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.
Understanding Queueing Theory (M/M/1) Reference
The two letters M in the model name describe memoryless randomness. For arrivals, memoryless means that knowing how long it has been since the last arrival does not predict the next one. For service, it means that a job already in service has no predictable remaining time.
This can feel unrealistic for a fixed task such as scanning one item. It can be reasonable for jobs with highly variable lengths, such as technical support calls or network packets.
The model uses rates, so students must keep time units consistent. If arrivals are counted per minute, service capacity must be counted per minute too.
Randomness is the reason a queue can form even when the server has enough average capacity. Customers do not arrive at perfectly even intervals. Several may arrive close together, then no one may arrive for a while.
Service times vary as well. A short job might be followed by an unusually long one. During that long service, new arrivals accumulate.
Averages smooth out this behavior, but they do not remove it. An M/M/1 result describes long-run average performance. It does not promise that every customer experiences the average wait.
Utilization has a strongly non-linear effect near full capacity. Suppose a server can complete ten jobs per hour. With eight arrivals per hour, the average total time in the system is one half hour.
With nine arrivals per hour, it rises to one hour. Adding only one extra arrival per hour doubles the average time. This sharp increase happens because there is less spare capacity to clear temporary buildups.
A system that appears busy but manageable can become slow very quickly. In practice, managers often keep some unused capacity because it protects customers from normal fluctuations.
Little’s Law is especially useful because it turns a count into a time, or a time into a count. Imagine a clinic receives twelve patients per hour and each patient spends an average of fifteen minutes in the full process. Fifteen minutes is one quarter of an hour.
The average number of patients in the system is therefore three. This relationship does not require exponential service times.
It works much more broadly when the system is stable and measurements use the same system boundary. For example, a count taken only in the waiting room must be paired with waiting time only, not total visit time.
When solving problems, first state exactly what counts as arrival, service, waiting, and departure. A call center may measure from the moment a caller joins the phone queue, while a hospital may measure from check-in. Check whether people ever leave before service, because then the effective arrival rate differs from the attempted arrival rate.
Check whether there is truly one server. A shared desk with two staff members needs a different model.
Finally, treat outputs as averages rather than schedules. The model is best used as a baseline for judging whether variability, limited space, priority rules, or extra servers need to be considered.