261 research outputs found
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 , 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 -safe
Petri nets communicating through a fixed set of buffers (modelled as
unbounded places). We identify a parameter , 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 to be a
fixed parameter, that is, the space requirement is ,
where is an exponential function and 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
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