Math Bridge: Idle Listening and Retry Energy

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Math BridgeWSNStruggle-friendly runway

Which invisible radio bill emptied the node?

Put idle listening and repeated transmissions into separate columns before blaming the battery.

Packet Pete, the guidePacket Pete guides
The one targetSeparate time-based listening waste from event-based retry waste.
The chapter case20 ms/s listening and three transmit attempts.
What it buys youA repair aimed at the actual battery bill.

A field team has a real problem to settle: Which invisible radio bill emptied the node? They must decide what happens before they change number of transmit attempts per reporting event on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is number of transmit attempts per reporting event. The middle card uses this page's rule. The green card is idle average. Follow the arrows: set the input, use the rule, then read the result and its unit.

The audit later on checks more than one number. Here, the added model uses the baseline named below and holds every other chapter value fixed. That sentence bridges the fixtures, so the numbers do not change without a reason.

Number of transmit attempts per reporting event changes idle average An input card leads through the page rule to the idle average result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. Attempts multiply event energy. The listen schedule runs on time whether or not an event occurs. The two outputs stay separate on purpose.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for number of transmit attempts per reporting event is 3.

  2. 2

    Name the rule. Iavg=DIlisten+(1-D)Isleep; Cusable=C0(1-k)^t(1-δ); Eevent=nVIt

  3. 3

    Put in the chapter value. Set number of transmit attempts per reporting event to 3. The page rule gives idle average as 0.102 mA.

  4. 4

    Read the result. Keep mA next to the value. Use it only within the limits on this page.

Predict, then change number of transmit attempts per reporting event

Try Predict what happens to idle average. Move one control, calculate, then check your idea.

3
Chapter baseline
Idle average

Observe Attempts multiply event energy. The listen schedule runs on time whether or not an event occurs. The two outputs stay separate on purpose. Reset to 3 and compare idle average.

Explain Only number of transmit attempts per reporting event moves here. The other chapter values stay fixed.

Check yourself

What should you do before you trust the result?
Answer: Predict its direction, use the shown rule, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only number of transmit attempts per reporting event moves. Field effects named in the page limits stay fixed.

Honesty note. This teaching model changes only number of transmit attempts per reporting event. It replays the chapter baseline; it is not a field prediction.

1. Silence can still draw current

A receiver consumes current while it listens, even when no useful packet arrives. A weak link creates another bill: each failed attempt repeats the transmit pulse. Both shorten life, but changing the listen schedule does not repair a poor antenna and improving the link does not erase an always-awake receiver.

Packet Pete: Diagnose by state and event, not by the single symptom “battery low.”

2. Name every algebra move

1

Find listen fractionD=tlisten/T.

2

Weight listen and sleep currentsIavg=DIlisten+(1−D)Isleep.

3

Discount capacityCusable=C0(1−k)^t(1−δ).

4

Divide capacity by average currentRuntime=Cusable/Iavg.

5

Multiply each transmit pulseEattempt=VIt; Eevent=attempts×Eattempt.

3. Work the two bills

Iavg=0.02(5)+0.98(0.002)=0.102 mA; Cusable=2000(0.98)²(0.85)=1633 mAh

The bounded listen-only runtime is about 16,010 h or 1.83 years. One 30 mA, 10 ms, 1.5 V attempt uses 0.450 mJ; three attempts use 1.35 mJ, or 3× the clean-attempt energy.

4. Try one controlled change

Iavg=DIlisten+(1−D)Isleep; Cusable=C0(1−k)^t(1−δ); Eevent=nVIt

TryMove only the number of transmit attempts. The listen schedule, cell assumptions, voltage, current, and pulse length stay fixed.

Listen fraction
Idle average
Usable charge
Listen-only runtime
Runtime years
One attempt
Event transmit energy
Retry waste
Energy multiplier

ObserveAt three attempts, event transmit energy is 1.35 mJ and the multiplier is 3×. Moving attempts does not change the 0.102 mA idle average or its bounded runtime.

ExplainAttempts multiply event energy. The listen schedule runs on time whether or not an event occurs. The two outputs stay separate on purpose.

Technical boundaries.

This ledger isolates two terms rather than modelling the full node.

Battery
Voltage sag, regulator efficiency, temperature, and load-dependent capacity are omitted
Radio
Receive windows, clear-channel checks, acknowledgements, backoff, startup, and payload airtime vary
Link
Retry count depends on interference, antenna mounting, topology, and protocol behaviour

Measure the complete state trace and retry distribution on the installed network.

5. Match the repair to the bill

To reduce idle waste, shorten or schedule receive windows and verify rendezvous reliability. To reduce retries, improve link margin, antenna clearance, channel choice, routing, contention, and acknowledgement behaviour.

6. Build a diagnostic record

Record state currents and durations, reporting rate, attempts per success, packet loss, channel occupancy, RSSI/SNR, antenna and enclosure, cell voltage under pulse, temperature, firmware, topology, and the change that reduced each bill.

7. Check yourself

Why does low traffic not remove the 0.102 mA bill?
Answer: The receiver still listens for 20 ms every second even when no payload arrives.
Why do three attempts use 1.35 mJ?
Answer: Each attempt uses 1.5×0.030×0.010 = 0.00045 J = 0.450 mJ; three multiply that by three.
Does 1.83 years predict the deployed node?
Answer: No. It is a listen-only, constant-condition ledger that omits all other loads and real battery behaviour.
Honesty boundary.

The 20 ms per second schedule comes from the chapter; cell, current, and retry values are catalog-typical teaching inputs.

0.102 mA
Illustrative idle-listen average
1.83 years
Bounded listen-only result
3×
Three attempts versus one clean attempt

Go deeper in the chapter, then replace every typical input with measured field evidence.

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