---
title: "Erlang delay model: notation and derivation"
subject: "Telecommunications"
chapter: 6
key: "semester_8/telecommunications:erlang-delay-kendall-derivation"
source: https://www.bctengineer.com/q/semester_8/telecommunications/erlang-delay-kendall-derivation
site: BCT Engineering
---

# Erlang delay model: notation and derivation

Telecommunications · Chapter 6: Telephone Traffic

- Asked in: 2076 Asoj, 7 marks
- Video: [DSS 17EC654: Queuing systems: 2. Probability of delay](https://www.youtube.com/watch?v=VcblcC6AxeE) — Santosh Shirol
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/telecommunications/erlang-delay-kendall-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-6

## Question

Provide the Kendall-Lee notation for Erlang's delay traffic model and derive its blocking probability.

## Short answer

Kendall-Lee notation: (M/M/N):(FCFS/∞/∞). The blocking (delay) probability is Erlang C: E2,N(A) = [ (A^N/N!) * N/(N-A) ] / [ Σ_{k=0}^{N-1} A^k/k! + (A^N/N!) * N/(N-A) ], derived from balance equations with arrival rate λ and service rate min(k,N)μ.

## Answer

#### Kendall–Lee notation
Erlang's delay model is
$$
(M/M/N):(FCFS/\infty/\infty)
$$
Poisson arrivals (rate $\lambda$), exponential holding time (mean $h = 1/\mu$), N servers, first-come-first-served, unlimited waiting room and infinite sources. Offered traffic $A = \lambda/\mu = \lambda h$ Erlangs, with $A < N$ for stability.

#### State diagram
State k = number of calls in the system (in service + waiting).
```text
   lam     lam           lam      lam      lam
 0 ---> 1 ---> 2 ... N-1 ---> N ---> N+1 ---> ...
   <---   <---           <---     <---     <---
    mu    2mu            N mu     N mu     N mu
```
Arrival rate is always $\lambda$. Service rate is $k\mu$ for $k \le N$ and $N\mu$ for $k > N$ (only N servers).

#### Derivation
Balance between neighbouring states: $\lambda P_{k-1} = k\mu P_k$ for $k \le N$, and $\lambda P_{k-1} = N\mu P_k$ for $k > N$.

$$
\begin{aligned}
P_k &= \frac{A^k}{k!}P_0, & k \le N \\
P_k &= \frac{A^N}{N!}\left(\frac{A}{N}\right)^{k-N}P_0, & k > N
\end{aligned}
$$

Normalisation $\sum P_k = 1$ (the geometric series converges since $A/N < 1$):
$$
\begin{aligned}
P_0^{-1} &= \sum_{k=0}^{N-1}\frac{A^k}{k!} + \frac{A^N}{N!}\sum_{j=0}^{\infty}\left(\frac{A}{N}\right)^j \\
&= \sum_{k=0}^{N-1}\frac{A^k}{k!} + \frac{A^N}{N!}\,\frac{N}{N-A}
\end{aligned}
$$

A call is delayed ("blocked") when it finds all N servers busy, i.e. $k \ge N$:
$$
\begin{aligned}
P(W>0) &= \sum_{k=N}^{\infty}P_k = \frac{A^N}{N!}\,\frac{N}{N-A}\,P_0
\end{aligned}
$$

**Erlang C (delay) formula:**
$$
E_{2,N}(A) = \frac{\dfrac{A^N}{N!}\dfrac{N}{N-A}}{\displaystyle\sum_{k=0}^{N-1}\frac{A^k}{k!} + \frac{A^N}{N!}\frac{N}{N-A}}
$$

It can also be written using Erlang B: $E_{2,N} = \dfrac{N\,E_{1,N}}{N - A(1 - E_{1,N})}$.

#### Related results
- Mean delay of all calls: $W = E_{2,N}(A)\,\dfrac{h}{N-A}$
- Mean delay of delayed calls: $\dfrac{h}{N-A}$
- $P(\text{wait} > t) = E_{2,N}(A)\,e^{-(N-A)t/h}$
- For N = 1: $E_{2,1} = A$, $W = \dfrac{A\,h}{1-A}$ (M/M/1).

## Check yourself

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

1. What does (M/M/N):(FCFS/inf/inf) denote?
   A. Deterministic arrivals, N servers, no queue
   B. Poisson arrivals, one server, finite queue
   C. General arrivals, exponential service, LIFO
   D. Poisson arrivals, exponential service, N servers, FCFS, unlimited queue and sources
   Answer: D. M/M/N with FCFS discipline and infinite waiting room and source population. (Key point: Kendall notation)

2. What condition must hold for the delay model to be stable?
   A. A < N
   B. A > N
   C. A = 1
   D. A < 1 regardless of N
   Answer: A. The geometric series converges only if A/N < 1. (Key point: Stability)

3. How does the service rate behave in the state diagram?
   A. mu always
   B. k mu up to N, then N mu
   C. N mu always
   D. k mu for all k
   Answer: B. Only N servers exist, so the rate saturates at N mu. (Key point: State probabilities)

4. What does summing the geometric tail give in P0^-1?
   A. A^N/N! x (N-A)/N
   B. A^N/N! x 1/(1-N)
   C. A^N/N! x N/(N-A)
   D. A^N/N!
   Answer: C. The sum of (A/N)^j is 1/(1-A/N) = N/(N-A). (Key point: Normalisation)

## Related in Telecommunications

- [Kendall-Lee notation for queuing systems](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Akendall-lee-notation) — Telecommunications, chapter 6, asked 2×
- [Queuing system and Kendall's notation](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aqueuing-system-kendall) — Telecommunications, chapter 6, asked 4×
- [GOS and blocking probability in a loss system](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Agos-blocking-loss-system) — Telecommunications, chapter 6
- [GOS vs blocking probability; E.164 plan](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Agos-vs-blocking-e164) — Telecommunications, chapter 6
- [Derive STS and TST blocking probabilities](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Asts-tst-blocking-derivation) — Telecommunications, chapter 4
- [Traffic intensity, GOS and call duration probability](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atraffic-intensity-call-duration-probability) — Telecommunications, chapter 6

## Related in other subjects

- [Numerical: trunked channels and blocking (Erlang B)](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Atrunking-five-channel-blocking) — Wireless Communication, chapter 2
- [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
- [Derivation of tunneling probability through barrier](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Atunneling-probability-derivation) — Electrical Engineering Material, chapter 1