A General Theory of Computational Scalability Based on Rational Functions

A General Theory of Computational Scalability Based on Rational Functions Neil J. Gunther∗ November 26, 2024 Abstract The universal scalability law of computational capacity is a rational function Cp = P(p)/Q(p) with P(p) a linear polynomial and Q(p) a second-degree polynomial in the number of physical processors p, that has been long used for statistical modeling and prediction of computer system performance. We prove that Cp is equivalent to the synchronous throughput bound for a machine- repairman with state-dependent service rate. Simpler rational functions, such as Amdahl’s law and Gustafson speedup, are corollaries of this queue-theoretic bound. Cp is further shown to be both necessary and sufficient for modeling all practical characteristics of computational scalability. 1 Introduction For several decades, a class of real functions called rational functions [1], has been used to represent throughput scalability as a function of physical processor configuration. In particular, Amdahl’s law [2], its modification due to Gustafson [3] and the Universal Scalability Law (USL) [4] have found ubiquitous application. In this context, the relative computing capacity, Cp, is a rational function of the number of physical processors p. It is defined as the quotient of a polynomial P(p) in the numerator and Q(p) in the denominator, i.e., Cp = P(p)/Q(p). Each of the above-mentioned scalability models is distinguished by the number of coefficients or fitting parameters associated with the polynomials in P(p) and Q(p). For example, Amdahl’s law and Gustafson’s modification are single parameter models, whereas the USL model contains two parameters. Despite their historical utility, these models have stood in isolation without any deeper physical interpretation. It has even been suggested that Amdahl’s law is not fundamental [5]. More importantly, the lack of a unified physical interpretation has led to the use of certain flawed scalability models [6]. In this note, we demonstrate that the aforementioned class of rational functions corresponds to certain performance bounds belonging to a queue-theoretic model. The idea that Amdahl’s law, which has most frequently been associated with the scalability of massively parallel systems, can be considered from a queue-theoretic standpoint, is not entirely new [See e.g., 7, 8]. However, quite apart from motivations entirely different from our own, those previous works employed open queueing models with an unbounded number of requests (See Appendix C), whereas we shall use a closed queueing model with a finite number of requests p corresponding to the number of physical processors. The USL function is associated with a state-dependent generalization of the machine repairman [9]. The organization of this paper is as follows. We briefly review the scalability models of interest in Sect. 2. The appropriate queueing metrics associated with the standard machine repairman and its state-dependent extension are discussed in Sect. 3. The performance characteristics associated with synchronous queueing are also presented there. The main theorem (Theorem 2) is established in Sect. 4. Amdahl’s law and Gustafson’s linear speedup are shown to be corollaries of this theorem. Finally, in Sect. 5 we prove an earlier conjecture that a rational function with Q(p) a second-degree polynomial is both necessary and sufficient to model all practical cases of computational scalability. ∗Performance Dynamics Company, 4061 East Castro Valley Blvd., Suite 110, Castro Valley, CA 94552, USA. Email: nj gunther @ perfdynamics . com 1 arXiv:0808.1431v2 [cs.PF] 25 Aug 2008 2 Parametric Models Although technically, we are discussing rational functions, we shall hereafter refer to them as parametric models, and the coefficients as parameters, since the primary application of these models is nonlinear statistical regression of performance data [See e.g., 4, 10, 11, 12, and references therein]. 0 20 40 60 80 100 p 5 10 15 20 Cp Figure 1: Parametric models: USL (red), Amdahl (green), Gustafson (blue), with parameter values exaggerated to distinguish their typical characteristic relative to ideal linear scaling (dashed). The horizontal line is the Amdahl asymptote at σ−1 . Definition 1 (Speedup). If an amount of work N is completed in time T1 on a uniprocessor, the same amount of work can be completed in time Tp < T1 on a p-way multiprocessor. The speedup Sp = T1/Tp is one measure of scalability. 2.1 Amdahl’s law For a single task that takes time T1 to execute on a uniprocessor (p = 1), Amdahl’s law [2] states that if the task can be equipartitioned onto p processors, but contains an irreducible fraction of sequential work σ ∈[0, 1], then only the remaining portion of the execution time (1 −σ)T1 can be executed as p parallel subtasks on p physical processors. The bound on the achievable equipartitioned speedup [13] is given by the ratio Sp(σ) = T1 σT1 + „1 −σ p « T1 (1) which simplifies to Sp(σ) = p 1 + σ(p −1) ; (2) a rational functio

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