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Equation 15 · Part 5 · The Economics, Energy, and Physical Limits of Tool Protocols and the Model Context Protocol

Symbol bart_tool

nmax⁡=⌊C−Tsystem−Treservetˉtool⌋.n_{\max} = \left\lfloor \frac{C - T_{\mathrm{system}} - T_{\mathrm{reserve}}}{\bar{t}_{\mathrm{tool}}} \right\rfloor.
tˉtool\bar{t}_{\mathrm{tool}}

What this part means

the number of tokens.

Its job in the formula

barttt_tool occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Where the article explains it

If every tool schema costs on average tˉtool\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 nmax⁡=⌊C−Tsystem−Treservetˉtool⌋n_{\max} = \left\lfloor \frac{C - T_{\mathrm{system}} - T_{\mathrm{reserve}}}{\bar{t}_{\mathrm{tool}}} \right\rfloor.

The passage around this formula

…usable context window, TsystemT_{\mathrm{system}} the fixed cost of operator instructions, and TreserveT_{\mathrm{reserve}} a reserved allowance for the user’s message and the model’s own answer. If every tool schema costs on average tˉtool\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 nmax⁡=⌊C−Tsystem−Treservetˉtool⌋n_{\max} = \left\lfloor \frac{C - T_{\mathrm{system}} - T_{\mathrm{reserve}}}{\bar{t}_{\mathrm{tool}}} \right\rfloor. The assumption doing the work in that formula — every schema injected in full, on every…

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Sources cited in the surrounding passage

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