Math Bridge: BLE Mesh Airtime and Polling

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Math BridgeBluetooth and BLEStruggle-friendly runway

Why does a 0.320 ms packet occupy 129.6 ms in this mesh?

Count bytes, bits, three advertising channels, relay transmissions, polls, and charge without changing arithmetic engines.

Radio Remi, the guideRadio Remi guides
The one targetConnect mesh airtime to a battery bound.
The chapter case135 relay transmissions and one Friend poll every 2 s.
What it buys youSeparate latency, channel occupancy, and polling charge.

A technician must decide whether three-channel daily charge is safe before changing friend poll interval on the real device. The result is unresolved until the rule and units are checked. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is friend poll interval. The middle card applies this page's rule. The green card is three-channel daily charge. 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 model keeps those stated values fixed and changes only friend poll interval, so the numeric fixture does not switch without explanation.

Friend poll interval changes three-channel daily charge An input card leads through the rule charge = 0.2304 mAh s/day / poll interval to the three-channel daily charge result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Longer poll intervals reduce how often the same packet cost repeats.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 2 s.

  2. 2

    Name the relationship. charge = 0.2304 mAh s/day / poll interval

  3. 3

    Substitute with units. 0.2304 / 2 = 0.1152 mAh/day

  4. 4

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

Predict, then change friend poll interval

Try Predict the direction of charge = 0.2304 mAh s/day / poll interval. Test another friend poll interval, then compare three-channel daily charge.

2 s
Chapter baseline
Three-channel daily charge

Observe Longer poll intervals reduce how often the same packet cost repeats. Reset friend poll interval to 2 and compare three-channel daily charge.

Explain Longer poll intervals reduce how often the same packet cost repeats.

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 friend poll interval moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only friend poll interval. It is not a field prediction.

1. Start with one radio transmission

Ten bytes contain 80 bits. A 250,000 bit/s radio needs 80/250,000=0.000320 s, or 0.320 ms, on one channel. BLE advertising sends the same PDU on three primary channels, so one advertising event occupies 0.960 ms.

Radio Remi: “Three channels” multiplies airtime; it does not triple the payload.

2. Name each algebra move

1

Convert bytes to bitsbits=8×bytes.

2

Divide by PHY ratet1ch=bits/RPHY; t3ch=3t1ch.

3

Count repeated workQday=(86,400/Tpoll)It3ch and tflood=Nrelayt3ch.

3. Keep time and charge units aligned

t3ch=3(8B/R); Qday=(86,400/Tpoll)It3ch/3.6 mAh

The 3.6 converts ampere-seconds to milliamp-hours. Relay airtime and LPN polling use the same event duration, but they answer different questions.

4. Try one controlled change

Qday=(86,400/Tpoll)It3ch/3.6

TryLengthen only the Friend poll interval from the chapter’s 2 s value. Packet size, channels, current, relays, and cell stay fixed.

Bits/PDU
One channel
Three channels
135 relays
Polls/day
One-channel charge
Three-channel charge
Polling-only upper bound

ObserveAt 2 s, 43,200 polls/day cost 0.0384 mAh/day on one channel but 0.1152 mAh/day across three, leaving a polling-only upper bound of 5.23 years.

ExplainLonger polling intervals reduce daily poll charge in direct proportion. They do not change the fixed 129.6 ms airtime of the chapter’s 135-transmission flood.

Technical boundaries.

This ledger counts ideal transmitted airtime and poll transmit charge.

Packet timing
Excludes inter-frame gaps, random backoff, and retransmission
LPN energy
Excludes receive windows, sleep current, processing, and regulator loss
Cell capacity
Nameplate charge is not guaranteed usable service charge

Measure the real bearer trace and current waveform before setting a service-life claim.

5. Reproduce the chapter values

80/250,000=0.000320 s. Three channels give 0.960 ms; 135 transmissions give 129.6 ms. At 2 s, 43,200×0.010 A×0.000960 s/3.6=0.1152 mAh/day. Then 220/0.1152/365=5.23 years before any other load.

6. Carry the evidence forward

Record PDU length, PHY, bearer, channel count, relay count, retransmit policy, TTL, poll interval, TX and RX current, receive-window time, sleep current, cell cutoff, temperature, and measured delivery rate.

7. Check yourself

Why is 0.320 ms not the full advertising event?
Answer: It covers one channel; the PDU is sent on three primary advertising channels.
Why is 5.23 years only an upper bound?
Answer: It spends the entire nameplate cell on poll transmission and ignores every other load and cutoff loss.
Does a longer poll interval reduce the 135-relay flood airtime?
Answer: No. It reduces poll frequency; the fixed flood still contains the same relay transmissions.
Honesty boundary.

The page makes the chapter’s single-channel shortcut and three-channel bearer cost visible.

129.6 ms
Ideal summed airtime, not an end-to-end latency prediction
5.23 years
Polling-only charge bound from a 220 mAh nameplate
Three channels
Primary advertising-channel repetitions in this teaching model

A deployed mesh needs packet captures, current traces, and field delivery evidence.

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