Martingale proofs of many-server heavy-traffic limits for Markovian queues

This is an expository review paper illustrating the ``martingale method'' for proving many-server heavy-traffic stochastic-process limits for queueing models, supporting diffusion-process approximations. Careful treatment is given to an elementary model -- the classical infinite-server model $M/M/\infty$, but models with finitely many servers and customer abandonment are also treated. The Markovian stochastic process representing the number of customers in the system is constructed in terms of rate-1 Poisson processes in two ways: (i) through random time changes and (ii) through random thinnings. Associated martingale representations are obtained for these constructions by applying, respectively: (i) optional stopping theorems where the random time changes are the stopping times and (ii) the integration theorem associated with random thinning of a counting process. Convergence to the diffusion process limit for the appropriate sequence of scaled queueing processes is obtained by applying the continuous mapping theorem. A key FCLT and a key FWLLN in this framework are established both with and without applying martingales.Comment: Published in at http://dx.doi.org/10.1214/06-PS091 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Finkel was right : counter-examples to several conjectures on variants of vector addition systems (invited talk)

Studying one-dimensional grammar vector addition systems has long been advocated by Alain Finkel. In this presentation, we shall see how research on those systems has led to the recent breakthrough tower lower bound for the reachability problem on vector addition systems, obtained by Czerwinski et al. In fact, we shall look at how appropriate modifications of an underlying technical construction can lead to counter-examples to several conjectures on one-dimensional grammar vector addition systems, fixed-dimensional vector addition systems, and fixed-dimensional flat vector addition systems

Decidability Issues for Petri Nets

This is a survey of some decidability results for Petri nets, covering the last three decades. The presentation is structured around decidability of specific properties, various behavioural equivalences and finally the model checking problem for temporal logics

Modelchecking counting properties of 1-safe nets with buffers in paraPSPACE

We consider concurrent systems that can be modelled as $1$-safe Petri nets communicating through a fixed set of buffers (modelled as unbounded places). We identify a parameter $ben$, which we call ``benefit depth\u27\u27, formed from the communication graph between the buffers. We show that for our system model, the coverability and boundedness problems can be solved in polynomial space assuming $ben$ to be a fixed parameter, that is, the space requirement is $f(ben)p(n)$, where $f$ is an exponential function and $p$ is a polynomial in the size of the input. We then obtain similar complexity bounds for modelchecking a logic based on such counting properties. This means that systems that have sparse communication patterns can be analyzed more efficiently than using previously known algorithms for general Petri nets

Small Vertex Cover makes Petri Net Coverability and Boundedness Easier

The coverability and boundedness problems for Petri nets are known to be Expspace-complete. Given a Petri net, we associate a graph with it. With the vertex cover number k of this graph and the maximum arc weight W as parameters, we show that coverability and boundedness are in ParaPspace. This means that these problems can be solved in space O(ef(k,W)poly(n)), where ef(k,W) is some exponential function and poly(n) is some polynomial in the size of the input. We then extend the ParaPspace result to model checking a logic that can express some generalizations of coverability and boundedness.Comment: Full version of the paper appearing in IPEC 201

Lectures on Nehari's Theorem on the Polydisk

We give a leisurely proof of a result of Ferguson--Lacey (math.CA/0104144) and Lacey--Terwelleger (math.CA/0601192) on a Nehari theorem for "little" Hankel operators on a polydisk. If H_b is a little Hankel operator with symbol b on product Hardy space we have || H_b || \simeq || b ||_{BMO} where BMO is the product BMO space identified by Chang and Fefferman. This article begins with the classical Nehari theorem, and presents the necessary background for the proof of the extension above. The proof of the extension is an induction on parameters, with a bootstrapping argument. Some of the more technical details of the earlier proofs are now seen as consequences of a paraproduct theory.Comment: 35 pages. 65 Reference