**Author details**

Gustavo Callou, Paulo Maciel, Julian Araújo, João Ferreira and Rafael Souza *Informatics Center, Federal University of Pernambuco - Recife, Brazil*

#### Dietmar Tutsch

22 Petri Nets

Equipment MTTF (hs) MTTR (hs) AC Source 4,380 8 Generator 2,190 8 STS 240,384 8 Subpanel 1,520,000 8 Transformer 1,412,908 8 UPS 250,000 8 Low Voltage Panel 1,520,000 8

the generator has increased the reliability of the architecture A2. Considering the availability results, similar behavior happened. The availability has increased from 5.47 to 7.96 (in number

This work considers the advantages of both Stochastic Petri Nets (SPN) and Reliability Block Diagrams (RBD) formalisms to analyze data center infrastructures. Such approach is supported by an integrated environment, ASTRO, which allows data center designers to estimate the dependability metrics before implementing the architectures. The methodology proposes that the system should be evaluated piecewisely to allow the composition of simpler models representing a data center infrastructure appropriately. Moreover, experiments demonstrate the feasibility of the environment, in which different architectures for a data

The authors would like to thank CNPQ for financing the project (290018/2011-0) and

Gustavo Callou, Paulo Maciel, Julian Araújo, João Ferreira and Rafael Souza

*Informatics Center, Federal University of Pernambuco - Recife, Brazil*

**Table 3.** MTTF and MTTR values for power devices.

**Figure 23.** Reliability Comparison of Architectures A1 and A2.

center power infrastructures have been adopted.

supporting the development of this work.

of 9's).

**6. Conclusion**

**Acknowledgments**

**Author details**

*Automation/Computer Science, University of Wuppertal, Wuppertal, Germany*

## **7. References**


URL: *http://doi.acm.org/10.1145/1773394.1773405*


[15] Patterson, D. [2002]. A simple way to estimate the cost of downtime, *Proceedings of the 16th USENIX conference on System administration*, LISA '02, USENIX Association, Berkeley, CA, USA, pp. 185–188. URL: *http://dl.acm.org/citation.cfm?id=1050517.1050538*

**Grammars Controlled by Petri Nets**

Additional information is available at the end of the chapter

*enumerable*, *context-sensitive*, *context-free* and *regular*.

cited.

http://dx.doi.org/10.5772/50637

**1. Introduction**

J. Dassow, G. Mavlankulov, M. Othman, S. Turaev, M.H. Selamat and R. Stiebe

**Chapter 15**

Formal language theory, introduced by Noam Chomsky in the 1950s as a tool for a description of natural languages [8–10], has also been widely involved in modeling and investigating phenomena appearing in computer science, artificial intelligence and other related fields because the symbolic representation of a modeled system in the form of strings makes its processes by information processing tools very easy: coding theory, cryptography, computation theory, computational linguistics, natural computing, and many other fields directly use sets of strings for the description and analysis of modeled systems. In formal language theory a model for a phenomenon is usually constructed by representing it as a set of words, i.e., a *language* over a certain alphabet, and defining a generative mechanism, i.e., a *grammar* which identifies exactly the words of this set. With respect to the forms of their rules, grammars and their languages are divided into four classes of *Chomsky hierarchy*: *recursively*

Context-free grammars are the most investigated type of Chomsky hierarchy which, in addition, have good mathematical properties and are extensively used in many applications of formal languages. However, they cannot cover all aspects which occur in modeling of phenomena. On the other hand, context-sensitive grammars, the next level in Chomsky hierarchy, are too powerful to be used in applications of formal languages, and have bad features, for instance, for context-sensitive grammars, the emptiness problem is undecidable and the existing algorithms for the membership problem, thus for the parsing, have exponential complexities. Moreover, such concepts as a derivation tree, which is an important tool for the analysis of context-free languages, cannot be transformed to context-sensitive grammars. Therefore, it is of interest to consider "intermediate" grammars which are more powerful than context-free grammars and have similar properties. One type of such grammars, called *grammars with regulated rewriting* (*controlled* or *regulated grammars* for short), is defined by considering grammars with some additional mechanisms which extract some subset of the generated language in order to cover some aspects of modeled phenomena. Due to the variety of investigated practical and theoretical problems, different additional mechanisms to grammars can be considered. Since Abraham [1] first defined matrix grammars in 1965, several grammars with restrictions such as programmed, random

> ©2012 Turaev et al., 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

©2012 Turaev et al., 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.

