$$T(n) \approx T_0 + n\,\bar{t}$$

Equation 9 · AI Infrastructure

Measuring Tool Protocols and the Model Context Protocol: Evidence, Benchmarks, and Uncertainty

This equation gives an approximation: it relates the quantities while allowing an approximation.

A simple linear model makes the assumption underneath that spread explicit. If a system prompt costs $T_0$ tokens before any tools are added, and each of n tools costs on average $\bar{t}$ tokens to describe, total schema overhead is approximately $T(n) \approx T_0 + n\,\bar{t}$. Under a fixed context budget B , the largest catalogue that fits is roughly $n_{\max}$ $\approx$ (B - $T_0$)/$\bar{t}$ . The model is a reasonable first approximation, and it is also the assumption a tenfold per-tool variance directly undermines: $\bar t$ is not a constant of the tool, it is a property of how verbosely that tool’s schema and description happen to be written, which is an editorial choice rather than a physical fact about the…

Meanings in this article

• $T_0$: the number of tokens.
• $\bar{t}$: the number of tokens.

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