Calculator inputs
Enter either the exponential rate or the mean waiting time. The page then computes expected wait, probability values, percentiles, interval probabilities, and memoryless interpretations.
Example data table
This worked example uses a restaurant queue with exponential waiting times. Rate λ = 0.12 per minute, so the expected wait is about 8.333333 minutes.
| Scenario | Value |
|---|---|
| Rate λ | 0.120000 per minute |
| Expected wait E[T] | 8.333333 minutes |
| Probability a guest waits more than 10 minutes | 30.1194% |
| Probability a guest is seated within 10 minutes | 69.8806% |
| Probability a wait falls between 5 and 15 minutes | 38.3513% |
| 90th percentile waiting time | 19.188209 minutes |
| Expected extra wait after already waiting 10 minutes | 8.333333 minutes |
| Expected total wait given a customer already waited 10 minutes | 18.333333 minutes |
Formula used
If waiting time T follows an exponential distribution with rate λ > 0, these formulas drive the calculator:
f(t) = λe-λt, for t ≥ 0
F(t) = P(T ≤ t) = 1 - e-λt
S(t) = P(T > t) = e-λt
E[T] = 1 / λ
Var(T) = 1 / λ², SD(T) = 1 / λ
Median = ln(2) / λ, Q(p) = -ln(1 - p) / λ
P(a < T ≤ b) = F(b) - F(a) = e-λa - e-λb
E[T - x | T > x] = 1 / λ, E[T | T > x] = x + 1 / λ
How to use this calculator
- Choose whether you know the exponential rate or the mean waiting time.
- Enter a time point to evaluate density, cumulative probability, and survival probability.
- Add interval bounds if you want the chance of waiting within a range.
- Enter a percentile such as 90 to estimate a service-level threshold.
- Set the graph maximum to control the displayed time horizon.
- Press the calculate button and review the result block above the form.
- Use the CSV and PDF buttons to export the computed metrics.
Interpretation notes
The exponential model assumes a constant hazard rate. That means the chance of service finishing in the next instant does not depend on how long the wait has already lasted.
This is useful for queues, arrivals, component failures, and response times when the memoryless assumption is reasonable.
Frequently asked questions
1) What does expected waiting time mean here?
It is the average waiting time predicted by an exponential model. If the rate is λ, the expected wait equals 1 divided by λ. Over many repeated cases, the long-run average wait should approach that value.
2) Should I enter rate or mean waiting time?
Enter whichever value you know. The page converts between them automatically because mean wait equals 1/λ. If your source gives average waiting time, choose mean mode. If it gives events per unit time, choose rate mode.
3) Why is the exponential model called memoryless?
The expected future wait does not depend on how long you already waited. If a process truly follows an exponential distribution, waiting 10 more minutes after already waiting 10 minutes has the same expected additional duration as waiting from the start.
4) At a certain restaurant, the distribution of wait times is exponential. How does this help?
You can estimate average wait, chance of seating within a target, chance of exceeding a threshold, and service-level percentiles. Restaurant managers can use those outputs to set expectations, staffing plans, and quoted wait windows.
5) What is the difference between CDF and survival probability?
The CDF gives the chance a wait ends by time t. Survival gives the chance the wait still exceeds t. They always add to 1 for nonnegative waiting times in this model.
6) What does the percentile output tell me?
A percentile gives the waiting time below which a chosen share of cases should fall. For example, the 90th percentile is a useful service-level threshold because about 90% of waits should finish by that time.
7) When should I avoid the exponential distribution?
Avoid it when the hazard rate clearly changes over time, when waits have strong minimum durations, or when data show heavier tails than the exponential curve. In those cases, gamma, Weibull, lognormal, or empirical methods may fit better.
8) Do units matter in this calculator?
Units matter for interpretation, not for the formulas themselves. If λ is per minute, the mean and percentiles are in minutes. Use one consistent time unit across every field for accurate results.