Simulator What is M/M/c? Use Cases
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QueueSim -- M/M/c Queuing Simulator

Predict wait times, queue lengths, and server utilization. Configure, simulate, compare.

Free and open. Built by ChiAha.

Arrivals Queue Servers
Speed: 1x
Service Time Distribution:

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Configure your inputs and click Run Simulation

8,760 simulated hours of queuing in the blink of an eye. Each run adds a scenario for comparison.

What is M/M/c?

M/M/c is the foundational model in queuing theory. The notation describes three things about the system:

M
Markovian arrivals
Poisson process -- memoryless, random inter-arrival times
M
Markovian service
Exponential service times -- also memoryless
c
Parallel servers
Multiple identical servers sharing a single queue

Little's Law & Key Relationships

The most important result in queuing theory. It holds for any stable queuing system, regardless of arrival or service distribution:

L = λ · W
Average number in system = arrival rate x average time in system

From this single law, all the standard metrics follow. If you know any two of L, λ, and W, you can derive the third.

ρ = λ / (c · μ)
Server utilization must be less than 1 for the system to be stable. As ρ approaches 1, wait times grow without bound.
W = Wq + 1/μ
Total time in the system is the wait in queue plus the service time. Reducing either one improves the customer experience.
L = Lq + λ/μ
Customers in system = those waiting + those being served. Adding servers reduces Lq but with diminishing returns.

When to Use This Tool

Any system with random arrivals and finite servers is a queuing system. Use QueueSim for:

Capacity planning -- How many servers do you need to meet a wait-time target?
Staffing decisions -- Where should you add or remove staff across the day?
Service level targets -- What utilization keeps wait times acceptable?
What-if analysis -- Compare scenarios with different arrival patterns or server counts.

QueueSim goes beyond steady-state formulas. It runs a discrete event simulation -- modeling every individual customer through arrival, queueing, service, and departure -- across a full year of simulated time. This captures time-varying demand, shift changes, and the natural variability that analytic formulas cannot.

Why simulate what you can calculate?

Closed-Form Math (Erlang-C)

  • Works perfectly for steady-state M/M/c
  • Assumes constant arrival rate, exponential service, identical servers
  • Breaks when reality doesn't match assumptions

Discrete Event Simulation

  • Models each customer individually through the system
  • Handles time-varying arrivals, non-exponential service, shift changes
  • 8,760 simulated hours produce statistically stable results
"The tool you need when the math runs out"

AI / LLM Estimation

  • Can estimate averages from training data
  • Cannot simulate cascading effects of variability
  • Cannot run 8,760 hours of stochastic flow
"AI uses QueueSim -- it doesn't replace it"

QueueSim starts with the M/M/c fundamentals, then goes beyond them. Every simulation run models individual customers through random arrivals, queuing, server assignment, and service -- producing results that account for the variability that formulas assume away.

Works for Any Queuing System

Call Centers

How many agents do you need per shift to keep hold times under 2 minutes?

IT Help Desks

Model ticket arrivals and technician capacity to predict resolution backlogs.

Retail Checkout

Find the right number of cashiers for peak hours without overstaffing off-peak.

Bank Tellers

Balance customer wait times against staffing costs across branch hours.

Government Offices

DMV, permit offices, passport agencies -- optimize counter staffing for citizen satisfaction.

Food Service

Restaurants, cafeterias, drive-throughs -- model customer flow through ordering and pickup.

Advanced: Time-Varying Analysis

Real systems have time-varying demand. Rush hours, shift changes, and seasonal patterns mean that a single arrival rate rarely tells the whole story. Time-varying analysis models each hour independently -- the analysis that closed-form M/M/c formulas cannot provide.

Coming soon to QueueSim -- hourly arrival rates and variable staffing.

Need Healthcare-Specific Simulation?

QSimHealth is our purpose-built staffing simulator for emergency departments. It models provider types (MD, PA/NP), treatment distributions, and staffing costs specific to healthcare operations.

Learn About QSimHealth