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Operations Management and Strategic Management · Scheduling and Queuing Models

Queuing Theory: Concepts and Structure of a Queuing System

Updated 10 October 2026 · Fact-checked

Queuing theory studies waiting lines. A queuing system has a calling population, arrivals, a queue, service channels and departures. You describe it by arrival pattern, service pattern, number of servers, capacity, population and queue discipline. Kendall notation (A/B/c/K/N/D) summarises these. In exams, identify each component and decode the notation.

Understand Queuing Theory: Concepts and Structure

A queue forms when demand for a service is more than the service can handle at that moment. Customers at a bank, vehicles at a toll plaza, machines waiting for a repair technician and calls at a helpline are all examples. Queuing theory uses probability to predict waiting time and queue length, so a manager can balance the cost of waiting against the cost of adding service capacity.

Every queuing system has a few parts. The calling population (input source) is the pool from which arrivals come; it can be infinite (general public) or finite (10 machines of a factory). Arrivals join the queue. The service facility has one or more channels (servers). After service, units depart. The number of channels is the number of customers who can be served at the same time. The number of phases is the number of service stages in sequence.

The arrival process says how units come in. The usual assumption is that arrivals are random and follow a Poisson distribution, which means the time between arrivals is exponential. The average arrival rate is denoted by λ (units per time period). The service process says how long service takes. The usual assumption is exponential service time, with average service rate μ (units served per time period, when the server is busy). Average service time is 1 ÷ μ.

Queue discipline is the rule for choosing who is served next. The common rules are FIFO (first in, first out), LIFO (last in, first out), SIRO (service in random order) and priority service (for example, emergency patients first). Other behaviours matter too: balking (a person sees the queue and does not join), reneging (a person leaves after waiting) and jockeying (a person switches to a shorter queue). A queue may also have a limited capacity, in which case arrivals beyond it are turned away.

Kendall notation gives a short code for a queuing model, written A/B/c, and sometimes extended. A is the arrival distribution, B the service-time distribution, c the number of servers. M stands for Markovian (Poisson arrivals or exponential service), D for deterministic (fixed) and G for general. So M/M/1 means Poisson arrivals, exponential service and one server. M/M/s means the same with s servers. In the extended form A/B/c/K/N/D, K is the system capacity, N the population size and D the discipline. If the last three are not shown, assume infinite capacity, infinite population and FIFO.

Key rules to remember

Average service time
Mean service time = 1 ÷ μ
μ is the service rate per unit time. If μ = 6 per hour, mean service time = 10 minutes.
Average inter-arrival time
Mean time between arrivals = 1 ÷ λ
λ is the arrival rate per unit time. Keep λ and μ in the same time unit.
Traffic intensity (utilisation) for one server
ρ = λ ÷ μ
For a single-server model, a steady state exists only if ρ < 1. If λ ≥ μ, the queue grows without limit.
Traffic intensity for s servers
ρ = λ ÷ (s × μ)
A steady state requires ρ < 1 here too.
Kendall notation
A / B / c / K / N / D
A = arrival distribution, B = service distribution, c = number of servers, K = system capacity, N = population size, D = queue discipline. Defaults when omitted: ∞, ∞, FIFO.

How to solve Queuing Theory: Concepts and Structure questions

Use this method for any conceptual or short numerical question on queuing structure.

  1. 1Read the situation and name the calling population, the arrivals, the queue, the service channels and the departures.
  2. 2Decide whether the population is finite or infinite and whether the queue has a limit.
  3. 3Identify the arrival pattern (usually Poisson with rate λ) and the service pattern (usually exponential with rate μ).
  4. 4Count the channels and phases. One counter is single-channel; two counters in parallel are multi-channel; stages in sequence are multi-phase.
  5. 5State the queue discipline (FIFO, LIFO, SIRO or priority) and note any balking, reneging or jockeying.
  6. 6Write the Kendall notation. Use M for Poisson or exponential, D for fixed and G for general.
  7. 7If numbers are given, convert λ and μ to the same time unit, then compute ρ and check that ρ < 1.
  8. 8Write a one-line conclusion, such as whether the system is stable or whether more servers are needed.

Quickest way: Decode and classify in 30 seconds

When to use it: Use this for MCQs that give a situation or a Kendall code and ask you to identify the model, a component or the discipline.

  1. Read the code left to right: arrival, service, servers, then optional capacity, population and discipline.
  2. Replace M with Poisson arrivals or exponential service, as per its position.
  3. For rates, convert to the same unit first, then use ρ = λ ÷ (s × μ).
  4. If ρ ≥ 1, the queue is not stable, so no steady-state answer exists.
  5. Match key words: left without joining is balking, left after waiting is reneging, switched lines is jockeying.

Common mistakes in Queuing Theory: Concepts and Structure

  • Mixing time units for λ and μ, such as arrivals per hour with service time in minutes.

    Data is given in different forms, and students plug it in directly.

    Fix: Convert both to the same unit first. Service time of 5 minutes means μ = 12 per hour.

  • Treating the mean service time as μ.

    Both words say 'service', so they look interchangeable.

    Fix: μ is a rate. Mean service time = 1 ÷ μ. Check which one the question gives.

  • Confusing balking with reneging.

    Both mean a customer is lost to the system.

    Fix: Balking happens before joining the queue. Reneging happens after joining and waiting.

  • Reading M in Kendall notation as 'multiple' or 'machine'.

    The letter looks like an abbreviation of a common word.

    Fix: M means Markovian: Poisson arrivals (first position) or exponential service (second position).

  • Counting parallel servers as phases.

    Both relate to the service facility.

    Fix: Channels are parallel servers. Phases are sequential stages that a unit passes through.

  • Ignoring the condition ρ < 1 and still computing queue measures.

    Students apply formulas mechanically.

    Fix: Always compute ρ first. If ρ ≥ 1, state that the queue grows without limit.

Worked examples

Example 1

A bank has one cash counter. Customers arrive randomly at an average of 12 per hour (Poisson). Service time is exponential with a mean of 4 minutes. Customers are served in order of arrival, and there is no limit on the queue. (a) Write the Kendall notation. (b) Find λ, μ and ρ. (c) Is the system stable?

Show the solution
  1. Arrivals are Poisson, service is exponential, there is one server, capacity and population are unlimited and discipline is FIFO.
  2. So the notation is M/M/1 (full form M/M/1/∞/∞/FIFO).
  3. λ = 12 per hour.
  4. Mean service time = 4 minutes, so μ = 60 ÷ 4 = 15 per hour.
  5. ρ = λ ÷ μ = 12 ÷ 15 = 0.8.
  6. Since ρ = 0.8 < 1, a steady state exists.

Answer: M/M/1; λ = 12 per hour, μ = 15 per hour, ρ = 0.8 (the counter is busy 80% of the time). The system is stable.

Example 2

A repair shop has 2 technicians working in parallel. Jobs arrive at an average of 9 per hour. Each technician takes an average of 10 minutes per job. Check whether the system is stable and identify the model in Kendall notation, assuming Poisson arrivals, exponential service, infinite queue and FIFO.

Show the solution
  1. The arrival rate λ = 9 per hour.
  2. Mean service time = 10 minutes, so μ = 60 ÷ 10 = 6 per hour per technician.
  3. Number of servers s = 2, so total capacity = 2 × 6 = 12 per hour.
  4. ρ = λ ÷ (s × μ) = 9 ÷ 12 = 0.75.
  5. Since 0.75 < 1, the system is stable.
  6. Poisson arrivals, exponential service and 2 servers give M/M/2.

Answer: The model is M/M/2 and ρ = 0.75, so the system is stable. Using a single technician would give λ ÷ μ = 9 ÷ 6 = 1.5, which is more than 1 and would make the queue grow without limit.

Exam tips

  • In MCQs, expect questions that ask you to decode a Kendall code or match a term (balking, reneging, jockeying) to a situation. Learn the definitions word for word in your own language.
  • Always write λ and μ with units before doing any calculation, and convert to the same unit.
  • In written answers, draw a small flow: calling population, arrivals, queue, service channels, departures. It earns marks quickly and shows structure.
  • Link the topic to the cost trade-off: more servers reduce waiting cost but raise service cost. State this in one line when asked for the objective of queuing analysis.
  • Revise this topic before M/M/1 and M/M/s, because those models start from the notation and ρ you learn here.

Practice questions from Scheduling and Queuing Models

Queuing Theory: Concepts and Structure: frequently asked questions

What does M mean in Kendall notation?

M stands for Markovian. In the first position it means Poisson arrivals, which is the same as exponential gaps between arrivals. In the second position it means exponential service times.

What is the difference between a channel and a phase?

A channel is a server working in parallel with others, so more channels serve more customers at once. A phase is a stage of service in sequence, such as billing followed by packing.

What is queue discipline?

It is the rule that decides which waiting unit is served next. Common rules are FIFO, LIFO, SIRO and priority. If a question does not state the rule, assume FIFO.

Why must ρ be less than 1?

If ρ is 1 or more, arrivals come at least as fast as the system can serve them. The queue then keeps growing and no steady-state measures exist.