The Shape of Code

Before microprocessor cost/performance wiped out (in the early 1990s) other cpu platforms (e.g., mainframes and minis), people argued that computer hardware benefited from economies of scale.

The claimed benefit was more bang for the buck, i.e., more compute for less money.

Checking this claim requires treating pre-microprocessor computer systems and the later microprocessor-based systems as two separate cases, because many of the factors driving costs and performance are very different.

Today’s large microprocessor-based computer systems achieve economies of scale through discounts from bulk purchases and spreading fixed costs across multiple systems. The data is available, and the economic analysis is straight forward.

A lack of reliable data on the costs of designing/building pre-microprocessor computer systems rules out an economic analysis of cost/performance from first principles. The data that was/is available is the cost of computer systems and some indicators of performance (such as instruction timings or benchmarks).

Now, the observed fact that the cost of compute was decreasing over time is unrelated to the claim that the cost of compute decreases as the size of the computer increases.

Assuming a power law relationship between computer cost, , and size, , at a point in time, we have: , where is some constant. Economies of scale occur when:

In his detailed cost/performance analysis of computers between 1944-1967, Kenneth Knight treated computers launched in the same year as effectively occurring at the same time. He also built a single model, with year included as an explanatory variable, which means the fitted rate of decrease is the same over all years (rather than varying between years).

The plot below uses Knight’s 1953-1961 data, and shows operations per second against seconds per dollar (a confusing combination, but what Knight used), with fitted regression lines for three years using Knight’s model (code and data)

The fitted exponent for this form of x/y axis maps to a value which has , i.e., there are economies of scale.

It so happens that the value of the Knight’s fitted exponent is close to that proposed in a 1953 paper (“High speed arithmetic: The digital computer as a research tool”, no online copy):

  It used to cost one cent to do a multiplication on a
  desk calculator; now it is more like four cents; but
  with these big machines we can do a million in an hour
  for $400, and that means twenty-five multiplications
  for a cent! I believe that there is a fundamental rule,
  which I modestly call Grosch's law, giving added
  economy only as the square root of the increase in
  speed-that is, to do a calculation ten times as cheaply
  you must do it one hundred times as fast.

which did indeed become widely known as Grosch’s law.

Having been given a lucky kick-start by Knight (fitted individually, years are not close to Grosch’s law), checking for agreement with Grosch’s law became a focus for later studies. While various papers highlighted problems with the later data analysis (e.g., the regression techniques and sample noise producing mathematical artifacts), Grosch’s law ceased being a thing because mainframes/minicomputers ceased being a thing.

Did mainframe/mincomputers have economies of scale in the years after Knight’s data? It’s difficult to tell, the publicly available data is too sparse to support reliable analysis.