Math Bridge: Relay Hotspot Budget

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Math BridgeWSNStruggle-friendly runway

Why does the relay die before identical edge nodes?

Count downstream traffic in current, then follow the same usable capacity into two lifetimes.

Packet Pete, the guidePacket Pete guides
The one targetExpose the near-sink relay hotspot in one battery ledger.
The chapter case50 µA edge node; relay forwards five downstream flows.
What it buys youA sizing decision tied to the first expected failure.

A field team has a real problem to settle: Why does the relay die before identical edge nodes? They must decide what happens before they change downstream nodes on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is downstream nodes. The middle card uses this page's rule. The green card is usable capacity. 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.

Downstream nodes changes usable capacity An input card leads through the page rule to the usable capacity result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. Each downstream share adds another edge-current unit in this model. Capacity stays fixed, so relay life shrinks in inverse proportion.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for downstream nodes is 5.

  2. 2

    Name the rule. Qusable=0.80x2000=1600 mAh tedge=1600/0.050=32,000 h=3.65 years Irelay=0.050(1+5)=0.300 mA=300 uA trelay=1600/0.300=5,333 h≈7.3 months 20log10(2400/868)=8.83 dB; ideal range=2.76x

  3. 3

    Put in the chapter value. Set downstream nodes to 5. The page rule gives usable capacity as 1600 mAh.

  4. 4

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

Predict, then change downstream nodes

Try Predict what happens to usable capacity. Move one control, calculate, then check your idea.

5
Chapter baseline
Usable capacity

Observe Each downstream share adds another edge-current unit in this model. Capacity stays fixed, so relay life shrinks in inverse proportion. Reset to 5 and compare usable capacity.

Explain Only downstream nodes 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 downstream nodes moves. Field effects named in the page limits stay fixed.

Honesty note. This teaching model changes only downstream nodes. It replays the chapter baseline; it is not a field prediction.

1. Identical hardware can carry unequal work

An edge node sends its own readings. A relay sends its own plus the traffic forwarded for downstream nodes, so identical cells can reach the same usable charge floor at very different times.

Packet Pete: Count load per role, not only the average node.

2. Name every algebra move

1

Derate nameplate capacityQusable=fQnominal.

2

Count the relay's traffic sharesIrelay≈Iedge(1+n).

3

Divide one capacity twicetedge=Qusable/Iedge; trelay=Qusable/Irelay.

4

Compare bands in dBΔFSPL=20log10(fhigh/flow).

5

Undo the distance logarithmrange ratio=10^(ΔFSPL/20).

3. Reproduce the hotspot case

Qusable=0.80×2000=1600 mAh
tedge=1600/0.050=32,000 h=3.65 years
Irelay=0.050(1+5)=0.300 mA=300 µA
trelay=1600/0.300=5,333 h≈7.3 months
20log10(2400/868)=8.83 dB; ideal range=2.76×

The lower-band range is a possible topology lever, not a promise that relay count will fall.

4. Try downstream load

TryChange how many downstream nodes the relay carries.

Downstream nodes
Usable capacity
Relay current
Edge runtime
Edge years
Relay runtime
Relay years
Relay months
Band advantage
Ideal range ratio

ObserveAt five downstream nodes, relay current is 300 µA and life falls to 5,333 h or about 7.3 months while edge life remains 3.65 years.

ExplainEach downstream share adds another edge-current unit in this model. Capacity stays fixed, so relay life shrinks in inverse proportion.

Technical boundaries.

The relay-current relation is a role-loading approximation.

Traffic
Payloads, aggregation, receive cost, retransmissions, routes, and reporting schedules vary
Battery
Pulse sag, self-discharge, temperature, cutoff, and rate effects are omitted
Range
The band comparison is free-space only and does not determine topology by itself

Measure per-role state currents and validate the actual relay tree.

5. Size for the hotspot, not the average

Mitigations include rotating relay roles, adding gateways, changing routes, aggregating traffic, reducing reporting load, using larger energy stores, or shortening service intervals.

6. Build the sizing record

Record role, downstream count, traffic schedule, state currents, route and retry evidence, capacity and cutoff, band and measured reach, maintenance target, mitigation, owner, and retest trigger.

7. Check yourself

Why is relay current 300 µA?
Answer: The simplified relay carries its own share plus five downstream shares: 50(1+5)=300 µA.
Why does the same cell last only about 7.3 months?
Answer: 1600 mAh divided by 0.300 mA is 5,333 h, about 0.609 years.
Does 868 MHz guarantee fewer relays?
Answer: No. The 2.76× figure is ideal free-space range; site, regulation, antenna, and link requirements decide topology.
Honesty boundary.

The battery, current, downstream-count, and band figures are the chapter's explicit representative case.

300 µA
Linear traffic-share approximation
7.3 months
Bounded relay ledger
2.76×
Ideal band range comparison

Correct, not complete: commissioning evidence must validate role load and coverage.

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