Queueing theory studies waiting lines using probability models, and the M/M/1 model is the standard starting point for many service systems. It applies to one server with random arrivals and random service times, such as a help desk, checkout lane, or packet router. This cheat sheet helps students quickly connect the assumptions, notation, stability condition, and performance formulas needed for homework and applied modeling.
It is especially useful when interpreting what each metric means in a real system.
In an M/M/1 queue, arrivals follow a Poisson process with rate lambda, service times are exponential with rate mu, and there is one server. The most important condition is stability: lambda < mu, or equivalently rho = lambda / mu < 1. Core results include L = rho / (1 - rho), Lq = rho^2 / (1 - rho), W = 1 / (mu - lambda), and Wq = lambda / (mu(mu - lambda)).
Little’s Law connects average number and average time through L = lambda W and Lq = lambda Wq.
Key Facts
- The M/M/1 model assumes Poisson arrivals, exponential service times, one server, infinite waiting room, and first-come, first-served service.
- The arrival rate lambda is the average number of customers arriving per unit time, and the service rate mu is the average number served per unit time when the server is busy.
- The utilization is rho = lambda / mu, which is the long-run fraction of time the server is busy.
- The queue is stable only when lambda < mu, or rho < 1, because the server must work faster on average than customers arrive.
- The average number in the system is L = rho / (1 - rho) = lambda / (mu - lambda).
- The average number waiting in line is Lq = rho^2 / (1 - rho) = lambda^2 / (mu(mu - lambda)).
- The average time in the system is W = 1 / (mu - lambda), and the average waiting time in line is Wq = lambda / (mu(mu - lambda)).
- Little’s Law states L = lambda W and Lq = lambda Wq for stable systems using the effective arrival rate.
Vocabulary
- Arrival rate
- The arrival rate lambda is the average number of customers or jobs entering the system per unit time.
- Service rate
- The service rate mu is the average number of customers or jobs a busy server can complete per unit time.
- Utilization
- Utilization rho is the fraction of time the server is busy, calculated as rho = lambda / mu.
- Stable queue
- A stable queue has lambda < mu so that long-run average queue lengths and waiting times remain finite.
- System time
- System time W is the total average time a customer spends waiting in line plus receiving service.
- Little’s Law
- Little’s Law states that the average number in a stable system equals the effective arrival rate times the average time in the system.
Common Mistakes to Avoid
- Using M/M/1 formulas when lambda >= mu is wrong because the queue is unstable and the steady-state averages do not exist.
- Confusing W with Wq is wrong because W includes both waiting time and service time, while Wq includes only time spent waiting in line.
- Mixing time units for lambda and mu is wrong because both rates must use the same time unit before calculating rho or any performance measure.
- Interpreting rho as the probability a customer waits is wrong because rho is server utilization, although in M/M/1 it also equals the probability the server is busy.
- Forgetting that service time mean is 1 / mu is wrong because mu is a rate, not the average service duration itself.
Practice Questions
- 1 Customers arrive at a help desk at lambda = 6 per hour, and the server can handle mu = 10 per hour. Find rho, L, W, and Wq.
- 2 A router receives packets at lambda = 80 packets per second and serves them at mu = 100 packets per second. Determine whether the queue is stable and find Lq.
- 3 For an M/M/1 system with average system time W = 0.5 hours and arrival rate lambda = 3 per hour, use Little’s Law to find L.
- 4 Explain why increasing utilization from 0.80 to 0.95 can cause a much larger increase in waiting time than the small change in utilization might suggest.