Queueing Theory Calculator (M/M/c)
Model a service queue using the M/M/c (Erlang-C) model: enter arrival rate λ, service rate μ and number of parallel servers c to get utilisation ρ, average queue length Lq, waiting time Wq, time in system W and the probability a customer must wait.
Fraction of time each server is busy — must be < 1 for a stable queue
- 1
Offered traffic (a = λ ÷ μ)
5 ÷ 8 = 0.625Total work offered to all servers per unit time, measured in Erlangs. - 2
Per-server utilisation (ρ = a ÷ c)
0.625 ÷ 1 = 0.6250
How does this calculator work?
M/M/c queue: ρ = λ/(cμ) must be < 1 for stability. Erlang-C formula gives P(customer waits). Key results: Lq = C·ρ/(1−ρ) customers in queue; W = L/λ average sojourn time. As ρ → 1, queue length → ∞. Design for ρ ≤ 0.7–0.8 for acceptable response times.
Formula
How this is calculated
The M/M/c model (Markovian arrivals, Markovian service, c servers) is the workhorse of basic queueing theory. "M/M" means both the inter-arrival times and service times follow exponential distributions — equivalently, arrivals form a Poisson process with rate λ and each server completes service at rate μ. With c parallel servers the system is stable only when ρ = λ/(cμ) < 1; otherwise the queue grows without bound.
The Erlang-C formula gives C(c, a) — the probability a customer must wait before being served — where a = λ/μ is the offered traffic in Erlangs: C = [a^c/(c!(1−ρ))] × P₀, and P₀ is the probability all servers are idle, computed from the steady-state balance equations. Once C is known, Little's law (L = λW) links the four performance measures: Lq = C·ρ/(1−ρ) is the average number waiting; L = Lq + a is the average in the whole system; Wq = Lq/λ is the average queue wait; W = L/λ is the average total sojourn time.
Assumptions and limitations: FCFS discipline, infinite waiting room, infinite population, and exponential distributions. Real queues often have bounded buffers, finite populations, non-exponential service times (M/G/c), or priorities (M/M/c/c+d). Use this model for order-of-magnitude capacity planning rather than precise prediction. For ρ close to 1, small measurement errors cause large swings in Lq — this is the "knee of the curve" effect where queueing performance degrades sharply.
Frequently asked questions
ρ = λ/(cμ) is the fraction of time each server is busy on average. ρ = 0.8 means each server is busy 80% of the time. As ρ → 1, average queue length Lq → ∞ — even a small imbalance between arrivals and service capacity causes explosive congestion. Practical systems are typically designed for ρ ≤ 0.7–0.8.
Erlang-C gives the probability C that an arriving customer finds all servers busy and must wait. It is used in telephone network capacity planning, call centre staffing, and any service system where customers queue rather than being turned away. (Erlang-B, by contrast, models loss systems where a customer is blocked and leaves if all servers are busy.)
M/M/1 is M/M/c with c = 1. For one server, ρ = λ/μ, the Erlang-C formula simplifies to C = ρ, P₀ = 1 − ρ, Lq = ρ²/(1−ρ), and W = 1/(μ − λ). The M/M/1 formulas are clean closed-form expressions and are widely used as baseline estimates for more complex systems.
Also known as
TG we-Calculate Editorial Team. (2026). Queueing Theory Calculator (M/M/c) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/queueing-theory-calculator
TG we-Calculate Editorial Team. "Queueing Theory Calculator (M/M/c)." TG we-Calculate. 2026. https://we-calculate.com/calculator/queueing-theory-calculator.
TG we-Calculate Editorial Team, "Queueing Theory Calculator (M/M/c)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/queueing-theory-calculator
@misc{wecalculate_queueing_theory_calculator, title = {Queueing Theory Calculator (M/M/c)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/queueing-theory-calculator}}, year = {2026}, note = {TG we-Calculate} }
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