Math Bridge: Path-Loss Exponents in a Packet Simulator

Try this in the Packet Tracer packet-walk lab, where you step packets through the routed network and see the ARP and ICMP headers and TTL.

1. Free space is the clean baseline

A wave spreads over a growing sphere. At twice the distance, the same power covers four times the area. In dB, that inverse-square law becomes a 20 log10 distance term.

Pete: Free space is useful because it is a best case, not because a factory floor behaves like empty sky.

2. Let the exponent describe clutter

Build wavelengthλ=c/f.

Find free-space lossFSPL=20log10(4πd/λ).

Charge for clutterExtra loss over free space is 10(n−2)log10(d/d0).

3. Turn loss into margin

PL=FSPL+10(n−2)log10(d/d0); M=Pt−PL−Srx

Path loss is subtracted from transmit power. Sensitivity is a negative threshold, so margin subtracts that negative number.

4. Reproduce the chapter values

At 2.4 GHz, λ=0.125 m. At 20 m, FSPL=66.0666… dB, reported as 66.1 dB. Moving from n=2 to n=3 adds 10log10(20)=13.0 dB, so margin falls from 28.9 to 15.9 dB. The 14-, 16-, and 18-byte packets cost about 0.00124, 0.00142, and 0.00160 µWh.

5. Prove the model rather than trust it

Record simulator model and seed, measured or justified n, reference loss, distance, antenna placement, interference, packet success, retries, latency distribution, energy accounting, warm-up, and confidence interval across independent runs.

6. Check yourself

Why is free space a lower-loss bound?

Answer: It includes ideal geometric spreading but none of the extra obstruction, absorption, scattering, or interference of a real indoor site.

Does n=3 mean every packet loses exactly 13 dB extra?

Answer: No. It describes the model's mean distance slope; individual locations and times still vary.

Why connect margin to energy?

Answer: A weak link often retries, and each retry pays the packet transmission energy again.

Honesty boundary.

The path comparison fixes 2.4 GHz, 20 m, transmit power, receiver threshold, and the packet byte counts while the path-loss exponent changes. It describes mean loss and ideal packet energy, leaving individual locations, interference, retries, and installed-site reliability to measurement.

10 mW/+10 dBm: Radio transmit power
250 kbps: Assumed radio bit rate
86,400 packets/day: Packet transmission count per day
2.4 GHz: Carrier frequency
20 m: Transmitter-to-receiver distance
−85 dBm: Receiver sensitivity threshold
n=3: Cluttered-path loss exponent
66.1 dB: Free-space path loss at stated distance
13.0 dB: Additional loss from exponent increase
28.9 dB: Link margin with free-space exponent
15.9 dB: Link margin with cluttered-path exponent
packet-energy: Transmission energy for compared packet sizes

They do not validate a particular industrial site.