**8. References**

Agha, G. A.; Cinindio, F. & Rozenberg, G. (2001). *Concurrent Object-Oriented Programming and Petri Nets: Advances in Petri Nets*. Springer, ISBN 978-3-540-41942-6, Berlin, Germany

Diaz, M. (2009). *Petri Nets: Fundamental Models, Verification and Applications*, John Willey & Sons, ISTE Ltd., ISBN: 978-0-470-39430-4, London, United Kingdom

**Chapter 10** 

© 2012 Mitrevski and Kotevski, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Mitrevski and Kotevski, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

**Fluid Stochastic Petri Nets:** 

Additional information is available at the end of the chapter

Pece Mitrevski and Zoran Kotevski

appropriately chosen boundary conditions.

http://dx.doi.org/10.5772/50615

**1. Introduction** 

**From Fluid Atoms in ILP Processor Pipelines** 

**to Fluid Atoms in P2P Streaming Networks** 

Fluid models have been used and investigated in queuing theory [1]. Recently, the concept of fluid models was used in the context of Stochastic Petri Nets, referred to as *Fluid Stochastic Petri Nets* (FSPNs) [2-6]. In FSPNs, the fluid variables are represented by fluid places, which can hold fluid rather than discrete tokens. Transition firings are determined by both discrete and fluid places, and fluid flow is permitted through the enabled timed transitions in the Petri Net. By associating exponentially distributed or zero firing time with transitions, the differential equations for the underlying stochastic process can be derived. The dynamics of an FSPN are described by a system of first-order hyperbolic *partial differential equations* (PDEs) combined with initial and boundary equations. The general system of PDEs may be solved by a standard discretization approach. In [6], the problem of immediate transitions has also been addressed in relation to the fluid levels, by allowing fluid places to be connected to immediate transitions. The transportation of fluid in zero time is described by

In a typical multiple-issue processor, instructions flow through pipeline and pass through separate pipeline stages connected by buffers. An open multi-chain queuing network can present this organization, with each stage being a service center with a limited buffer size. Considering a machine that employs multiple execution units capable to execute large number of instructions in parallel, the service and storage requirements of each individual instruction are small compared to the total volume of the instruction stream. Individual instructions may then be regarded as *atoms of a fluid* flowing through the pipeline. The objective of this approach is to approximate large buffer levels by continuous fluid levels and decrease state-space complexity. Thus, in the first part of this chapter, we employ an

