Analysis of Petri Net Models through Stochastic Differential Equations

It is well known, mainly because of the work of Kurtz, that density dependent Markov chains can be approximated by sets of ordinary differential equations (ODEs) when their indexing parameter grows very large. This approximation cannot capture the stochastic nature of the process and, consequently, it can provide an erroneous view of the behavior of the Markov chain if the indexing parameter is not sufficiently high. Important phenomena that cannot be revealed include non-negligible variance and bi-modal population distributions. A less-known approximation proposed by Kurtz applies stochastic differential equations (SDEs) and provides information about the stochastic nature of the process. In this paper we apply and extend this diffusion approximation to study stochastic Petri nets. We identify a class of nets whose underlying stochastic process is a density dependent Markov chain whose indexing parameter is a multiplicative constant which identifies the population level expressed by the initial marking and we provide means to automatically construct the associated set of SDEs. Since the diffusion approximation of Kurtz considers the process only up to the time when it first exits an open interval, we extend the approximation by a machinery that mimics the behavior of the Markov chain at the boundary and allows thus to apply the approach to a wider set of problems. The resulting process is of the jump-diffusion type. We illustrate by examples that the jump-diffusion approximation which extends to bounded domains can be much more informative than that based on ODEs as it can provide accurate quantity distributions even when they are multi-modal and even for relatively small population levels. Moreover, we show that the method is faster than simulating the original Markov chain

Continuization of Timed Petri Nets: From Performance Evaluation to Observation and Control

Abstract. State explosion is a fundamental problem in the analysis and synthesis of discrete event systems. Continuous Petri nets can be seen as a relaxation of discrete models allowing more efficient (in some cases polynomial time) analysis and synthesis algorithms. Nevertheless computational costs can be reduced at the expense of the analyzability of some properties. Even more, some net systems do not allow any kind of continuization. The present work first considers these aspects and some of the alternative formalisms usable for continuous relaxations of discrete systems. Particular emphasis is done later on the presentation of some results concerning performance evaluation, parametric design and marking (i.e., state) observation and control. Even if a significant amount of results are available today for continuous net systems, many essential issues are still not solved. A list of some of these are given in the introduction as an invitation to work on them.

Postoperative complications after procedure for prolapsed hemorrhoids (PPH) and stapled transanal rectal resection (STARR) procedures

Procedure for prolapsing hemorrhoids (PPH) and stapled transanal rectal resection for obstructed defecation (STARR) carry low postoperative pain, but may be followed by unusual and severe postoperative complications. This review deals with the pathogenesis, prevention and treatment of adverse events that may occasionally be life threatening. PPH and STARR carry the expected morbidity following anorectal surgery, such as bleeding, strictures and fecal incontinence. Complications that are particular to these stapled procedures are rectovaginal fistula, chronic proctalgia, total rectal obliteration, rectal wall hematoma and perforation with pelvic sepsis often requiring a diverting stoma. A higher complication rate and worse results are expected after PPH for fourth-degree piles. Enterocele and anismus are contraindications to PPH and STARR and both operations should be used with caution in patients with weak sphincters. In conclusion, complications after PPH and STARR are not infrequent and may be difficult to manage. However, if performed in selected cases by skilled specialists aware of the risks and associated diseases, some complications may be prevented

Analysis of shared heritability in common disorders of the brain

Paroxysmal Cerebral Disorder

An efficient Kronecker representation for PEPA models

Abstract. In this paper we present a representation of the Markov process underlying a PEPA model in terms of a Kronecker product of terms. Whilst this representation is similar to previous representations of Stochastic Automata Networks and Stochastic Petri Nets, it has novel features, arising from the definition of the PEPA models. In particular, capturing the correct timing behaviour of cooperating PEPA activities relies on functional dependencies.