Introduction

In this chapter we develop a very strong notion of equivalence between PEPA components called isomorphism. This is a condition on the derivation graphs of components and it ensures that components are only considered equivalent if there is a one-to-one correspondence between their derivatives and they are capable of carrying out exactly the same activities. It is not an observation based notion of equivalence in the style of bisimulation which is usual for process algebras. It is structural, in the style of the equivalence between Markov processes introduced in Section 5.3. Isomorphism is defined in Section 6.2.

In Sections 6.3 to 6.5 we examine some properties of this notion of equivalence, from the perspectives of a process algebra, the modelled system components and the underlying Markov processes. As we might expect from such a strong notion of equivalence, we can derive strong properties for isomorphism. The relation is a congruence for PEPA. The relationship between isomorphism and the Markov processes underlying the PEPA components is found to be a close one—isomorphic components generate equivalent Markov processes.

In the remainder of the chapter we develop a weaker form of this equivalence called weak isomorphism. This equivalence reflects the hidden nature of τ type activities. We will consider two components equivalent in this way if they only differ in their capabilities to carry out such activities. A definition of this notion of equivalence is presented in Section 6.6.

This book is, in essence, the dissertation I submitted to the University of Edinburgh in early January 1994. My examiners, Peter Harrison of the Imperial College, and Stuart Anderson of the University of Edinburgh, suggested some corrections and revisions. Apart from those changes, most chapters remain unaltered except for minor corrections and reformatting. The exceptions are the first and final chapter.

Since the final chapter discusses several possible directions for future work, it is now supplemented with a section which reviews the progress which has been made in each of these directions since January 1994. There are now many more people interested in stochastic process algebras and their application to performance modelling. Moreover, since these researchers have backgrounds and motivations different from my own some of the most interesting new developments are outside the areas identified in the original conclusions of the thesis. Therefore the book concludes with a brief overview of the current status of the field which includes many recent references. This change to the structure of the book is reflected in the summary given in Chapter 1. No other chapters of the thesis have been updated to reflect more recent developments. A modified version of Chapter 8 appeared in the proceedings of the 2nd International Workshop on Numerical Solution of Markov Chains, January 1995.

I would like to thank my supervisor, Rob Pooley, for introducing me to performance modelling and giving me the job which brought me to Edinburgh initially.

This text is devoted to the statics of rigid laminas on a plane and to the first-order instantaneous kinematics (velocities) of rigid laminas moving over a plane. Higher-order instantaneous kinematic problems, which involve the study of accelerations (second-order properties) and jerk (third-order properties) are not considered.

This text is influenced by the book Elementary Mathematics from an Advanced Standpoint: Geometry, written by the famous German geometer Felix Klein. It was published in German in 1908 and the third edition was translated into English and published in New York by the Macmillan Company in 1939. The book was part of a course of lectures given to German High School Teachers at Göttingen in 1908. Klein was admonishing the teachers for not teaching geometry correctly, and he was essentially providing a proper foundation for its instruction.

The present text stems from the undergraduate course “The Geometry of Robot Manipulators,” taught in the Mechanical Engineering Department at the University of Florida. This course is based on Klein's development of the geometry of points and lines in the plane and upon his elegant development of mechanics: “A directed line-segment represents a force applied to a rigid body. A free plane-segment, represented by a parallelogram of definite contour sense, and the couple given by two opposite sides of the parallelogram, with arrows directed opposite to that sense, are geometrically equivalent configurations, i.e., they have equal components with reference to every coordinate system.”