Serguei Foss

An introduction to heavy-tailed and subexponential distributions

Preface.- Introduction.- Heavy- and long-tailed distributions.- Subexponential distributions.- Densities and local probabilities.- Maximum of random walks.- References.- Index

Asymptotics for sums of random variables with local subexponential behaviour

We study distributions F on [0,∞) such that for some T ≤ ∞, F*2(x, x+T] ∼ 2F(x, x+T]. The case T = ∞ corresponds to F being subexponential, and our analysis shows that the properties for T < ∞ are, in fact, very similar to this classical case. A parallel theory is developed in the presence of densities. Applications are given to random walks, the key renewal theorem, compound Poisson process and Bellman–Harris branching processes.

ON THE SATURATION RULE FOR THE STABILITY OF QUEUES

This paper focuses on the stability of open queueing systems under stationary ergodic assumptions. It defines a set of conditions, the monotone separable framework, ensuring that the stability region is given by the following saturation rule : saturate the queues which are fed by the external arrival stream, look ate the intensity [??] of the departure stream in this saturated system, then stability holds whenever the intensity of the arrival process, say l satisfies the condition [??], whereas the network is unstable if [??]. Whenever the stability condition is satisfied, it is also shown that certain state variables associated with the network admit a finite stationary regime which is constructed pathwise using a Loynes type bacward argument. This framework involves two main pathwise properties, external monotonicity and separability, which are satisfied by several classical queueing networks. The main tool for the proof of this rule is sub-additive ergodic theory.

Tail Asymptotics for the Supremum of a Random Walk when the Mean Is not Finite

We consider the sums Sn=ξ1+⋯+ξn of independent identically distributed random variables. We do not assume that the ξs have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability P{M>x} as x→∞, provided that M=sup {Sn,n≥1} is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity. We provide some sufficient conditions for the integrated weighted tail distribution to be subexponential. We supplement these conditions by a number of examples which cover both the infinite- and the finite-mean cases. In particular, we show that the subexponentiality of distribution F does not imply the subexponentiality of its integrated tail distribution FI.

Moments and tails in monotone-separable stochastic networks

A network belongs to the monotone separable class if its state variables are homogeneous and monotone functions of the epochs of the arrival process. This framework, which was first introduced to derive the stability region for stochastic networks with stationary and ergodic driving sequences, is revisited. It contains several classical queueing network models, including generalized Jackson networks, max-plus networks, polling systems, multiserver queues, and various classes of stochastic Petri nets. Our purpose is the analysis of the tails of the stationary state variables in the particular case of i.i.d. driving sequences. For this, we establish general comparison relationships between networks of this class and the GI/GI/1/\\infty queue. We first use this to show that two classical results of the asymptotic theory for GI/GI/1/\\infty queues can be directly extended to this framework. The first one concerns the existence of moments for the stationary state variables. We establish that for all \\alpha\\geq 1, the (\\alpha+1)-moment condition for service times is necessary and sufficient for the existence of the \\alpha-moment for the stationary maximal dater (typically the time to empty the network when stopping further arrivals) in any network of this class. The second one is a direct extension of Veraverbeke\s tail asymptotic for the stationary waiting times in the GI/GI/1/\\infty queue.

Ergodicity of Jackson-type queueing networks

This paper gives a pathwise construction of Jackson-type queueing networks allowing the derivation of stability and convergence theorems under general probabilistic assumptions on the driving sequences; namely, it is only assumed that the input process, the service sequences and the routing mechanism are jointly stationary and ergodic in a sense that is made precise in the paper. The main tools for these results are the subadditive ergodic theorem, which is used to derive a strong law of large numbers, and basic theorems on monotone stochastic recursive sequences. The techniques which are proposed here apply to other and more general classes of discrete event systems, like Petri nets or GSMPs. The paper also provides new results on the Jackson-type networks with i.i.d. driving sequences which were studied in the past.

Lower limits and equivalences for convolution tails

Suppose F is a distribution on the half-line [0, oo). We study the limits of the ratios of tails F * F(x)/F(x) as x→ oo. We also discuss the classes of distributions &,& (y) and &*.

On Sums of Conditionally Independent Subexponential Random Variables

The asymptotic tail behaviour of sums of independent subexponential random variables is well understood, one of the main characteristics being the principle of the single big jump. We study the case of dependent subexponential random variables, for both deterministic and random sums, using a fresh approach, by considering conditional independence structures on the random variables. We seek sufficient conditions for the results of the theory with independent random variables to still hold. 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 demonstrate a variety of effects owing to the dependence, and are also interesting in their own right.

Free-choice Petri nets-an algebraic approach

In this paper, we give evolution equations for free-choice Petri nets which generalize the [max, +]-algebraic setting already known for event graphs. These evolution equations can be seen as a coupling of two linear systems, a (min, +)-linear system and a quasi-(+, x)-linear one. This leads to new methods and algorithms to: 1) in the untimed case, check liveness and several other basic logical properties; 2) in the timed case, establish various conservation and monotonicity properties; and 3) in the stochastic case, check stability, i.e., the fact that the marking remains bounded in probability, and constructs minimal stationary regimes. The main tools for proving these properties are graph theory, idempotent algebras, and ergodic theory.

On the stability of a partially accessible multi-station queue with state-dependent routing

We consider a multi‐station queue with a multi‐class input process when any station is available for the service of only some (not all) customer classes. Upon arrival, any customer may choose one of its accessible stations according to some state‐dependent policy. We obtain simple stability criteria for this model in two particular cases when service rates are either station‐ or class‐independent. Then, we study a two‐station queue under general assumptions on service rates. Our proofs are based on the fluid approximation approach.