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Equation 13 · Part 1 · Serving a Frontier Model: The KV Cache, Batching, and What a Token Actually Costs

Symbol b_max

bmax⁡≈Mdevice−MweightsMkv(s),b_{\max} \approx \frac{M_{\mathrm{device}} - M_{\mathrm{weights}}}{M_{\mathrm{kv}}(s)} ,
bmax⁡b_{\max}

What this part means

bmb_max is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Its job in the formula

bmb_max is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

The passage around this formula

What binds is memory, and specifically the cache. Weights are shared across the batch; the key–value cache is not. Each concurrent request carries its own, and each grows with every token it generates. Achievable batch size is therefore bmax⁡≈Mdevice−MweightsMkv(s)b_{\max} \approx \frac{M_{\mathrm{device}} - M_{\mathrm{weights}}}{M_{\mathrm{kv}}(s)} . which falls as contexts lengthen. This is the mechanism behind an effect users notice without explanation: long-context workloads cost disproportionately more, because they crowd out the concurrency that made short-context serving cheap.

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