Math Bridge: When do shorter hops save energy?

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When do shorter hops save energy?

Separate the physical saving from shorter radio links from the fixed cost every transmission and relay still pays.

Packet Pete, the guidePacket Pete guides
The one targetCompare distance-scaled transmit energy with a fixed per-hop radio ledger.
The chapter casen=3, 0.5 mJ transmission, 0.2 mJ relay reception, six hops versus three.
What it buys youAn honest energy-aware routing claim.

See the relationship before changing it

The figure reads from left to right. The blue card is equal radio hops. The middle card applies this page's rule. The green card is ideal distance-power ratio. 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 equal radio hops, so the numeric fixture does not switch without explanation.

Equal radio hops changes ideal distance-power ratio An input card leads through the rule ideal ratio = 1 / hops^(3 - 1) to the ideal distance-power ratio result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Shorter hops reduce ideal path loss while each relay still pays fixed energy.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 6 hops.

  2. 2

    Name the relationship. ideal ratio = 1 / hops^(3 - 1)

  3. 3

    Substitute with units. 1 / 6^2 = 0.0278

  4. 4

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

Predict, then change equal radio hops

Try Predict the direction of ideal ratio = 1 / hops^(3 - 1). Test another equal radio hops, then compare ideal distance-power ratio.

6 hops
Chapter baseline
Ideal distance-power ratio

Observe Shorter hops reduce ideal path loss while each relay still pays fixed energy. Reset equal radio hops to 6 and compare ideal distance-power ratio.

Explain Shorter hops reduce ideal path loss while each relay still pays fixed energy.

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 equal radio hops moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only equal radio hops. It is not a field prediction.

1. Start with two different energy stories

Shortening a distance-scaled radio link can save transmit energy, yet every added hop still wakes a transmitter and usually a receiver. Those effects can point in opposite directions.

Packet Pete: Never let “multi-hop saves power” hide which energy model produced the claim.

2. Name every algebra move

1

Split the spanN equal hops make each hop D/N long.

2

Apply the distance powerEach ideal transmit cost scales as (D/N)^n.

3

Count all transmittersN(D/N)^n=D^n/N^(n−1).

4

Build the fixed ledgerEfixed=N×Etx+(N−1)×Erx.

5

Compare like with likeReport both models instead of mixing their assumptions.

3. Reproduce the chapter case

Eideal(N)/Eideal(1)=1/N^(n−1); Efixed(N)=N(0.5)+(N−1)(0.2) mJ
n=3 and N=6: ideal ratio=0.0278, saving=36.00×; fixed six-hop=4.00 mJ; fixed three-hop=1.90 mJ

At n=3, doubling distance costs 8.00× transmit power, but the chapter's fixed ledger makes the six-hop route 2.11× the three-hop cost.

4. Try the hop count

TryMove the hop count. Compare the ideal distance-scaled saving with the fixed radio ledger.

Hop count
Double-distance cost
Ideal energy ratio
Ideal saving
Fixed path energy
Three-hop energy
Fixed-cost ratio
Saving by using three

ObserveAt six hops, the pure distance model says 36.00× less transmit energy, while the fixed ledger says 4.00 mJ versus 1.90 mJ—52.50% less when using three hops.

ExplainThe first result assumes power is reduced for each shorter link. The second keeps a flat cost per transmission and reception. Neither may silently stand in for the other.

Technical boundaries.

This compact engine compares two teaching models; it does not predict a deployed route.

Distance model
Equal hops and one exponent omit fading, interference, retries, and antenna differences
Fixed ledger
0.5 and 0.2 mJ omit idle listening, channel access variation, and payload size
Routing
Energy alone does not satisfy latency, delivery, or node-lifetime requirements

Measure the real path and reopen the decision when traffic or radio settings change.

5. Decide which claim is allowed

Use the distance model only when transmit power actually tracks hop length. Use the fixed ledger when measured packet costs stay roughly constant. A production record may include both.

6. Keep the routing evidence

Record hop lengths, path exponent, power-control policy, transmit and receive energy, retries, latency, payload, battery state, owner, and the condition that triggers a retest.

7. Check yourself

Why is the n=3, six-hop ideal ratio 1/36?
Answer: 1/N^(n−1)=1/6²=1/36, because six shortened transmissions replace one long transmission.
Why does the chapter's six-hop ledger cost 4.0 mJ?
Answer: Six transmissions cost 3.0 mJ and five relay receptions cost 1.0 mJ.
Does the ideal saving prove six hops are better?
Answer: No. It excludes fixed radio cost, retries, latency, and whether the radio turns power down on short links.
Honesty boundary.

Every fixed value is traceable to the chapter's worked alert case.

n=3
The chapter's obstructed teaching exponent
0.5 and 0.2 mJ
The chapter's approximate per-hop transmit and receive ledger
Three and six hops
The chapter's competing critical-alert routes

Correct, not complete: measured radio energy and service evidence still decide acceptance.

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