---
title: "Kendall-Lee notation for queuing systems"
subject: "Telecommunications"
chapter: 6
key: "semester_8/telecommunications:kendall-lee-notation"
source: https://www.bctengineer.com/q/semester_8/telecommunications/kendall-lee-notation
site: BCT Engineering
---

# Kendall-Lee notation for queuing systems

Telecommunications · Chapter 6: Telephone Traffic

- Asked in: 2080 Bhadra, 8 marks; 2076 Chaitra, 4 marks (asked 2 times)
- Video: [Kendall's Notation in Queueing Theory | Operation Research in hindi](https://www.youtube.com/watch?v=O3vfHQSAe0s) — Start Practicing
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/telecommunications/kendall-lee-notation
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-6

## Question

Write the Kendall-Lee Notation for Queuing Systems. Explain with example.

## Short answer

Kendall-Lee notation is (a/b/c):(d/e/f) where a: arrival distribution, b: service distribution, c: number of servers, d: queue discipline, e: maximum system capacity, f: calling population size. For example, (M/M/1):(FCFS/∞/∞) represents a single-server queue with Poisson arrivals, exponential service, and infinite population.

## Answer

**Kendall–Lee notation** is a compact six-part code that describes the main features of a queuing system. Kendall gave the first three symbols; Lee added the last three.

$$
(a\,/\,b\,/\,c) : (d\,/\,e\,/\,f)
$$

| Symbol | Meaning | Typical codes |
|---|---|---|
| a | Arrival (inter-arrival time) distribution | M (Poisson/exponential), D (constant), $E_k$ (Erlang-k), GI (general independent) |
| b | Service (holding) time distribution | M, D, $E_k$, G |
| c | Number of parallel servers | 1, 2, ..., N |
| d | Queue discipline | FCFS/FIFO, LCFS/LIFO, SIRO (random), PR (priority), GD (general) |
| e | Maximum number allowed in the system (waiting + in service) | finite K or ∞ |
| f | Size of the calling population (sources) | finite or ∞ |

When the last three are FCFS/∞/∞ they are often left out, so M/M/1 means $(M/M/1):(FCFS/\infty/\infty)$.

#### Meaning of the codes
- **M (Markovian):** Poisson arrivals with rate $\lambda$, i.e. exponential inter-arrival times; or exponential service with mean $1/\mu$. Memoryless.
- **D:** every inter-arrival or service time is the same.
- **$E_k$:** sum of k exponential phases; less random than M.
- **G/GI:** any distribution.

#### Examples
1. **(M/M/1):(FCFS/∞/∞)** – single public call office with Poisson callers and exponential call durations; callers wait in line, infinite population.
   - Utilisation $\rho = \lambda/\mu$, mean number in system $L = \rho/(1-\rho)$, mean time in queue $W_q = \rho/(\mu-\lambda)$.
   - If $\lambda = 10$ calls/h and mean call = 3 min ($\mu = 20$/h): $\rho = 0.5$, $L = 1$, $W_q = 0.5/10$ h $= 3$ min.
2. **(M/M/N):(FCFS/∞/∞)** – Erlang delay (Erlang C) model: N operators in a call centre with queuing.
3. **(M/M/N):(GD/N/∞)** – Erlang loss (Erlang B) model: N trunks, no waiting space, blocked calls lost.
4. **(M/M/N):(GD/N/S)** – Engset model: finite number S of sources, no waiting.
5. **(M/D/1):(FCFS/K/∞)** – router output link with fixed-size packets and a buffer of K packets.

The notation lets an engineer pick the correct formula (Erlang B, Erlang C, Engset, M/M/1) by simply reading the system description.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In the Kendall-Lee notation (a/b/c):(d/e/f), what do d, e, f represent?
   A. Queue discipline, max number in system, calling population size
   B. Servers, arrivals, service
   C. Arrival, service, servers
   D. Utilisation, delay, loss
   Answer: A. The last three symbols added by Lee are discipline, system capacity and source population. (Key point: Notation form)

2. What does M/M/1 mean in full?
   A. (M/M/1):(FCFS/1/1)
   B. (M/M/1):(FCFS/∞/∞)
   C. (M/M/1):(LCFS/∞/1)
   D. (M/M/1):(GD/N/∞)
   Answer: B. When the last three are FCFS/∞/∞ they are omitted. (Key point: Default omission)

3. For λ = 10 calls/h and mean call 3 min, what is ρ?
   A. 0.2
   B. 0.3
   C. 0.5
   D. 2
   Answer: C. μ = 20/h, so ρ = λ/μ = 0.5. (Key point: M/M/1 numerical)

4. Which notation represents the Erlang loss (Erlang B) model?
   A. (M/M/N):(FCFS/∞/∞)
   B. (M/M/N):(GD/N/∞)
   C. (M/D/1):(FCFS/K/∞)
   D. (M/M/N):(GD/N/S)
   Answer: B. Capacity N with infinite sources gives blocked-calls-cleared: Erlang B. The last option is Engset. (Key point: Erlang B)

## Related in Telecommunications

- [Queuing system and Kendall's notation](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aqueuing-system-kendall) — Telecommunications, chapter 6, asked 4×
- [Queuing theory in delay systems](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aqueuing-theory-delay-systems) — Telecommunications, chapter 6
- [Erlang delay model: notation and derivation](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aerlang-delay-kendall-derivation) — Telecommunications, chapter 6

## Related in other subjects

- [Queuing system, characteristics, Kendall notation](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aqueuing-characteristics-role-kendall) — Simulation and Modeling, chapter 4, asked 2×
- [Queuing model, Kendall, M/D/8/15/LIFO](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aqueuing-model-mdlifo-8) — Simulation and Modeling, chapter 4
- [Queuing notation, default values, elements](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aqueuing-notation-defaults-elements) — Simulation and Modeling, chapter 4
- [Queuing characteristics; M/D/6/10/FIFO](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aqueuing-characteristics-mdfifo-6) — Simulation and Modeling, chapter 4