$$n_{\max} = \left\lfloor \frac{C - T_{\mathrm{system}} - T_{\mathrm{reserve}}}{\bar{t}_{\mathrm{tool}}} \right\rfloor$$

Equation 15 · AI Infrastructure

The Economics, Energy, and Physical Limits of Tool Protocols and the Model Context Protocol

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Put the three costs together and a deployment question comes into focus: given a fixed context window, how many tools can actually be offered before something has to give? A simple ceiling model makes the trade-off explicit. Let C be the model’s usable context window, $T_{\mathrm{system}}$ the fixed cost of operator instructions, and $T_{\mathrm{reserve}}$ a reserved allowance for the user’s message and the model’s own answer. If every tool schema costs on average $\bar{t}_{\mathrm{tool}}$ tokens and all of them are injected in full on every turn, the maximum number of tools a deployment can nominally offer before it must start pruning or retrieving is $n_{\max} = \left\lfloor \frac{C - T_{\mathrm{system}} - T_{\mathrm{reserve}}}{\bar{t}_{\mathrm{tool}}} \right\rfloor$. The assumption doing the…

Meanings in this article

• $n_{\max}$: the maximum number of tools a deployment can nominally offer before it must start pruning or retrieving.
• $C$: the model’s usable context window.
• $T_{\mathrm{system}}$: the fixed cost of operator instructions.
• $T_{\mathrm{reserve}}$: the reserved allowance for the user’s message and the model’s own answer.
• $\bar{t}_{\mathrm{tool}}$: the number of tokens.

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