M/M/c Queueing System Simulator

M/M/c Queue

⚠ ρ ≥ 1 — unstable, queue grows without bound

Arrival rate λ 1.0 /min

Service rate μ 1.0 /min (per server)

Servers c 2

Sim speed 1×

Sim clock

0.0 min

Jobs served

In queue now

ρ = λ/(cμ)

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Empirical vs. Erlang-C

Metric	Simulated	Theory
Avg wait Wq	–	–
Avg queue Lq	–	–
Utilization	–	–

Stats accumulate from the moment you last hit Reset — the longer it runs, the closer "Simulated" tracks "Theory".

How it works

This is a genuine discrete-event simulation, not a formula readout. Jobs arrive one at a time with inter-arrival gaps drawn from a real exponential distribution with rate λ (a Poisson arrival process). Each job is handed to the first free server out of c parallel servers; if all c are busy it waits in a FIFO queue. Every server draws its own exponential service time with rate μ per job. The engine tracks each job's real arrival time, service-start time and departure time, and only from those timestamps does it compute the empirical average wait, average queue length and server utilization — nothing is looked up from a table.

Erlang-C gives the closed-form theoretical prediction for the same M/M/c system, computed independently from λ, μ, c:

a = λ/μ   (offered load)
ρ = a/c   (traffic intensity per server)
P0 = [ Σ(k=0..c-1) aᵏ/k! + aᶜ/(c!·(1−ρ)) ]⁻¹
C(c,a) = (aᶜ/(c!·(1−ρ)))·P0   (Erlang-C: prob. an arrival waits)
Wq = C(c,a) / (cμ − λ)
Lq = λ·Wq

λ (arrival rate) — average number of jobs arriving per minute; higher λ means a busier system.
μ (service rate) — average jobs one server can finish per minute; higher μ means faster processing.
c (servers) — number of parallel automation workers/agents handling jobs simultaneously.
ρ = λ/(cμ) — traffic intensity. When ρ < 1 the queue reaches statistical equilibrium and the simulated averages converge to the Erlang-C prediction. When ρ ≥ 1, arrivals outpace total service capacity and the queue genuinely grows without bound — this is a real, emergent result of the simulation loop, not a scripted effect.

Real-world relevance: M/M/c is the foundational capacity-planning model for automated service systems — RPA bot pools, ticket-routing queues, call-center staffing, API gateway worker counts — anywhere jobs arrive randomly and a fixed number of parallel workers process them.

Queued job Busy server Idle server