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Equation 14 · The Real Economics of Shipping a Model to a Device

What does this equation mean?

TCdevice=Cdev+U+Q⋅cd,TCcloud=Q⋅ccloud,TC_{\mathrm{device}} = C_{\mathrm{dev}} + U + Q \cdot c_{d}, \qquad TC_{\mathrm{cloud}} = Q \cdot c_{\mathrm{cloud}},

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Inputs and operationsC_dev + U + Q × c_d, qquad TC_cloud = Q × c_cloud
Result or conditionTC_device
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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TT

Symbol T

T is part of the quantity the equation computes from the expression on the right.

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CdeviceC_{\mathrm{device}}

Symbol C_device

CdC_device is part of the quantity the equation computes from the expression on the right.

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CdevC_{\mathrm{dev}}

Symbol C_dev

the fixed, per-release engineering and QA cost of producing and validating the device-tier variant matrix described above — the harness time, the pass/fail sign-off per tier, per quantization scheme, per chip target.

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UU

Symbol U

the cost of distributing model updates to the installed fleet, expanded in the next section.

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QQ

Symbol Q

Q is one of the signed contributions combined to compute the quantity on the left.

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cdc_{d}

Symbol c_d

the marginal cost of one additional on-device inference and ccloudc_{\mathrm{cloud}} as the marginal cost of one additional cloud-served inference of the same model.

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CcloudC_{\mathrm{cloud}}

Symbol C_cloud

CcC_cloud is one of the signed contributions combined to compute the quantity on the left.

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ccloudc_{\mathrm{cloud}}

Symbol c_cloud

ccc_cloud is one of the signed contributions combined to compute the quantity on the left.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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multiplication

multiplication

Multiply the quantities on either side.

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addition

addition

Add the term after the plus sign to the term or group before it.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

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What the article says around this equation

Define CdevC_{\mathrm{dev}} as the fixed, per-release engineering and QA cost of producing and validating the device-tier variant matrix described above — the harness time, the pass/fail sign-off per tier, per quantization scheme, per chip target. Define U as the cost of distributing model updates to the installed fleet, expanded in the next section. Define cdc_{d} as the marginal cost of one additional on-device inference and ccloudc_{\mathrm{cloud}} as the marginal cost of one additional cloud-served inference of the same model. Total cost over Q queries under each placement is TCdevice=Cdev+U+Q⋅cd,TCcloud=Q⋅ccloudTC_{\mathrm{device}} = C_{\mathrm{dev}} + U + Q \cdot c_{d}, \qquad TC_{\mathrm{cloud}} = Q \cdot c_{\mathrm{cloud}}. and the break-even query volume — the point at which on-device placement stops costing more than cloud…
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Define CdevC_{\mathrm{dev}} as the fixed, per-release engineering and QA cost of producing and validating the device-tier variant matrix described above — the harness time, the pass/fail sign-off per tier, per quantization scheme, per chip target. Define U as the cost of distributing model updates to the installed fleet, expanded in the next section. Define cdc_{d} as the marginal cost of one additional on-device inference and ccloudc_{\mathrm{cloud}} as the marginal cost of one additional cloud-served inference of the same model. Total cost over Q queries under each placement is TCdevice=Cdev+U+Q⋅cd,TCcloud=Q⋅ccloudTC_{\mathrm{device}} = C_{\mathrm{dev}} + U + Q \cdot c_{d}, \qquad TC_{\mathrm{cloud}} = Q \cdot c_{\mathrm{cloud}}. and the break-even query volume — the point at which on-device placement stops costing more than cloud placement — is

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