$$n_{\max} \approx (B - T_0)/\bar{t}$$

Equation 11 · 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.

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 function. A recent controlled study of exactly this trade-off tested fourteen models from 1.5B to 32B parameters plus one frontier API model across 6,566 controlled tool calls at three context budgets, and found that at an 8K-token budget uncompressed JSON schemas…

Meanings in this article

• $B$: the fixed context budget.
• $T_0$: the number of tokens.
• $\bar{t}$: the number of tokens.

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