T beingasymptotically Weibullian, that is

P(T > t) = Ctγ exp(−βtα)(1 + o(1)),

as u→∞, where α, β, C > 0, γ ∈ R.We apply obtained asymptotics to analyze of extremal behavior of fractional Laplace motion process.

In the second part of the presentation we analyze asymptotic behavior of P(supt∈[0,T ]X(t) > u)in case of X(t) being a stationary Gaussian process. We distinguish three scenarioses (related tothe heaviness of T ) leading to three qualitatively di�rent asymptotics.

Joint work with Krzysztof D¦bicki.

5

IMPORTANCE SAMPLING FOR FAILUREPROBABILITIES IN COMPUTING AND DATATRANSMISSION

SØREN ASMUSSEN Aarhus University, Denmark, [email protected]

A job like the execution of a computer program or a �le transfer may fail and then needs to berestarted. Let T be the ideal time needed and X the actual time. We study e�cient simulation ofP (X > x) for large x. The standard scheme for importance sampling suggests to do importancesampling on T with an importance distribution so close as possible to the conditional distributiongiven X > x. We identify the limits and present a number of limit theorems for the e�ciency

6

TRANSMISSION TIMES IN MOBILE AD HOCNETWORKS WITH RANDOM FADING

FRANCOIS BACCELLI INRIA Unité de Recherche de Rocquencourt, ENS Départementd'Informatique , [email protected]

We consider a mobile ad hoc network where

* nodes are randomly located according to some Poisson point process in the Euclidean plane;* nodes transmit packets using spatial Aloha;* each transmitter-receiver channel has some random fading.

We analyze the random delay for a packet to be successfully transported from a typical trans-mitter to its receiver when success is de�ned by signal to noise and interference ratio.

Joint work with B. Bªaszczyszyn.

7

STOCHASTIC GAMES FOR SPECTRALLY ONE-SIDEDLÉVY PROCESSES

ERIK BAURDOUX LSE, [email protected]

We study various methods for solving stochastic games for spectrally one-sided Lévy processes. Inparticular, we present the solution to the stochastic games [1], [2] and [3] by using techniques in-cluding �uctuation theory, stochastic analytic methods associated with martingale characterisationsand the reduction of the stochastic game to an optimal stopping problem.

References

[1] E.J. Baurdoux and A.E. Kyprianou. The McKean stochastic game driven by a spectrally negativeLévy process. Electronic Journal of Probability (2008) no. 8, pages 173-197.[2] E.J. Baurdoux and A.E. Kyprianou. The Shepp�Shiryaev stochastic game driven by a spectrallynegative Lévy process. To appear in Theory of Probability and Its Applications.[3] E.J. Baurdoux, A.E. Kyprianou and J.C. Pardo. The Gapeev�Kühn stochastic game driven byspectrally positive Lévy processes. Working paper.

8

EXTENDED RANDOMSIGNAL-TO-INTERFERENCE-AND-NOISE-RATIOGRAPHS WITH FADING

BART�OMIEJ B�ASZCZYSZYN INRIA Unité de Recherche de Rocquencourt, ENSDépartement d'Informatique and University of Wrocªaw , [email protected]

We study the asymptotic properties of a random geometric graph on a homogeneous Poissonpoint process on the plane, in which a directed link exists between two nodes if the so called signal tointerference and noise ratio is above a certain threshold. We �rst prove an almost sure upper boundon the maximum received interference. This allows us to choose an asymptotic spread parameterso as to bound the maximum received interference. Subsequently, under the assumption that theinterference e�ects are uniformly bounded, we study the critical power required to ensure that thegraph does not possess isolated nodes with high probability in the presence of fading e�ects alone.This work is related to previous studies of the percolation problem in a similar model.

work in progress with Srikanth K. Iyer (IIS Bangalore, India).

9

LOAD OPTIMIZATION IN A PLANAR NETWORK

CHARLES BORDENAVE CNRS & University of Toulouse ,[email protected]

We will analyze the asymptotic properties of an Euclidean optimization problem on the plane.Speci�cally, we consider a network with 3 bins and n objects spatially uniformly distributed, eachobject being allocated to a bin at a cost depending on its position. Two allocations are considered:the allocation minimizing the bin loads and the allocation allocating each object to its less costlybin. This model is motivated by current issues in wireless cellular networks. We will aim at theasymptotic properties of these allocations as the number of objects grows to in�nity. Using thesymmetries of the problem, we will derive a law of large numbers, a central limit theorem and alarge deviation principle for both loads with explicit expressions. In particular, we prove that thetwo allocations satisfy the same law of large numbers, but they do not have the same asymptotic�uctuations and rate functions. This is a joint work with G.-L. TORRISI.

10

ON THE WEBER PROBLEM IN STOCHASTICNETWORKS

HANS DADUNA University of Hamburg, Department of Mathematics,[email protected]

For a set of service stations in the plane with prescribed coordinates we determine the location of acentral station to build a star-like network. In this network a �xed number of customers cycle fromthe center to the exterior nodes and back. The location of the central station is chosen such thatthe throughput at the exterior nodes is maximized.We show that the problem can be reduced to the classical Weber problem with generalized distances.Our methods utilize an interplay of product form network theory and location theory.

11

NETWORK QUESTIONS IN LILYPOND PROTOCOLMODELS

DARYL DALEY ANU and Melbourne University, [email protected]

The germ�grain model for lilypond leaves (`grains') has spawned a variety of models in othergeometric settings. The simplest entails Poisson distributed points (`germs') on the line, where itleads to tractable algebra providing explicit answers to a range of questions. In other settings, mostquestions have been answered only via simulation or inequalities. The talk will use `classic' germ�grain models in 1, 2 and 3-D, and grains that are �nite line-segments in the plane, in illustratinganswers to `network' questions such as those that follow.

First, we observe that explicit constructions based on large �nite numbers of germs describethe growth of grains about the germs until the growth is stopped by contact of the growing grainwith another which in turn may have already stopped, or be stopped simultaneously, or may evencontinue to grow. The resulting model of grown grains can also be described algebraically (work ofLast and Heverling), and this description enables us to address the basic question of whether ornot there exists an in�nite `chain' (or chains) of touching grains. By analogy with graph-theoreticquestions such an in�nite chain means that the model percolates. If there is no percolation whatis the distribution of cluster sizes? What is the grain-size distribution? What is the distribution ofthe number of grains touching a given grain? What is the role of the Poisson distribution? What isthe role of di�erent granular shapes? Are there properties that persist across di�erent dimensions?

12

MULTIDIMENSIONAL AFFINE RECURSIONS ANDLIMIT THEOREMS

EWA DAMEK Wroclaw University , [email protected]

The talk is based on the joint research of Dariusz Buraczewski (Wroclaw), Yves Guivarc'h(Rennes) and myself (Wroclaw). We study the matrix recursion:

Xn+1 = An+1Xn +Bn+1, (1)

where Bn+1 ∈ Rd, An+1 ∈ R+ × O(d) i.e. An+1 is a product of a positive scalar and an orthogonalmatrix. (Bn+1, An+1) are i.i.d with the law µ supported by Rd × (R+ ×O(d)).

We prove that under appropriate assumptions the stationary measure has not only a heavy taili.e.

ν{|x| > t}tα → C,

when t→∞, but also that is asymptotically behaves at ∞ as the homogeneous measure Λ:∫Rd

f(x)dΛ(x) =

∫Sd−1×R+

f(rω)dσ(ω)dr

r1+α.

Namely, we prove that

lima→∞

aα∫

Rn

f(a−1kx) dν(x) =

∫Rn

f(x) dΛ(x)

for every k ∈ O(d) ∩ supp G. The method is �exible enough to give hope to generalize to othersituations.

The shape of Λ says that ν belongs to the domain of attraction of the stable i.e. the sumX1 + ...+Xn appropriately normalized should converge to stable law of index α. Indeed, this is thecase:

for α < 1X1 + · · ·+Xn

n1/α,

for 1 < α < 2X1 + · · ·+Xn − nm

n1/α

converges to the stable law of index αand for α > 2

X1 + · · ·+Xn − nm√n

converges to the gaussian law, m =∫

Rn x dν(x). I am looking forward to discuss the above resultswith the participants of the conference and to �nd possibly other situations to pursue a joint study.

References

[1] D. Buraczewski, E. Damek, Y. Guivarc'h. Convergence to stable laws for a class multidi-mensional stochastic recursions, to appear in Probab. Theory Related Fields., arxiv.org/abs/0809.4349[2] D.Buraczewski, E.Damek, Y.Guivarc'h, A.Hulanicki, R.Urban. Tail-homogeneity ofstationary measures for some multidimensional stochastic recursions, to appear in Probab. TheoryRelated Fields.

13

ON SUMS OF CONDITIONALLY INDEPENDENTSUBEXPONENTIAL RANDOM VARIABLES

SERGUEI FOSS Heriot-Watt, Edinburgh, [email protected]

The asymptotic tail behaviour of sums of independent subexponential random variables is wellunderstood, one of the main characteristics being the principle of the single big jump. We study thecase of dependent subexponential random variables, for both deterministic and random sums, usinga fresh approach, by considering conditional independence structures on the random variables. Weseek su�cient conditions for the results of the theory with independent random variables still tohold. For a subexponential distribution, we introduce the concept of a boundary class of functions,which we hope will be a useful tool in studying many aspects of subexponential random variables.The examples we give in the paper demonstrate a variety of e�ects owing to the dependence, andare also interesting in their own right. This is join work with Andrew Richards.

14

MARKOV ADDITIVE PROCESSES WITH ONE-SIDEDJUMPS AND QUEUES

JEVGENIJ IVANOVS Eindhoven University of Technology, [email protected]

In this talk I will consider a Markov additive process (MAP) with one-sided jumps. This processcan be seen as a Levy process modulated by an external Markov chain. It is a non-trivial general-ization of the standard Levy process, with many analogous properties and characteristics, as wellas new mathematical objects associated to it, posing new challenges.

In the absence of positive jumps the record process of a MAP is a MAP itself. We identify thegenerator of the record process, and in particular the generator of the associated Markov chain.This result allows us to solve a number of important problems. For example, we determine thestationary distribution of a queue fed by a MAP. This is done for MAPs with no positive jumpsand MAPs with no negative jumps. We make no additional assumptions, such as the eigenvalues ofthe generator of a MAP being distinct.

Note: joint work with Bernardo D`Auria, O�er Kella and Michel Mandjes.

15

Quantile hedging for an insider (Co-authors: Z. Palmowski,K. Zwierz)

PRZEMYSLAW KLUSIK University of Wroclaw [email protected]

We consider the problem of the quantile hedging from the point of view of a better informedagent acting on the market. The additional knowledge of the agent is modelled by a �ltrationinitially enlarged by some random variable. By using equivalent martingale measures introduced byJ. Amendinger (2000) we solve the problem for the complete case, by extending the results obtainedby Foellmer and Leukert (1999) to the insider context. Finally, we consider the examples with theexplicit calculations within the standard Black-Scholes model.

16

CHARACTERISTICS OF REFLECTED LOCAL TIMES OFLÉVY PROCESSES

TAKIS KONSTANTOPOULOS School of Mathematical Sciences Heriot-WattUniversity, Edinburgh UK, [email protected]

We construct the stationary version of a processes obtained by re�ecting the local time of aLévy process and study various characteristics, such as marginal distribution and length of time theprocess stays at zero by using a combination of Palm theory and �uctuation theory. This is basedon past and current joint work with Andreas Kyprianou, Paavo Salminen and Marina Sirviö [1].

References

[1] Konstantopoulos, T., Kyprianou, A., Sirviö, M., and Salminen, P. (2008). Analysis of stochastic�uid queues driven by local time processes. Advances in Applied Probability 40, 1072-1103.

17

POLLING MODELS WITH LÉVY INPUT

KAMIL KOSI�SKI EURANDOM, TU Eindhoven, [email protected]

We consider a multi-class single-server polling model with switchover times. Throughout thevast polling literature, it is almost always assumed that customers arrive at the queues accordingto independent Poisson processes, having independent service requirements. The resulting inputprocesses in the queues thus constitute independent compound Poisson processes (CPP). We relaxthis assumption in several ways. First, we let the input process W change at polling and switchinginstants; in classical polling models, the arrival rates and service requirement distributions aretypically �xed once and for all. Second, we consider Lévy-driven, possibly correlated input, i.e., weassume that the input process is an N -dimensional Lévy subordinator (increasing process), whereN ≥ 1 corresponds to the number of queues. The generalization from CPPs to Lévy input impliesthat we can no longer speak of customers. Instead, the focus will now be on another key performancemeasure of polling systems, viz., the (joint) workload process.

By considering the input as an N -dimensional Lévy process W instead of N one-dimensionalprocesses Wi, we accomplish an easy incorporation of correlation between input to di�erent queues.This stems from the fact that every Lévy process is uniquely characterized by its characteristicexponent, which in the multidimensional case also contains the correlation structure between thecoordinates.

Resing [2] has shown that for a large class of classical polling models, including those withexhaustive or gated visit discipline at all queues, the evolution of the queue-length process atsuccessive polling instants at a �xed queue can be described as a multi-type branching process(MTBP) with immigration. Models that satisfy this MTBP-structure allow for an exact analysis,whereas models that violate the MTBP-structure are often much more intricate. Analogously, weidentify a class of visit disciplines (again including exhaustive and gated) that allows to describe themultidimensional workload in the system at successive polling instants at a �xed queue as a multi-type continuous state space (discrete time) branching process. This branching process is referredto in the sequel as MTJBP due to Ji°ina [1], who introduced the notion of continuous state spacebranching processes and gave special attention to discrete time processes (called Ji°ina processes inthe literature).

We are thus able to obtain the LST of the joint steady-state workload at polling instants. Fol-lowing a martingale approach we �nally use that LST to obtain the LST of the joint workload atarbitrary instants.

Note: This is joint work with Onno Boxma, Jevgenijs Ivanovs, and Michel Mandjes

References

[1] Reference 1. Ji°ina, M. (1958). Stochastic branching processes with continuous state space.Czechosl. Math. J. 8, 292�313.[2] Reference 2. Resing, J.A.C. (1993). Polling systems and multitype branching processes. QueuingSyst. 13, 409�426.

18

HEAVY TRAFFIC LIMITS FOR G/G/1 SRPT QUEUES

�UKASZ KRUK Institute of Mathematics, Maria Curie-Skªodowska University, Lublin,Poland, [email protected]

This is a report on a joint work with H. Christian Gromoll and Amber Puha.We present a heavy tra�c analysis for a G/G/1 queue in which the server uses the shortest

remaining processing time (SRPT) policy. We provide heavy tra�c limit theorems for measure-valued state descriptor processes. Three outlines of proofs of these theorems will be discussed. The�rst one uses a recent result on �uid limits for SRPT queues [1], together with the modular Bramson-Williams approach. The main idea of the second one is to approximate the SRPT system underconsideration by a sequence of suitable earliest deadline �rst (EDF) queues, for which heavy tra�climits are known [2]. This approach carries over to the longest remaining processing time (LRPT)queues. Finally, the third one uses priority analysis and methods developed for priority queues.

References

[1] D. G. Down, H. C. Gromoll and A. L. Puha, Fluid limits for shortest remaining processing timequeues, preprint.[2] �. Kruk, Di�usion approximation for a G/G/1 EDF queue with unbounded lead times, AnnalesUMCS Mathematica A, 61 (2007), 51-90.

19

EXPLICIT REPRESENTATIONS OF POISSONMARTINGALES

GUENTER LAST University of Karlsruhe, [email protected]

We consider a Poisson process η on the product of [0,∞) and a Borel space with a di�use (andσ-�nite) intensity measure. We derive an explicit representation of square integrable martingales(de�ned with respect to the natural �ltration associated with η.) This Clark-Ocone type formulahas been known in case the intensity measure of η is the product of Lebesgue measure and a σ-�nitemeasure. Our proof is based on only a few basic properties of Poisson processes and stochasticintegrals. As applications we shall discuss the classical chaos expansion of Poisson functionals aswell as Poincaré's and related inequalities. The talk is based on joint work with Mathew Penrose(Bath).

References

[1] Last, G. and Penrose, M.: Martingale representation and chaos expansion for Poisson processes.in preparation.

20

RESOLVENT OF LARGE RANDOM GRAPHS

MARC LELARGE INRIA-ENS, [email protected]

We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit in�nitegraph. We apply these results to graphs converging locally to trees and derive a new formula for theStieljes transform of the spectral measure of such graphs. We illustrate our results on the uniformregular graphs, Erdös-Rényi graphs and graphs with a given degree sequence. We give examples ofapplication for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices.

joint work with Charles BORDENAVE.

21

ON A MARKOVIAN GROWTH COLLAPSE PROCESS

ANDREAS LÖPKER Technical University and EURANDOM, Eindhoven, TheNetherlands, [email protected]

We consider a Markovian growth collapse process Xt, that increases according to the di�erentialequation

d

dtXt = rXα

t , t > 0, r > 0

between random jumps at times T1, T2, . . ., which are governed by a random intensity such that theprobability of a jump during [t, t+ ∆t] given that Xt = x is given by

λXβt ∆t+ o(∆t), t > 0, λ > 0.

If XTk− = x at a jump time, then the process is reduced to a fraction Qk ·XTk, where Q1, Q2, . . . is

a sequence of i.i.d. random variables with values in (0, 1).There is a vast quantity of literature about related processes. We try to give a (naturally limited)

overview over existing results and connections to other topics, such as linear stochastic recursiveequations, exponential functionals of Levy processes, perpetuities, TCP.

We describe the behavior of Xt as t→∞ for di�erent values of α and β. In particular we showthat for β > α − 1 the process is stable, in the sense that Xt ⇒ X∞, whereas for β < α − 1 theprocess is unstable. We also discuss the critical case β = α− 1. We present some results about thethe asymptotics of the �rst hitting time τy of y as y →∞ and discuss a Markov modulated variantof the α = 0, β = 0 case.

Joint work withDavid Goldberg, O�er Kella, Johan van Leeuwaarden, Teunis Ott, Zbyszek Palmowski and WolfgangStadje.

22

ON THE CORRELATION FUNCTION OF REFLECTEDLÉVY PROCESSES

MICHEL MANDJES University of Amsterdam, [email protected]

In this talk I will consider a single-server queue with Lévy input, and in particular its workloadprocess (Qt)t, focusing on its correlation structure. With the correlation function de�ned as r(t) :=Cov(Q0, Qt)/Var(Q0) (assuming that the workload process is in stationarity at time 0), I �rst showhow powerful results for both spectrally positive and spectrally negative Lévy processes can be usedto �nd its Laplace transform.

This expression allows us to prove that r(·) is positive, decreasing, and convex � propertiesthat are likely to hold, but far from straightforward to prove directly from �rst principles. Relyingon the machinery of completely monotone functions, however, a rather short and elegant proof canbe given.

We also show that r(·) can be represented as the complementary distribution function of aspeci�c random variable. These results are used to estimate the asymptotics of r(t), for t large.

The last part of the talk is devoted to estimating r(t) by using simulation. A coupling procedureis given that leads to substantial variance reduction; using importance sampling on top of this, theresulting simulation scheme is even asymptotically optimal.

This talk includes joint work with Peter Glynn (Stanford) and Abdel Es-Saghouani (Amster-dam).

23

ON STOCHASTIC INTEGRALS WITH NON-GAUSSIANα-STABLE NOISE

ZBIGNIEW MICHNA Department of Mathematics and CyberneticsWroclaw University of EconomicsWroclaw, Poland, [email protected]

We use a series representation of the α-stable Lévy process to condition on the length of the largestjump. Then we get a certain decomposition into a simple process and a Lévy process which has�nite exponential moments. Using this decomposition we evaluate the expected value of X(t) =∫ t

0Z(s−) dZ(s) where Z is an α-stable Lévy process with 0 < α < 2. We show that EX(t) = 0 for

1 < α < 2 and this expectation is equal in�nity for α ≤ 1 irrespective of the value of the skewnessparameter.As a second application of this decomposition we �nd an exact asymptotic behavior of the taildistribution of supremum for an integral of stochastic process with respect to an α-stable Lévyprocess. We consider two examples with integrands being fractional Brownian motion and gammaprocess.

References

[1] Z. Michna (2009) On the mean of a stochastic integral with non-Gaussian α-stable noise. Stochas-tic Analysis and Applications 27, pp. 258-269.[2] Z. Michna (2008) Asymptotic behavior of anomalous di�usions driven by α- stable noise. ActaPhysica Polonica B 39, pp. 1825-1847.

24

PPREDICTION OF CLAIMS RESERVES IN A POISSONCLUSTER MODEL

THOMAS MIKOSCH University of Copenhagen, [email protected]

This is joint work with A> H. Jessen and G. Samorodnitsky.At the arrival times of a claim we start a Poisson cluster process which describes the payment

process for the claim. Then, given the past annual payment numbers for all claims arriving in agiven year, we calculate the expectation of payment numbers and amounts in future years. We alsocalculate the prediction error and give asymptotics for the case of large numbers of payments. Weapply the prediction formulas to real-life data and compare with the industrial standard, the chainladder method.

25

EXACT ASYMPTOTICS FOR A LÉVY-DRIVEN TANDEMQUEUE WITH AN INTERMEDIATE INPUT

MASAKIYO MIYAZAWA Tokyo University of Sceince, [email protected]

We consider a Lévy-driven tandem queue with an intermediate input assuming that its bu�ercontent process obtained by a re�ection mapping has the stationary distribution. For this queue, noclosed form formula is known, not only for its distribution but also for the corresponding transform.In this paper we consider only light-tailed inputs, and derive exact asymptotics for the tail distri-bution of convex type combination of two bu�er contents. This includes the marginal stationarydistributions of the bu�er contents and their sum as special cases. These results generalize thoseof Lieshout and Mandjes from the recent paper Lieshout and Mandjes [1] for the correspondingtandem queue without an intermediate input. This talk is based on the joint paper with TomaszRolski, which will appear in Queueing Systems.

References

[1] P. Lieshout and M. Mandjes (2008) Asymptotic analysis of Lévy-driven tandem queues, QueueingSystems 60, 203-226.

26

ROUTING AND TRAFFIC IN INFINITE-VARIANCEPOWER-LAW RANDOM GRAPHS

NORROS ILKKA VTT Technical Research Centre of Finland, ilkka.norros@vtt.�

We consider power-law random graphs with in�nite variance degree distribution as a (very) abstractmodel for `Internet-like' networks. Three speci�c topics are discussed:

1. an adaptation of a compact routing scheme by Carmi, Cohen and Dolev to our model

2. a `mean-�eld' modi�cation of the model, with some applications

3. a model for injecting tra�c to the network with a gravity rule tra�c matrix

References

[1] S. Carmi, R. Cohen, and D. Dolev. Searching complex networks e�ciently with minimal infor-mation. Europhysics Letters, 74(6), 2006.[2] S. Janson, T. �uczak, and I. Norros. Large cliques in a power-law random graph. 2009. Submittedfor publication. Preprint: arXiv 0905.0561.[3] I. Norros and H. Reittu. On a conditionally Poissonian graph process. Adv. Appl. Prob., 38:59�75,2006.[4] I. Norros. Powernet: Compact routing on Internet-like random networks. In NGI 2009, Aveiro,July 2009.[5] I. Norros. A mean-�eld approach to some Internet-like random networks. In ITC 21, Paris,September 2009.

27

First Passage for stochastic volatility models

MARTIJN PISTORIUS Imperial College London, [email protected]

Barrier options are �nancial contracts that are activated or de-activated when the underlyingprice process crosses a speci�c level; they are among the most widely traded of exotic contracts.The pay-o�s of barrier options are path dependent and their valuation requires the speci�cation ofthe �rst-hitting-time distribution. In this talk, we present a new approach to obtain �rst passageprobabilities for stochastic volatility models (i.e. di�usions whose coe�cients are functions of a one-dimensional di�usion). We derive a matrix Wiener-Hopf factorization, and use this to obtain theLaplace transforms of the running supremum and in�mum over time. We illustrate the results bycalculating the values and Greeks of barrier options, and compare the outcomes with Monte Carlosimulation results. The talk is based on joint work with Marc Jeannin.

References

[1] S. Heston. A closed form solution for options with stochastic volatility with applications to bondand currency options. Review of Financial Studies, 6:327�343, 1993.[2] Z. Jiang and M. Pistorius. On perpetual American put valuation and �rst passage in regimeswitching model with jumps. FINANC STOCH, 2008, Vol: 12, Pages: 331 - 355,[3] M. Rubinstein. Displaced di�usion option pricing. Journal of Finance, 38(1):213�217, 1983.[4] L.E. Blumenson K. S. Miller, R. I. Bernstein. Generalized rayleigh processes. Quaterly of AppliedMathematics, 16:137�145, 1958.

28

SCALING LIMITS FOR A CAT AND MOUSE MARKOVCHAIN

PHILIPPE ROBERT INRIA, FRANCE, [email protected]

Motivated by an original on-line page-ranking algorithm, the asymptotic behavior of a cat andmouse Markov chain is presented. Its equilibrium properties are �rst analyzed in the context of a�nite state space where a representation of its invariant distribution is obtained. When the statespace is in�nite, it turns out that this Markov chain is in some cases null recurrent. In this context,scaling results for the location of the mouse are presented in various situations: random walks onZ, re�ected random walks on N and, in a continuous time setting, a discrete Ornstein-Uhlenbeckprocess, the M/M/∞ queue. For some of these limiting results, a time scaling with rapid growth isused giving rise to unusual asymptotic behaviors.

29

LARGE DEVIATIONS FOR POINT PROCESSES BASEDON STATIONARY SEQUENCES WITH HEAVY TAILS

GENNADY SAMORODNITSKY Cornell University, [email protected]

In many applications in queuing theory and other areas, that involve functional large devia-tions for partial sums of stationary, but not iid, processes with heavy tails, a curious phenomenonarises: closely grouped together large jumps coalesce together in the limit, leading to loss of infor-mation of the order in which these jumps arrive. In particular, many functionals of interest becomediscontinuous. To overcome this problem we move from the functional large deviations to the point-process-level large deviations. We develop the appropriate topological framework and prove largedeviations theorems for point processes based on stationary sequences with heavy tails. We showthat these results are useful in many situations where functional large deviations are not.

30

SCALE-FREE AND SPECTRAL PROPERTIES OFSPIKE-FLOW GRAPHS FOR RECURRENT NEURALNETS

TOMASZ SCHREIBER Nicolaus Copernicus University, Torun, [email protected]

A class of functional graphs characterising the mesoscopic spatio-temporal correlation structureof large-scale recurrent neural networks, including real-world systems such as human brain studiedby fMRI techniques, have recently been reported in experimental literature to exhibit scale-freebehaviour with ubiquitous power law exponent 2. No theoretical explanation of these �ndings hasbeen known so far. In our recent joint work with Filip Piekniewski we have proposed a mathemati-cal spin-glass type model, formally somewhat reminiscent of the celebrated Sherrington-Kirkpatrickmodel yet exhibiting a completely di�erent behaviour, which sheds new light on theoretical aspectsof this phenomenon and yields precisely the desired exponent 2. We were also able to study fur-ther properties of the model, including the asymptotics of spectra of the corresponding adjacencymatrices. In its basic form our model is of a mean-�eld nature, yet we also report on our currentwork in progress where the network gets embedded into a geometric random-connection graph withpolynomial connectivity law. Even though a number of important di�erences arise upon embeddingour model into a geometric space, strikingly its scale-free behaviour remains una�ected.

31

A UNIFIED APPROACH TO HEAVY TRAFFIC FORRANDOM WALKS

SEVA SHNEER Eindhoven University of Technology and EURANDOM,[email protected]

For families of random walks Sk(a) with ESk(a) = -ka < 0 we consider their maxima M(a) =supk ≥ 0 Sk(a). We investigate the asymptotic behaviour of M(a) as a→ 0 for asymptotically stablerandom walks. This problem appeared �rst in the 1960's in the analysis of a single-server queuewhen the tra�c load tends to 1 and since then is referred to as the heavy-tra�c approximationproblem. Kingman and Prokhorov suggested two di�erent approaches which were later followed bymany authors. We give two elementary proofs of the main result, using each of these approaches.It turns out that the main technical di�culties in both proofs are rather similar and may beresolved via a generalisation of the Kolmogorov inequality to the case of an in�nite variance. Sucha generalisation will also be discussed during the talk. This is a joint work with Vitali Wachtel(University of Munich).

32

ASYMPTOTICS OF HYBRID FLUID QUEUEINGMODELS WITH LÉVY INPUT

IWONA SIERPI�SKA Mathematical Institute, University of Wrocªaw,[email protected]

We consider a �uid queue with in�nite bu�er, fed by superposition of two independent stochasticprocesses with stationary increments:

• {X(t) : t ∈ R}: an integrated On-O� process with peak rate r > 0 and regularly varyingOn-times;

• {Y (t) : t ∈ R}: an α-stable Lévy process with α ∈ (1, 2).

The queue is drained with a constant rate c > E[X(1) + Y (1)].

We study the exact asymptotics of the steady state bu�er content Q, i.e.

P (Q > u) = P

(supt≥0{X(t) + Y (t)− ct} > u

), u→∞.

It appears that the relation between r and c leads to three qualitatively di�erent scenarios, in�u-encing the form of the asymptotics.

The talk is based on joint work with Krzysztof D�ebicki (Wroclaw University, Poland) and BertZwart (CWI, Amsterdam).

33

TIGHTNESS IN HEAVY TRAFFIC INVARIANCEPRINCIPLE

MAREK CZYSTO�OWSKI and W�ADYS�AW SZCZOTKAMathematical Institute, University of Wrocªaw, [email protected]

Let for each n ≥ 1, {(vn,k, un,k), k ≥ 1} be a stationary and ergodic sequence of pairs of nonnegative

random variables vn,k, un,k such that adf= Evn,k − Eun,k < 0 and ωn = sup0≤k<∞

∑kj=1(vn,j − un,j).

Under some interpretation of vn,k, un,k the random variable ωn can be interpreted as the stationarywaiting time for a G/G/1 queue. It is well known, that ωn

p→ ∞ as an ↑ 0 and this situation iscalled the heavy tra�c situation.

Representation of ωn in the form ωn = sup0≤t<∞(Xn(t) − βn(t)), where Xn(t) =∑[nt]

j=1(vn,j −un,j − an) and βn(t) = |an|[nt], t ≥ 0, n ≥ 1, allows us to consider an asymptotic of ωn as an ↑ 0using the approach of the Heavy Tra�c Invariance Principle formulated as follows.

HTIP: If there exists a sequence of positive numbers cn ↑ ∞ such that the following conditionshold:(I) 1

cnXn

D→ X in the J1 Skorochod topology in D[0,∞), where X is stochastically continuous,

(II) n|an|/cn → β, 0 < β <∞, X(t)− βt p→ −∞,(III) {ωn/cn} is tight,then ωn/cn

D→ sup0≤t<∞(X(t)− βt).In the talk will be given an example of a sequence of M/GI/1 queues for which conditions (I)-(II)

hold but condition (III) does not hold. The talk is based on the papers [1]-[3] from References.

References

[1] W. Szczotka, W.A. Woyczy«ski, W.A. (2003). Distributions of Suprema of Lévy Processes viaHeavy Tra�c Invariance Principle, Probab. Math. Stat. 23, pp 251-172.[2] M.Czystoªowski, W.Szczotka, (2007) Tightness of stationary waiting times in Heavy Tra�c forGI/GI/1 queues with thick tails, Probab. Math. Stat. 27, pp 109-123.[3] M.Czystoªowski, W.Szczotka, Queueing approximation of suprema of spectrally positive Lévyprocess (in reviewing).

34

EXTREMES FOR INTEGRAL MEAN FOR STATIONARYGAUSSIAN PROCESSES WITH APPLICATION TOQUEUES AND COLLISIONS

KAMIL TABI� Mathematical Institute, University of Wroclaw, Poland,[email protected]

Let {Z(t), t ≥ 0} be a centered stationary Gaussian process with covariance function R(t) =Cov(Z(t), Z(0)). We analyze asymptotic behavior of:

P

(supt≥0

∫ t0Z(s)ds

t> u

)as u→∞.

As an application of the obtained axact asymptotic we consider:

� the asymptotic of bu�er emptiness in Gaussian �uid model under many source regime,

� collision probabilities for integrated Gaussian processes.

The talk is based on a joint work with K. D¦bicki.

35

A LÉVY INPUT MODEL WITH ADDITIONAL STATEDEPENDENT SERVICES

MARIA VLASIOU Eindhoven University of Technology, [email protected]

We consider a queuing model with the workload evolving between consecutive i.i.d. exponentialtimers {e(i)q }i=1,2,... according to a spectrally positive Lévy process Y (t) which is re�ected at 0.When the exponential clock e(i)q ends, the additional state-dependent service requirement modi�esthe workload so that the latter is equal to Fi(Y (e

(i)q )) at epoch e

(1)q + . . . + e

(i)q for some random

nonnegative i.i.d. functionals Fi. In particular, we focus on the case when Fi(y) = (Bi − y)+,where {Bi}i=1,2,... are i.i.d. nonnegative random variables. We analyse the steady-state workloaddistribution for this model. This work has been carried out jointly with Zbigniew Palmowski.

36

FCFS INFINITE BIPARTITE MATCHING OF SERVERSAND CUSTOMERS

GIDEON WEISS

We consider an in�nite sequence of customers of types C = 1, 2, . . ., I and an in�nite sequenceof servers of types S = 1, 2, . . ., I, where a server of type j can serve a subset of customer types C(j)and where a customer of type i can be served by a subset of server types S(i). We assume the typesof customers and servers in the in�nite sequences are random, independent identically distributed,and customers and servers are matched according to their order in the sequence, on a �rst come�rst served (FCFS) basis. We investigate this process of in�nite bipartite matching. In particular,we are interested in the rate r(i,j) that customers of type i are assigned to servers of type j. Wepresent a countable state Markov chain to describe this process, we prove ergodicity and existenceof limiting rates, and calculate r(i,j) for some previously unsolved instances.

Joint work with Rene Caldentey and Ed Kaplan

37

Nonparametric inference for inverse queueing problems

Cornelia Wichelhaus University of Heidelberg, [email protected]

The talk is based on a joint work with Michael Schmälzle.

For the application �elds of queueing systems statistical inference of the service time distributionsbased on incomplete observations of the systems is of great importance in order to classify theperformance behavior. For example, a unimodal service time density shows a homogeneous servicebehavior whereas a bimodal density may indicate that there are two distinct customer populations orbreakdowns of the server. Surprisingly, statistical inference (parameter identi�cation) for queueingsystems has not been developed properly yet. In the statistical literature there are up to now onlyresults for single node systems where no dependencies between the components of the system mustbe taken into consideration. With this talk we try to close this gap and present an approach for astatistical analysis study of general open networks of queues with Poisson arrivals. We assume thatat each node we observe the external input process and the external departure process of customers.Our aim is to estimate in a nonparametric way the service time distributions at the nodes as well asthe arrival rates and the routing probabilities according to which customers move in the network.In case of feedforward networks of queues we are able to construct explicitly estimators for theservice time distribution functions which converge P-a.s. uniformly to the true functions. In caseof stochastic networks of general topology the analysis leads to an inverse problem. By means of aminimum-distance approach de�ning equations for P-a.s. converging estimators can be derived.

38

COMPETITIVE SCHEDULING AND LARGEDEVIATIONS

BERT ZWART CWI, [email protected]

For a GI/GI/1 queue it is well known that, in a large deviations setting, FIFO is optimal forlight-tailed service times, and service disciplines such as PS (Processor Sharing) and SRPT (ShortestRemaining Processing Time) are optimal for heavy-tailed service times. It is also known that PSand SRPT do not perform well for light-tailed service times, while FIFO does not perform well forheavy-tailed service times.

The goal of this talk is to answer the natural question whether it is possible to construct a singlescheduling discipline that is competitive with FIFO for light-tailed service times and competitivewith SRPT for heavy tailed service times.

Joint work with Adam Wierman (Caltech)

39

List of participants

40

Venkat AnantharamUniversity of California, Berkeley

Marek ArendarczykUniversity of Wrocªaw

Søren AsmussenAarhus University

Francois BaccelliINRIA

Erik BaurdouxLSE

Barbara BobikauUniversity of Wrocªaw

Charles BordenaveCNRS & University of Toulouse

Onno BoxmaEURANDOM and TU Eindhoven

Bartªomiej BªaszczyszynINRIA and University of Wrocªaw

Hans DadunaUniversity of Hamburg

Daryl DaleyANU and Melbourne University

Ewa DamekUniversity of Wrocªaw

Krzysztof D¦bickiUniversity of Wrocªaw

Serguei FossHeriot-Watt University

Jevgenij IvanovsEURANDOM

Barbara JasiulisUniversity of Wrocªaw

Przemysªaw KlusikUniversity of Wrocªaw

Takis KonstantopoulosHeriot-Watt University, Edinburgh

Kamil Kosi«skiEURANDOM and University of Amsterdam

�ukasz KrukUMCS, Lublin

Guenter LastUniversity of Karlsruhe

Marc LelargeINRIA-ENS

Andreas LopkerEURANDOM and TU/e Eindhoven

Józef �ukaszewiczUniversity of Wrocªaw

Michel MandjesUniversity of Amsterdam

Zbigniew MichnaWrocªaw University of Economics

Thomas MikoschUniversity of Copenhagen

Masakiyo MiyazawaTokyo University of Science

41

Ilkka NorrosVTT Technical Research Centre of Finland

Zbigniew PalmowskiUniversity of Wrocªaw

Martijn PistoriusImperial College London

Anna PochyªaUniversity of Wrocªaw

Philippe RobertINRIA

Tomasz RolskiUniversity of Wrocªaw

Gennady SamorodnitskyCornell University

Tomek SchreiberNicolaus Copernicus University, Toru«

Seva ShneerEindhoven University of Technologyand EURANDOM

Iwona Sierpi«skaUniversity of Wrocªaw

Wªadysªaw SzczotkaUniversity of Wrocªaw

Ryszard SzekliUniversity of Wrocªaw

Kamil Tabi±University of Wrocªaw

Krzysztof TopolskiUniversity of Wrocªaw

Maria VlasiouEindhoven University of Technology

Gideon WeissHaifa University

Cornelia WichelhausUniversity of Heidelberg

Bert ZwartCWI

Piotr �ebrowskiUniversity of Wrocªaw

42