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Equation 13 · Part 2 · Serving a Frontier Model: The KV Cache, Batching, and What a Token Actually Costs

Symbol M_device

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

What this part means

MdM_device occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Its job in the formula

MdM_device occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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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A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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