Abstract
The Müller-Wichards model (MW) is an algebraic method that quantitatively estimates the performance of sequential and/or parallel computer applications. Because of category theory’s expressive power and mathematical precision, a category theoretic reformulation of MW, i.e., CMW, is presented in this paper. The CMW is effectively numerically equivalent to MW and can be used to estimate the performance of any system that can be represented as numerical sequences of arithmetic, data movement, and delay processes. The CMW fundamental symmetry group is introduced and CMW’s category theoretic formalism is used to facilitate the identification of associated model invariants. The formalism also yields a natural approach to dividing systems into subsystems in a manner that preserves performance. Closed form models are developed and studied statistically, and special case closed form models are used to abstractly quantify the effect of parallelization upon processing time vs. loading, as well as to establish a system performance stationary action principle.
Highlights
Motivated by category theory’s mathematical precision and expressive power, this paper introduces a new category theoretic application useful for numerical systems engineering modelling and analysis via a very simple straightforward categorification of MW for the case that the application monoid H is a free monoid generated by a finite set of basis processes, i.e., H is the set of all finite sequences of basis processes, including the empty sequence-where each sequence represents a system and each basis process corresponds to either an arithmetic process, a data movement process, or a delay process-and catenation of systems serves as the associative binary operation
This paper has presented a category theoretic reformulation of the Müller-Wichards system performance model
This defines for each basis process its type, rate, and weight
Summary
Müller-Wichards proposed a concise novel approach for estimating the total performance of computer-based (but machine independent) applications by algebraically combining the known performance estimates of the individual arithmetic, data movement, and delay elements that comprise the applications in a manner that accounts for various degrees of parallelism that can occur during processing [1]. The total numerical performance estimate for the application results naturally from the associative binary operations in H and P and the homomorphism property of φ, which algebraically combines the performance estimates for each application element in H. Many common mathematical concepts occur naturally with only slight variation in the various areas of mathematics. Category theory is that branch of mathematics which identifies and studies these common concepts and provides formal mechanisms for mapping them from one area of mathematics to another.
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