M/M/c Queue: Multi-Server Queueing Model
M/M/c Multi-Server Queue · 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.
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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.
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
- Gross, D., & Harris, C. M. (1998). Fundamentals of Queueing Theory (3rd ed.). Wiley. ISBN: 978-0-471-17083-9
How to cite this page
ScholarGate. (2026, June 2). M/M/c Multi-Server Queue. ScholarGate. https://scholargate.app/en/operations-research/mmc-queue
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Erlang C ModelOperations Research↔ compare
- Little's LawOperations Research↔ compare
- M/M/1 QueueOperations Research↔ compare