Inputs
Choose a model, set rates, and compute queue performance.
Example Data Table
These scenarios show typical inputs and key outputs.
| Scenario | λ (/hr) | μ (/hr/server) | c | ρ | Pw | Wq (min) | W (min) | Lq | L |
|---|---|---|---|---|---|---|---|---|---|
| Balanced workload | 18 | 24 | 2 | 0.375 | 0.205 | 0.41 | 2.91 | 0.123 | 0.873 |
| Moderate load | 30 | 20 | 3 | 0.500 | 0.237 | 0.47 | 3.47 | 0.237 | 1.737 |
| High load | 45 | 30 | 2 | 0.750 | 0.643 | 2.57 | 4.57 | 1.929 | 3.429 |
Formula Used
Common definitions
- λ: arrival rate.
- μ: service rate per server.
- c: number of parallel servers.
- ρ (utilization per server): ρ = λ / (cμ).
M/M/c (infinite queue, Erlang C)
- Offered load: a = λ / μ.
- P0 = 1 / [ Σ(n=0 to c−1) (a^n / n!) + (a^c / (c! (1−ρ))) ].
- Pw = (a^c / (c! (1−ρ))) · P0.
- Lq = (Pw · ρ) / (1−ρ).
- Wq = Lq / λ, W = Wq + 1/μ, L = λW.
- Optional service level for a target wait t: P(Wq ≤ t) = 1 − Pw · exp(−(cμ − λ)t).
M/M/c/K (finite capacity)
- States n = 0..K represent customers in system.
- Arrival rate into state n: λn = λ for n < K, else 0.
- Service rate out of state n: μn = min(n, c) · μ.
- Compute probabilities using recursion: Pn = P(n−1) · (λ / μn), then normalize.
- Blocking probability: Pblock = PK, and effective throughput: λeff = λ(1 − Pblock).
- L = Σ nPn, Ls = Σ min(n,c)Pn, Lq = L − Ls, W = L/λeff, Wq = Lq/λeff.
How to Use This Calculator
- Select a queue model based on whether capacity is limited.
- Enter servers c and your arrival rate λ with its time basis.
- Provide service information as μ or as mean service time.
- Optional: set a target wait time to estimate service level.
- Optional: enter costs to compare staffing versus delay tradeoffs.
- Press Submit to view results above the form, then export.
Arrival Rate and Service Capacity
The calculator treats arrivals as a Poisson flow with rate λ and service as exponential with rate μ per server. After unit conversion, both rates are handled per hour. Total service capacity is cμ, so λ=30/hr, μ=12/hr, c=3 implies capacity 36/hr and offered load a=λ/μ=2.5 server-hours. Mean service time equals 1/μ; with μ=12/hr, average service takes 5 minutes. When arrivals and service are given per minute, the tool converts them to the same hourly basis before computing probabilities and times.
Utilization Targets for Stable Operation
Utilization per server is ρ=λ/(cμ). In the infinite-queue model, stability requires ρ<1, but practical designs aim lower. Many service systems plan around ρ=0.70–0.85 to limit delay spikes. For example, moving from ρ=0.90 to ρ=0.80 can cut waiting time sharply because the term 1/(1−ρ) grows rapidly. For contact centers, a 10% demand surge can push ρ above targets quickly.
Interpreting Probability of Waiting
The probability of waiting Pw (Erlang C) estimates the chance an arrival finds all servers busy. With c=4 and ρ near 0.80, Pw is often between 0.20 and 0.50 depending on a. A Pw of 0.35 means roughly 35 out of 100 arrivals queue before service, even though the system remains stable.
Waiting Time and Queue Length Metrics
Expected waiting time Wq and expected queue length Lq are linked by Little’s Law, Lq=λWq, using consistent time units. If λeff=28/hr and Wq=1.5 minutes, then Wq=0.025 hours and Lq=28×0.025=0.70 customers. Total time in system is W=Wq+1/μ, and L=λeff W summarizes overall congestion.
Finite Capacity and Blocking Effects
For M/M/c/K, the system holds at most K customers, so new arrivals may be blocked when the state reaches K. The blocking probability Pblock is PK, and effective throughput becomes λeff=λ(1−Pblock). If λ=50/hr and Pblock=0.04, then λeff=48/hr. Blocking lowers queues but represents lost demand or diverted work.
Staffing and Cost Sensitivity Planning
Cost fields translate performance into an hourly tradeoff. Staffing cost is c×(cost per server-hour), waiting cost is Lq×(cost per customer-hour), and lost-demand cost uses λPblock×(cost per lost customer). Example: cost server=18, c=3 gives 54/hr. If Lq=0.90 and cost wait=40, delay cost is 36/hr. Adding one server may raise staffing to 72/hr but reduce Lq to 0.30, lowering delay cost to 12/hr.
FAQs
Which model should I choose?
Use M/M/c for systems with unlimited waiting space or no hard cap. Use M/M/c/K when only K customers can be in the system, such as limited seats, buffers, or call slots, and arrivals may be rejected.
What units should I enter for λ and μ?
Enter rates in any supported time basis. The calculator converts both to per-hour internally. If you provide mean service time, it converts to μ as 1 divided by the mean time, then uses the same unit conversion rules.
Why do I see an instability warning?
In the infinite-queue model, stability requires ρ=λ/(cμ) to be below 1. When ρ is 1 or higher, expected waiting and queue length grow without bound. Reduce λ, increase μ, or increase c.
What does Pw mean in practice?
Pw is the chance an arrival must wait because all servers are busy. It does not mean the customer waits long, only that they join the queue. Pair Pw with Wq to understand both frequency and duration of waiting.
How are Wq and Lq related?
They are connected by Little’s Law: Lq = λeff × Wq, using consistent time units. If Wq is in minutes, it is converted to hours before multiplying by λeff. This relationship helps validate results and spot input mistakes.
How should I use the cost fields?
Enter hourly staffing cost per server and an hourly penalty per waiting customer to compare staffing versus delay. For finite capacity, add a cost per blocked customer to value lost demand. The total cost is reported per hour for quick scenario testing.