Coverage Probability
From Median Signal to a Reliability Number
A radio link does not deliver a fixed signal level. As a receiver moves through a cell, the local mean power follows a smooth distance trend set by the path-loss exponent, but it is also perturbed by shadowing from buildings, terrain, and foliage. Empirically this slow variation is log-normal: expressed in dB, the received power is Gaussian about its distance-predicted median with a standard deviation σ (sigma) of roughly 4 dB in open suburban areas, 6 to 8 dB in dense urban macrocells, and 10 to 12 dB indoors. Coverage probability is the simple question that follows from this model: given the median at a point and the random spread σ, what is the chance that the instantaneous (or short-term-averaged) signal lands above the receiver threshold?
At the cell edge the median is, by convention, set equal to the threshold, so the edge coverage probability is exactly 50% with zero margin. Adding a fade margin shifts the median upward and pushes the edge probability above 50%. The widely used Jakes formulation then integrates this point probability over a circular cell, weighting by area, to produce the area coverage probability that operators actually commit to in service-level agreements. Because inner locations enjoy a much stronger median, the area figure always exceeds the edge figure, which is why a 75% edge design can satisfy a 90% area requirement.
Coverage probability is distinct from outage probability, though they are complementary. Outage counts the fraction of locations or time below threshold (one minus coverage), and it usually also folds in fast multipath fading and co-channel interference. Coverage analysis that ignores fast fading assumes a diversity or averaging mechanism removes it; when that assumption fails, an extra margin against Rayleigh fading must be stacked onto the shadowing margin.
Edge versus Area Coverage
The two reported quantities answer different design questions. Edge coverage probability sizes the worst-case boundary user and therefore drives base-station spacing in a regular grid. Area coverage probability describes the average subscriber experience and is the number that appears in regulatory and contractual coverage obligations. The ratio between them depends only on σ and the path-loss exponent n through the dimensionless parameter σ / (n × 10/ln10); a steeper path loss or a tighter shadowing distribution narrows the gap and lets a lower edge target meet the same area goal.
Coverage Probability Equations
Pedge = Q( (Pth − Pmed) / σ ) = 1 − Φ( −M / σ )
Required fade margin for target P:
M = σ × Q−1(1 − P) (e.g. P=0.95, σ=8 dB → M ≈ 13.2 dB)
Jakes area coverage (circular cell, exponent n):
Farea = ½[ 1 + erf(a) + e(1−2ab)/b²(1 − erf((1−ab)/b)) ]
where a = (Pth − Pmed(R)) / (σ√2), b = (10 n log10e) / (σ√2)
Q(·) is the Gaussian tail, Φ the standard normal CDF, Pth the threshold, Pmed the median received power (dBm), σ the shadowing standard deviation (dB), R the cell radius, n the path-loss exponent. Example: edge P=0.75, σ=8 dB, n=3.5 → Farea ≈ 0.90.
Edge Target to Area Coverage Mapping
| Edge coverage | Fade margin (σ=8 dB) | Area coverage (n=3.5) | Area coverage (n=4.0) | Typical use case |
|---|---|---|---|---|
| 50% | 0 dB | ~71% | ~74% | Boundary reference only |
| 75% | 5.4 dB | ~90% | ~91% | Standard macrocell planning |
| 90% | 10.2 dB | ~97% | ~98% | Public-safety / mission links |
| 95% | 13.2 dB | ~99% | ~99.4% | High-reliability fixed wireless |
| 99% | 18.6 dB | ~99.8% | ~99.9% | Critical telemetry / SCADA |
Frequently Asked Questions
What is the difference between edge coverage probability and area coverage probability?
Edge coverage is the reliability at the cell boundary, where the median signal equals the threshold by design; area coverage is the fraction of the whole cell served above threshold and is always higher, since inner locations have a stronger median. For n=3.5 and σ=8 dB, a 75% edge target maps to roughly 90% area coverage. The Jakes formula links the two through the ratio σ / (n × 10/ln10), so steeper path loss or lower shadowing narrows the gap.
How much fade margin do I need for 95 percent coverage probability?
The required margin equals the standard normal inverse of the target times the shadowing σ. With σ=8 dB, 90% edge coverage needs about 1.28 × 8 = 10.2 dB and 95% needs about 1.645 × 8 = 13.2 dB. If fast Rayleigh fading is not averaged out, add another 10 to 30 dB depending on the allowed outage rate. Most planners set a high area target such as 95% and back-calculate the lower edge probability that achieves it.
How does the path-loss exponent affect coverage probability?
A larger exponent n makes signal decay faster with distance, shrinking the served radius for a given power but improving the area-to-edge coverage ratio because the median rises steeply moving inward. Free space gives n=2, urban macrocells run 3.5 to 4, and dense indoor reaches 4 to 6. An error of 0.5 in n can shift the predicted cell radius by 20 to 40%, so the propagation model and exponent must match the deployment.