Math Bridge: Gateway Energy to Service Days

← Back to Energy and Latency Optimization
Math BridgeEdge & FogStruggle-friendly runway

How many gateway windows can one battery buy?

Carry the chapter's local and offload millijoules into a bounded service-interval estimate.

Edge Eddie, the guideEdge Eddie guides
The one targetTurn energy per window into comparable runtime.
The chapter case6.00 mJ local, 2.70 mJ offload, 3 Ah at 3.7 V.
What it buys youA cadence-sensitive visit estimate with its assumptions visible.

A field team faces an unresolved physical question: How many gateway windows can one battery buy? They must answer it before changing analysis windows per minute on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is analysis windows per minute. The middle card applies this page's relationship. The green card is cell energy. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Analysis windows per minute changes cell energy An input card leads through the page relationship to the cell energy result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Both strategies pay once per window. Cadence scales both daily ledgers; only changing the per-window costs moves their ratio.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for analysis windows per minute is 60.

  2. 2

    Name the relationship. Eday=Ewindowxwindows/day; Eusable=CV(0.98)^6(0.90); days=Eusable/Eday

  3. 3

    Substitute the chapter fixture. Set analysis windows per minute to 60. The page ledger gives cell energy as 11.10 Wh.

  4. 4

    Read the result. Keep Wh beside the value. Use it only inside the technical boundary on this page.

Predict, then change analysis windows per minute

Try Predict the direction of cell energy. Move one control, calculate, then check your prediction.

60
Chapter baseline
Cell energy

Observe Both strategies pay once per window. Cadence scales both daily ledgers; only changing the per-window costs moves their ratio. Reset the control to 60 and compare cell energy.

Explain Only analysis windows per minute moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only analysis windows per minute moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only analysis windows per minute. It replays the chapter fixture; it is not a field prediction.

1. Keep rate and amount apart

A window costs energy. Cadence says how often that cost repeats. Battery charge becomes energy only after multiplying by voltage; self-discharge and reserve then reduce what operations may spend.

Edge Eddie: Millijoules per event become days only after the event schedule is named.

2. Name each move

1

Count daily windowsWindows/min × 60 × 24.

2

Convert millijoulesDaily Wh = mJ/window × windows/day ÷ 3,600,000.

3

Convert chargeCell Wh = Ah × V.

4

Apply retention and reserveUsable Wh = cell Wh × 0.98^months × 0.90.

5

Divide budget by daily spendRuntime days = usable Wh / Wh per day.

3. Work one window per second

60 windows/min → 86,400/day; 3 Ah × 3.7 V = 11.1 Wh

Six months of 2% monthly self-discharge and 10% derating leave 8.85 Wh. Local processing costs 0.144 Wh/day for 61.5 days; offload costs 0.0648 Wh/day for about 137 days. Their ideal runtime ratio is 6.00/2.70 = 2.22×.

4. Try one controlled change

Eday=Ewindow×windows/day; Eusable=CV(0.98)^6(0.90); days=Eusable/Eday

TryMove only the analysis cadence. Per-window costs, cell, retention period, and reserve stay fixed.

Windows/day
Cell energy
Usable energy
Local daily
Offload daily
Local runtime
Offload runtime
Runtime ratio

ObserveAt 60 windows/min, usable energy is 8.85 Wh and the two ideal runtimes are 61.5 and 136.6 days. Lower cadence lengthens both by the same factor, so their 2.22× ratio stays fixed.

ExplainBoth strategies pay once per window. Cadence scales both daily ledgers; only changing the per-window costs moves their ratio.

Technical boundaries.

This is an energy-only comparison of the chapter's two window paths.

Baseline
Sleep, sensors, regulators, logging, and updates are not included
Cell
Capacity varies with temperature, age, discharge rate, and cutoff voltage
Radio
Retries, coverage, attach time, and network availability change offload cost

Measure the whole gateway duty cycle and use a cell model across field conditions before scheduling visits.

5. Read 2.22× correctly

The ratio follows directly from the two per-window costs only because every other modeled term is shared. Adding a fixed sleep or sensor baseline pulls the whole-system lifetime ratio closer to one.

6. Carry a field energy ledger

Record cadence, local compute, sensor, radio success and retry energy, idle states, regulator loss, cell curve, temperature, reserve, service target, and the latency or privacy reason for placement.

7. Check yourself

Why divide millijoules by 3,600,000?
Answer: One watt-hour equals 3600 joules, or 3,600,000 millijoules.
Why does reducing cadence not change the 2.22× ratio?
Answer: The same daily-window multiplier appears in both modeled energy paths and cancels.
Are 61.5 and 137 days field promises?
Answer: No. They omit important shared loads and cell/environment effects.
Honesty boundary.

This page reproduces a bounded teaching comparison, not a battery qualification.

6.00 mJ
Chapter local-window example
2.70 mJ
Chapter offload-window example
8.85 Wh
Assumption-dependent usable cell energy

Go deeper in the chapter, add all loads, then bench and field-test the chosen schedule.

Return to Energy and Latency Optimization with the maths unpacked →