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Modeling Outages: Poisson Processes and Verizon’s Refunds Explained

UUnknown

2026-02-23

9 min read

Estimate outage odds and expected refunds using Poisson and exponential models—step-by-step and ready for simulation.

Hook: When a network outage costs you time — and sometimes money — what are the real odds?

Network outages feel random and infuriating: missed calls, failed homework uploads, and a customer service loop that takes forever. If you've ever wondered whether that $20 refund offered after a Verizon outage is fair — or what the probability really is that you'll be affected again — this guide turns those frustrations into numbers you can use. We'll walk through how to model outages with a Poisson process, use the exponential distribution for time between failures and repair durations, and compute the expected value of compensation. By the end you'll have a repeatable exercise you can run with your own parameters (or classroom assignment) and practical tips to claim refunds.

Why this matters in 2026: trends and context

Late 2025 and early 2026 saw growing regulatory pressure on carriers to improve transparency and compensation after major outages. Telecom companies are piloting automated refund systems and status APIs, while advocacy groups push for clearer Service Level Agreements (SLAs) at the consumer level. That makes mathematical modeling more than academic — it helps consumers and regulators quantify service reliability and expected compensation. For students and teachers, outage modeling also links probability theory to a pressing real-world problem.

Key concepts you'll use

Poisson process: Models the count of outages in time, assuming events occur independently at a constant average rate.
Exponential distribution: Models the time between events (interarrival times) or repair durations when the memoryless property is reasonable.
Mean time between failures (MTBF): The average interval between outages; for a Poisson process, MTBF = 1/λ.
Expected value: Average compensation or downtime you can expect per unit time.
Queueing / repair modeling: Models how many repair crews and their rates affect outage duration.

Inverted pyramid: most important results first

Here are the headline formulas you can use right away:

Probability of at least one outage in time t: P(≥1 in t) = 1 − exp(−λt).
Expected number of outages in time t: E[N(t)] = λt (Poisson property).
Probability outage lasts longer than threshold τ (if durations ~ Exp(1/μ)): P(T > τ) = exp(−μτ).
Expected compensation per year (fixed amount C per qualifying outage): E[Compensation/year] = λ × P(duration>threshold) × C.
Mean time between failures (MTBF): MTBF = 1/λ.

Step-by-step practical exercise: estimating outage probabilities and expected compensation

We’ll walk through an example with realistic numbers you can change to match your situation or assignment.

Step 0 — Set assumptions (fill these in)

Average outage rate: λ = 3 outages per year (change to your estimate).
Mean outage duration: mean = 120 minutes (2 hours). If durations are exponential, μ = 1/mean = 1/2 hours⁻¹ = 0.5 hour⁻¹.
Compensation policy: $20 per qualifying outage (Verizon offered a $20 credit in some cases during 2025 outages; check your bill/policy for exact eligibility).
Qualification threshold: credits issued only if outage > 60 minutes (1 hour).

Step 1 — Compute MTBF

MTBF = 1/λ. For λ = 3/year:

MTBF = 1/3 year ≈ 0.333 years ≈ 122 days.

Step 2 — Probability at least one outage within 30 days

Compute t in years: t = 30/365 ≈ 0.08219 years. Then

P(≥1 in 30 days) = 1 − exp(−λt) = 1 − exp(−3 × 0.08219) ≈ 1 − exp(−0.2466) ≈ 0.2185.

Interpretation: With these assumptions, there's ≈ 21.9% chance of at least one outage in any 30-day window.

Step 3 — Probability a given outage exceeds the compensation threshold

Model outage durations as exponential with mean 2 hours → μ = 1/2 hour⁻¹ = 0.5.

P(duration > 1 hour) = exp(−μτ) = exp(−0.5 × 1) ≈ 0.6065.

Interpretation: About 60.65% of outages last more than one hour under this model.

Step 4 — Expected number of qualifying outages per year

Expected number of outages per year = λ = 3. Fraction qualifying = exp(−μτ) ≈ 0.6065.

So expected qualifying outages per year = λ × P(duration > threshold) = 3 × 0.6065 ≈ 1.8195.

Step 5 — Expected compensation per year

If compensation is a fixed $20 per qualifying outage:

E[Compensation/year] = 1.8195 × $20 ≈ $36.39 per year.

That's the long-term average a customer could expect under our assumptions.

Step 6 — Probability of k outages in a year (Poisson)

Poisson formula: P(N = k) = exp(−λ) × λ^k / k!. With λ = 3:

P(0 outages) = exp(−3) ≈ 0.0498 (≈ 5%).
P(1 outage) = exp(−3) × 3 ≈ 0.1494 (≈ 15%).
P(≥1 outage) = 1 − P(0) ≈ 0.9502 (≈ 95%).

Interpretation: Under λ = 3/year, it's very likely you get at least one outage annually, but qualifying outages are fewer since they must exceed the duration threshold.

Adding repair crew dynamics: a simple queueing view

So far we used the exponential for duration. If repair capacity matters (many overlapping outages, or limited crews), a queueing model helps. A basic way to approximate is to treat repair as a service process with rate μ per crew and c crews working in parallel.

Single-server approximation

If there's effectively one team and mean repair time = 2 hours (μ = 0.5 /hour), expected downtime per outage is 2 hours. Expected downtime per year = λ × mean repair time = 3 × 2 = 6 hours/year.

Proportion of time service is down: 6 hours / (365 × 24) ≈ 0.0685% of the year.

Multiple crews (c servers)

If there are c identical crews and repair times are exponential, the system behaves like an M/M/c queue for service—complex formulas give waiting probabilities. For many practical purposes, increasing c decreases expected waiting time and thus reduces qualifying outages (if threshold-sensitive).

Advanced strategies and 2026 developments

In 2026, expect these trends to shape outage modeling and compensation:

Automated compensation pipelines: Carriers pilot APIs that auto-detect qualifying outages and issue credits, reducing friction for customers and offering clearer data for model validation.
Better public datasets: Community-driven outage trackers and telecommunication transparency dashboards launched in late 2025 provide empirical λ and duration estimates you can plug into models.
Machine learning predictions: Carriers use predictive maintenance to reduce MTBF; students can compare Poisson assumptions (constant rate) to nonhomogeneous Poisson processes that allow time-varying λ.
Regulatory attention: Policymakers increasingly ask for consumer-facing SLA-like disclosures for average outage rates and compensation policies.

Practical tips: how to use these calculations to claim refunds

Turning math into action: here's a checklist you can follow when you're affected by a Verizon outage (or any carrier outage):

Document times precisely: log start and end times, the region affected, and device evidence (screenshots, error logs).
Check carrier policy: confirm the compensation threshold and whether credits are automatic or require a claim.
Calculate your expected compensation using your own λ and mean duration estimates; this gives you a negotiation baseline.
File the claim promptly with supporting logs. If you get an automated $20 credit, save the confirmation for records.
If denied, escalate: request a supervisor, cite your logs, and reference consumer protection policies (document dates of communications).
Use public outage trackers to corroborate your complaint—screenshots from these services are evidence of a region-wide disruption.

Classroom and homework extension: simulate this in Python or a spreadsheet

Simulating a Poisson process and exponential durations is a great assignment. Here is a short pseudocode you can translate to Python, R, or Excel.

Pseudocode (Python-like)

# parameters
lambda_per_year = 3
mean_duration_hours = 2
C = 20 # compensation per qualifying outage
threshold_hours = 1
T_years = 1

# simulate many years
N_sims = 100000
total_comp = 0
for s in range(N_sims):
    # number of outages this year ~ Poisson(lambda)
    k = poisson(lambda_per_year)
    comp = 0
    for i in range(k):
        duration = exponential(mean_duration_hours)
        if duration > threshold_hours:
            comp += C
    total_comp += comp

avg_comp = total_comp / N_sims
print('Simulated expected compensation/year =', avg_comp)

This will converge to the analytic value λ × P(duration>threshold) × C as N_sims grows.

Limitations and when models break down

Important caveats:

Nonhomogeneous behavior: Real outage rates can vary by season, infrastructure upgrades, or cascading failures; a constant λ is a simplification.
Duration distributions: Not all outage durations are memoryless; heavy-tailed distributions can change qualifying probabilities.
Compensation rules: Carriers may cap total credits, require multiple outages, or limit refunds to business accounts—always check policy language.
Independent events: Poisson assumes independent events; correlated failures (e.g., regional storms) violate that assumption and require cluster models.

Actionable takeaways

Use Poisson + exponential models to estimate outage risk and expected refunds quickly — they give transparent, interpretable results.
For a fixed credit C and threshold τ, expected annual compensation = λ × exp(−μτ) × C.
Validate assumptions against public outage data and carrier disclosures; in 2026, more data sources are becoming available for this purpose.
Document outages carefully — math helps you know what to claim, but evidence gets you the credit.

“Modeling outages turns anger into actionable numbers: you can estimate your expected compensation and make smarter claims.”

Wrapping up — why this helps students, teachers, and consumers

This exercise connects probability theory and queueing to a real consumer problem: understanding risk and expected compensation from network failures. For students, it's a practical application of Poisson processes and exponential distributions. For consumers, it gives a framework to evaluate carrier offers and prepare claims. For teachers, it’s a classroom lab: change λ, mean duration, or policy parameters and compare results.

Call to action

Try the exercise with your own data: estimate λ from public outage trackers or your past 2–3 years of logs, simulate in Python or a spreadsheet, and compute your expected yearly compensation. If you want a ready-to-run notebook or a classroom worksheet, click to download our free template and guide — then share your findings and questions in the comments so we can build better models together.

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