M/M/c Queue: Multi-Server Queueing Model
Also known as: Multi-Server Erlang Queue, c-Server Markovian Queue, Erlang-C Queue, Çok Sunuculu M/M/c Kuyruğu
The M/M/c queue is a multi-server stochastic model in which customers arrive according to a Poisson process at rate λ, are served by c identical servers each with exponentially distributed service times at rate μ, and wait in a single common queue when all servers are busy. Systematized within classical queueing theory and thoroughly treated by Gross and Harris (1998), it extends the simpler M/M/1 model to settings with parallel servers, making it the foundational tool for capacity planning in service systems.
Key highlights
- Exact closed-form steady-state formulas require no simulation, enabling rapid what-if analysis.
- Directly generalizes the M/M/1 model, preserving tractability while handling realistic multi-server settings.
- The Erlang-C waiting probability provides a single, interpretable metric widely accepted in service-level agreements.
- Scales naturally: increasing c in the formula immediately shows the marginal effect of each additional server.
Intuition
This section is available to Pro members. Upgrade to Pro
How it works
This section is available to Pro members. Upgrade to Pro
When to use it
Use M/M/c when arrivals can be modelled as Poisson, service times are approximately exponential, multiple parallel servers share a single queue, and the system operates under steady-state conditions (ρ < 1). It is appropriate for telephone call centers, hospital admissions desks, airport check-in counters, and similar facilities. Limitations include the requirement of infinite waiting room, no customer balking or reneging, and the exponential service-time assumption; for non-exponential service, M/G/c or simulation should be preferred.
Strengths & limitations
- Exact closed-form steady-state formulas require no simulation, enabling rapid what-if analysis.
- Directly generalizes the M/M/1 model, preserving tractability while handling realistic multi-server settings.
- The Erlang-C waiting probability provides a single, interpretable metric widely accepted in service-level agreements.
- Scales naturally: increasing c in the formula immediately shows the marginal effect of each additional server.
- Assumes Poisson arrivals and exponential service times, which may not hold in heavy-tailed or batch-arrival settings.
- Requires strict stationarity and ρ < 1; time-varying demand or near-saturation conditions violate these assumptions.
- Does not accommodate finite waiting-room capacity, customer impatience (balking/reneging), or priority classes without model extensions.
- All c servers must be identical in speed; heterogeneous server pools require more complex models.
Common pitfalls
This section is available to Pro members. Upgrade to Pro
Applications
This section is available to Pro members. Upgrade to Pro
Frequently asked
How does M/M/c differ from the Erlang-C model?
They are mathematically identical. 'Erlang-C' is the name given to the waiting probability formula P(W > 0) derived from the M/M/c steady-state equations, and is widely used in call-center contexts. The M/M/c label is the Kendall-notation name for the same queueing system. Using either term refers to the same underlying model with c parallel servers, Poisson arrivals, and exponential service.
What happens when ρ approaches 1?
As traffic intensity ρ → 1 from below, both the mean queue length L_q and the mean waiting time W_q grow without bound in a nonlinear (hyperbolic) fashion. Even small increases in arrival rate near saturation cause disproportionately large increases in congestion. This sensitivity makes accurate estimation of λ and μ critical when operating near full utilization.
Can M/M/c handle time-varying arrival rates?
The standard M/M/c model assumes stationary (time-homogeneous) Poisson arrivals and does not directly handle non-stationary demand. Practitioners often apply it piecewise over short time intervals where demand is approximately constant, or use the Pointwise Stationary Approximation. For systems with strong time-of-day patterns, fluid approximations or simulation are generally more accurate.
Sources
- 1.Gross, D., & Harris, C. M. (1998). Fundamentals of Queueing Theory (3rd ed.). Wiley.ISBN 978-0-471-17083-9
You have read it. What now?
Cite this page
ScholarGate. (2026, June 2). M/M/c Queue. ScholarGate. https://scholargate.app/operations-research/mmc-queue