Part II · The future chip · Chapter 16 of 18

Calibration at scale

Thousands of units sharing one calibration controller: availability, start-up time and logic-heat storms, from Part I's loop scaled up.

A fleet of units sharing one calibration controllerpatentillustrative

Each square is a unit. Failure rates are the simulator’s (experiment 15: Part I’s control loop with its tracking shared ⌈U/P⌉ ways, plus drift from logic heat); storms arrive at the logic’s busy/idle changes and knock out 0.0625 of the units each. Your browser runs the fleet as an event simulation with the simulator’s rules; its long-run average is tested against the simulator’s queueing result.

A_1 = \frac{1}{1 + \lambda\,T_{\text{rec}}}

p_{n+1} = p_n\,\frac{(U - n)\,\lambda}{\min(n + 1, P)\,/\,T_{\text{rec}}},\qquad \text{in service} = 1 - \frac{\sum_n n\,p_n}{U}

What this stage does

Every unit drifts like the module of chapter 6: its heaters wander, pilot tracking holds them, and when tracking loses a unit it stops and recalibrates. With thousands of units the question is whether one calibration controller can keep up. It can serve 4 illustrative units at a time by default; this chapter sweeps that parallelism P, the units U and the drift.

How it’s modelled

Two things change at scale. Tracking is shared: the controller dithers and reads pilots for P units at a time, so each unit gets P/U of the pilot updates it would get alone, which is Part I’s loop with its pilot period stretched ⌈U/P⌉ times. Logic heat adds 0.005 rad/√s illustrative of drift per heater. Running that loop gives each unit’s recalibration rate λ. Recalibrations queue: a failed unit waits for a free channel, then is out of service for 1 s (MEMS 0.75 s, non-volatile 50.5 s). The fleet is a machine-repair queue, solved exactly; a SimPy model adds fixed recalibration times and logic-heat storms.

It reduces to Part I. At U = 1 without logic heat, the queue model gives 0.6483 in service, and running the loop of chapter 6 directly at the same settings (0.1 rad/√s, 100,000 frames between pilots) gives an availability of 0.6483: the same number, as it must be.

What we found

Calibration scales if the controller does. With P = 4 the fleet keeps at least nine in ten units in service at every swept U up to 16,384 at the default drift (0.01 rad/√s illustrative), but falls below that from 4,096 units at three times the drift and from 1,024 at ten times. The collapse is sharp: once shared tracking is too thin, units fail faster than the controller can recalibrate them, and almost none are left in service. Raising P moves the cliff (from 256 units with P = 1 to 16,384 with the widest swept controller, P = 64, at the fastest drift).

Start-up takes time. Calibrating every unit once takes 4,096 s for 16,384 units with P = 4. Storms cost a tenth of the fleet. With hot logic, every busy/idle change (0.1 /s) knocks out a share of the units at once; at U = 64 and P = 1 the in-service fraction falls from 0.9821 to 0.8831.

Units in service vs U (drift 0.1 rad/√s)illustrativepatent

Below the dashed line calibration, not compute, limits throughput.

Data table

Units in service vs U (drift 0.1 rad/√s)
units U	P = 1	P = 4	P = 16	P = 64
1	1	1	1	1
4	1	1	1	1
16	1	1	1	1
64	0.975	1	1	1
256	0.006284	0.9878	1	1
1,024	0.0007611	0.006284	0.9899	1
4,096	0.0001998	0.0007611	0.006284	0.99
16,384	4.757 × 10⁻⁵	0.0001998	0.0007611	0.006284

Units in service vs U (drift 0.03 rad/√s)illustrativepatent

Data table

Units in service vs U (drift 0.03 rad/√s)
units U	P = 1	P = 4	P = 16	P = 64
1	1	1	1	1
4	1	1	1	1
16	1	1	1	1
64	1	1	1	1
256	1	1	1	1
1,024	0.02566	1	1	1
4,096	0.002873	0.02566	1	1
16,384	0.0005112	0.002873	0.02566	1

Effective units U × in-service (drift 0.1)illustrative

Units actually doing work: past the cliff, adding units adds almost nothing.

Data table

Effective units U × in-service (drift 0.1)
units U	P = 1	P = 4	P = 16	P = 64
1	1	1	1	1
4	4	4	4	4
16	16	16	16	16
64	62.4	64	64	64
256	1.609	252.9	256	256
1,024	0.7794	6.435	1,014	1,024
4,096	0.8182	3.118	25.74	4,055
16,384	0.7794	3.273	12.47	103

Start-up calibration timeillustrative

Data table

Start-up calibration time
units U	P = 1	P = 4	P = 16	P = 64
1	1	1	1	1
4	4	1	1	1
16	16	4	1	1
64	64	16	4	1
256	256	64	16	4
1,024	1,024	256	64	16
4,096	4,096	1,024	256	64
16,384	16,380	4,096	1,024	256

What would need measured data

The drift rates, recalibration time and logic-heat figures are Part I’s illustrative values or extensions of them; the failure rates come from a finite run of the loop (zero where none were seen). What would pin the cliff down: measured drift of many units on one die, under a real workload’s heat, how a real controller shares its dither and pilot measurements, and whether recalibrating one unit disturbs its neighbours (the model assumes not).