30 WSN Coverage Worked Examples

30.1 Start With the Field Story

Worked examples are useful when they keep the decision visible. For each coverage case, identify the monitoring goal, the assumed range, the active set, the failure case, and the record that would convince a reviewer.

30.2 In 60 Seconds

WSN coverage examples are useful only when they show how a decision is reviewed. A worked example should start with a claim, name the coverage type and redundancy level, inspect the active operating state, compare the claim with field evidence, record the finding, choose a repair or acceptance decision, and define the retest trigger.

This chapter uses examples to practice that review habit. It avoids treating a formula, coverage drawing, or unit-cost comparison as final proof. The learner should leave with a repeatable way to work through coverage decisions without overstating what the evidence proves.

30.3 Learning Objectives

By the end of this chapter, you will be able to:

• Work through WSN coverage examples using claim, evidence, finding, and decision records.
• Identify when a redundant area claim needs repair rather than approval.
• Review duty-cycle choices without letting energy savings break coverage.
• Compare sensing-range options using deployment evidence, not unit price alone.
• State limits and retest triggers for every example decision.

30.4 Quick Check: WSN Coverage Examples

30.5 How to Use Worked Examples

Worked examples should make reasoning visible. They should not imply that one example number, one drawing, or one rule proves every future deployment.

30.6 Example Review Route

A hidden assumption between Scenario and case boundary could overturn example review route. The chart in Figure 30.1 brings that assumption into the review.

Use case boundary as the pivot in the diagram at Figure 30.1. Before it, Scenario adds a distinct review condition; after it, and purpose defines what the design promises, while the pivot itself adds a distinct review condition. The resulting structure is not decorative: Wireless sensor network coverage worked example review route connecting the scenario, coverage claim, coverage type and level, active state, field evidence, finding, decision, known limit, and retest trigger. It sets the usable limit for example review route.

The route keeps examples from drifting into calculation theater. A calculation can support a decision, but the final approval still depends on the deployed state and evidence.

30.7 Example 1: Redundant Area Monitoring

A water treatment room has an area coverage claim: important process conditions should be detected anywhere inside the monitored room during normal operation. The risk is high enough that the design asks for backup sensing at weak points.

The learning point is not a fixed sensor count. The learning point is that redundant coverage must be true at the weak locations and in the active operating state.

30.8 Example 2: Duty-Cycle Evidence

A remote wildlife monitoring deployment needs long battery life. The team proposes a low-duty-cycle schedule so sensors wake briefly, sample, communicate, and then sleep again.

Duty cycling is a coverage decision, not only an energy decision. A sleeping sensor does not count as active coverage unless the accepted rule explains when it wakes and what latency remains.

30.9 Example 3: Sensing Range Tradeoff

A building automation team compares short-range and longer-range occupancy sensors. The longer-range device costs more per unit, but it may reduce installation count and maintenance work.

30.10 Barrier Coverage for Crossings

A facility wants to detect crossings at a perimeter line. The requirement is crossing detection, not full interior tracking.

Barrier examples should be explicit about weak versus strong claims. Detecting a crossing is not the same as tracking the object continuously after it crosses.

30.11 Example Evidence Record

Before the example evidence record argument advances, use Figure 30.2 to reconcile Scenario with area, point,. The diagram turns that reconciliation into something the team can inspect.

At Scenario, Figure 30.2 adds a distinct review condition. The visual next names area, point, to show where the review fixes the spatial boundary; barrier, or path finally adds a distinct review condition. Those concrete stages express the following claim: Compact wireless sensor network coverage worked example record linking the scenario, coverage claim, test evidence, finding, decision, known limit, owner, retest trigger, and reusable method boundary. They keep example evidence record tied to observable evidence.

The record should be short enough to update after maintenance. If it becomes long, split the example into separate claims by area, target list, route, or boundary.

30.12 Common Worked-Example Mistakes

30.13 Review Checklist

Before accepting a worked example, check:

Is the monitoring claim specific enough to approve or reject? Does the coverage type match the scenario? Is the redundancy level stated separately from the type? Which sensors count in the active operating state? Does the evidence include weak locations and reporting paths? Does the decision say accept, narrow, repair, or retest? Are limits visible to future readers? Is there an owner for repair or evidence updates? What exact change reopens the review?

30.14 Knowledge Check: Redundant Area Example

30.15 Knowledge Check: Duty-Cycle Example

30.16 Matching: Example Evidence

30.17 Ordering: Worked Example Review

30.18 Every Number Needs a Sizing Boundary

The body examples teach how to move from claim to evidence to decision. The deeper layer is the arithmetic boundary behind those decisions. A worked example may include a density formula, a node count, a duty-cycle estimate, or a k-coverage target, but none of those numbers is meaningful until the review names the scope and state that the number describes. The same calculation can be honest for a fenced corridor, misleading for a whole room, and useless after a gateway move.

Two observations govern this part of every number needs a sizing boundary: scope and state and Count driver. Their arrangement in the diagram at Figure 30.3 reveals whether the proposed boundary is complete.

Compare scope and state and Count driver first in the illustration at Figure 30.3: one adds a distinct review condition, while the other adds a distinct review condition. Only then bring in area, target, route, which defines what the design promises. This three-part reading supports the stated result: WSN deployment sizing record linking claim, count driver, candidate size, gateway evidence, power evidence, pilot finding, decision, known limit, owner, and retest trigger. It is the chapter’s bridge back to every number needs a sizing boundary.

That is why a count should be written as a review statement rather than as a bare answer. For example: “For a 100 m by 100 m area, with 10 m sensing radius, random placement, disk-sensing assumption, and 95% expected one-coverage, the planning count is about 96 nodes before gateway, power, and field-margin checks.” This statement is longer than “96 nodes,” but it says what must be true for the answer to be reusable. If the deployment changes to 99%, k=2, triangular planned placement, or a corridor/barrier claim, the earlier number is not simply adjusted by intuition; it is recalculated under the new boundary.

The review also separates arithmetic precision from field certainty. A formula can be evaluated exactly while still relying on a rough sensing radius, a simplified boundary, or a random-placement model that does not match installation practice. The point of the depth layer is not to make the worked example mathematical for its own sake. It is to show where a number came from, what it excludes, and which evidence would make it safe enough to accept.

30.19 Work the Random-Deployment Count

For nodes scattered randomly under a Poisson model, with density lambda nodes per square metre and sensing radius Rs, the expected fraction of points covered by at least one sensor is p = 1 - e^(-lambda*pi*Rs^2). Rearranging gives lambda = -ln(1-p) / (pi*Rs^2). The planning count is then N = lambda*A for area A. This is a model-based estimate, not a field guarantee.

Use a concrete worked example. Suppose the area is 100 m by 100 m, so A = 10,000 m^2, and the accepted sensing radius is 10 m, so pi*Rs^2 = 314.16 m^2. For 95% expected one-coverage, -ln(0.05) = 2.996, so lambda = 2.996 / 314.16 = 0.00954 nodes per square metre. Multiplying by 10,000 gives 95.4 nodes, so the planning count rounds up to 96 nodes before margin and deployment constraints.

The table shows the diminishing return that a bare diagram hides. Raising the target from 95% to 99% does not mean four percent more nodes. It changes the mean coverage term from about 3.0 to about 4.6, which raises the planning count from 96 to 147 nodes in this example. The reviewer should then ask whether the higher target is justified by the missed-event consequence, and whether field constraints make random placement a fair model.

30.20 k-Coverage Placement Meaning

Two refinements change what the worked number means. First, k-coverage is not a small label added after the count. If the requirement says each point must be sensed by at least two independent sensors, the review is now about the probability of fewer than two sensors at a point, not merely the probability of zero sensors. Under the same Poisson model, the number of sensors covering a point has mean mu = lambda*pi*Rs^2. To make under-coverage unlikely, the mean usually needs to sit above the required k value. The example must say whether redundancy is a resilience requirement, a localization requirement, or just a comfort phrase.

Second, placement model changes the count. Random scattering creates accidental overlaps and gaps, so the formula estimates expected coverage rather than a deterministic guarantee. A planned grid or triangular placement can use the field more efficiently, but only if the team can actually install nodes at those positions and maintain their range assumptions. A warehouse with fixed mounting points, a forest with dropped sensors, and a fenced perimeter with line-of-sight constraints are different worked examples even if their area is similar.

The safe review habit is to keep the count driver visible. Area coverage counts scale with region size and sensing footprint. Point coverage counts scale with target list and redundancy. Barrier coverage counts scale with boundary segments and crossing tests. Path coverage counts scale with route segments and timing windows. A worked example that mixes those drivers should be split before the arithmetic is accepted.

30.21 Knowledge Check: Coverage Count Meaning

30.22 Summary

WSN coverage worked examples should teach review discipline. The strongest examples start with a precise claim, name the coverage type and redundancy level, check the active operating state, compare the claim with field evidence, and record a decision with limits and retest triggers.

Do not let an example drift into unsupported certainty. A useful example shows how to reason from evidence, how to handle weak locations, how to keep energy schedules tied to coverage, and how to prevent one scenario from being reused outside its proof.

30.23 Key Takeaway

Worked coverage examples should connect the claim, weak-location evidence, active state, decision, limit, owner, and retest trigger before any number or drawing is treated as accepted coverage.

30.24 Concept Relationships

WSN Coverage: Problem Types gives the detailed area, point, barrier, path, and hybrid formulations. WSN Coverage Algorithms explains how verification, active-set, rotation, and repair algorithms support example decisions. WSN Coverage Implementations turns accepted example decisions into implementation records. WSN Deployment Sizing expands sizing arithmetic, pilot evidence, and count-driver review.

30.25 What’s Next

WSN Coverage Implementations for turning worked-example decisions into deployment records. WSN Coverage: K-Coverage and Rotation for redundant coverage and active-set scheduling. WSN Energy Duty Cycling for schedule evidence when sleep states affect coverage.