Queue Wait Time SLA Calculator
Estimate the probability that customers wait less than a target time using M/M/1 queueing theory. Compare against your service level agreement and find the service rate needed to meet your SLA target.
Configure Your Queue Parameters
Enter your arrival rate, service rate, and SLA target to calculate the probability that customers wait less than your threshold.
Quick Tips
- Arrival rate (λ): How many customers arrive per time unit
- Service rate (μ): How many customers can be served per time unit
- Keep μ > λ: Service rate must exceed arrival rate for a stable system
- Try a preset example to see typical configurations
P(Wq ≤ t) = 1 - ρ × e^{−(μ - λ)t}
M/M/1 Wait Time CDF Formula
Understanding Queue Wait Time SLA
What is a Wait Time SLA?
A Service Level Agreement (SLA) for wait times is a commitment that a certain percentage of customers will be served within a target time. For example, "80% of customers should wait no more than 20 seconds" is a common call center SLA.
The M/M/1 Queue Model
This calculator uses the M/M/1 queueing model, which is a foundational model in operations research with these assumptions:
- M (Markovian arrivals): Customers arrive according to a Poisson process with rate λ (memoryless arrivals)
- M (Markovian service): Service times follow an exponential distribution with rate μ (memoryless service)
- 1 (single server): There is exactly one server handling the queue
- FCFS: Customers are served in first-come-first-served order
- Infinite capacity: The queue can hold unlimited customers
Key Formulas
Stability Condition
For the queue to be stable (not grow indefinitely), the service rate must exceed the arrival rate:
ρ = λ/μ < 1 (i.e., μ > λ)
When ρ ≥ 1, the queue grows without bound and wait times are infinite.
Interpreting the Results
- Utilization (ρ): The fraction of time the server is busy. Higher utilization means longer waits.
- P(Wq ≤ t): The probability a customer waits at most t time units before service begins.
- SLA Met: If P(Wq ≤ t) ≥ target probability, your SLA is being met.
- Suggested Service Rate: The minimum μ needed to achieve the target SLA.
Real-World Applications
Call Centers
"80% of calls answered within 20 seconds" is a common industry standard for call center SLAs.
Web Services
API response time SLAs often target "95% of requests served in under 200ms".
Retail & Food Service
Drive-through and checkout lane wait time targets drive staffing decisions.
Healthcare
Emergency department triage targets and appointment scheduling optimization.
Limitations
The M/M/1 model makes simplifying assumptions that may not hold in real systems:
- Real arrivals may be bursty or scheduled, not Poisson
- Service times may vary widely or have minimum thresholds
- Most systems have multiple servers, not just one
- Customers may balk (leave before joining) or renege (leave while waiting)
- Service rates may vary by time of day or customer type
Frequently Asked Questions
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