---
title: "Queuing theory in delay systems"
subject: "Telecommunications"
chapter: 6
key: "semester_8/telecommunications:queuing-theory-delay-systems"
source: https://www.bctengineer.com/q/semester_8/telecommunications/queuing-theory-delay-systems
site: BCT Engineering
---

# Queuing theory in delay systems

Telecommunications · Chapter 6: Telephone Traffic

- Asked in: 2081 Bhadra, 8 marks
- Video: [DSS 17EC654 module 3 Queuing System or Delay System](https://www.youtube.com/watch?v=-OXWEs32-qo) — Basavaraj Chougula
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/telecommunications/queuing-theory-delay-systems
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-6

## Question

Describe the principles of queuing theory and its application in delay systems within telecommunication networks. How does queuing theory help in managing network delays and improving service quality?

## Short answer

Queuing theory studies waiting lines: key principles include Poisson arrivals, exponential service, servers, queue discipline, and traffic intensity ρ<1. In delay systems, calls queue when busy. It predicts wait time via M/M/1 or Erlang C, aiding in dimensioning links, sizing buffers, setting priorities, and planning capacity to meet delay targets and improve QoS.

## Answer

**Queuing theory** is the mathematical study of waiting lines. In a **delay system**, a call or packet that finds all servers busy waits in a queue instead of being lost. Queuing theory predicts how long it waits and how many wait, given the arrival rate, service rate and number of servers.

#### Principles of queuing theory
```text
 arrivals   +-------------+    +-----------+
 --lambda-->| queue/buffer|--->| N servers |--> out
            +-------------+    | rate mu   |
                               +-----------+
```
1. **Arrival process:** usually Poisson with mean rate $\lambda$ (exponential inter-arrival times).
2. **Service process:** holding/transmission time, often exponential with mean $1/\mu$.
3. **Servers:** N trunks, operators, processors or output links.
4. **Queue capacity and discipline:** finite or infinite buffer; FIFO, priority, etc.
5. **Kendall notation** A/B/C/K/N/D summarises the system (e.g. M/M/1, M/M/N).
6. **Traffic intensity** $A = \lambda/\mu$ Erlangs; utilisation per server $\rho = A/N$ must be below 1, otherwise the queue grows without limit.
7. **Little's law:** $L = \lambda W$ and $L_q = \lambda W_q$ (mean number = arrival rate × mean time).

#### Key delay-system results
- **M/M/1:** $W_q = \dfrac{\rho}{\mu(1-\rho)}$, $L_q = \dfrac{\rho^2}{1-\rho}$, $P(\text{wait}) = \rho$.
- **M/M/N (Erlang C):** probability of delay $C(N,A)$; mean delay of all calls $W = \dfrac{C(N,A)\,h}{N-A}$; probability of waiting longer than t: $C(N,A)\,e^{-(N-A)t/h}$.
- **M/D/1** (fixed packet length): $W_q = \dfrac{\rho}{2\mu(1-\rho)}$, half that of M/M/1.

#### Applications in telecommunication networks
- **Common-control exchanges:** callers wait for dial tone (register/processor queue); designed so that, e.g., not more than 1.5% wait over 3 s.
- **Call centres and operator services:** number of agents chosen with Erlang C for a target service level.
- **Packet networks (routers, switches, ATM):** packets queue in output buffers; queuing delay, jitter and buffer overflow (loss) are predicted with M/M/1, M/D/1 or M/M/1/K.
- **Signaling links (SS7)** and processors: message delay versus link load.

#### How queuing theory helps manage delay and improve service
1. **Dimensioning:** finds the minimum number of servers or link capacity for a target mean delay or probability of delay.
2. **Load limits:** shows that delay rises sharply as $\rho \to 1$ (e.g. M/M/1 $W/h$ = 2 at $\rho = 0.5$ but 10 at $\rho = 0.9$), so links are kept at moderate utilisation.
3. **Buffer sizing:** chooses queue length K to balance packet loss and delay.
4. **Scheduling and priority:** priority queues give voice/video low delay while data waits (QoS).
5. **Trunking efficiency:** shows one large pooled server group gives less delay than several small separate groups.
6. **Capacity planning and SLAs:** predicts performance as traffic grows, guiding upgrades before service degrades.

## Check yourself

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

1. What condition is needed for a stable queue?
   A. ρ > 1
   B. ρ < 1
   C. ρ = 2
   D. ρ = 0
   Answer: B. If utilisation reaches or exceeds 1 the queue grows without limit. (Key point: Stability)

2. Little's law states:
   A. L = λ W
   B. L = W / λ
   C. L = μ W
   D. L = λ + W
   Answer: A. Mean number = arrival rate × mean time. (Key point: Little's law)

3. Why are links kept at moderate utilisation?
   A. Delay rises sharply as ρ approaches 1
   B. Delay falls as ρ approaches 1
   C. Buffers become smaller
   D. Erlang C stops working
   Answer: A. For M/M/1, W/h is 2 at ρ = 0.5 but 10 at ρ = 0.9. (Key point: Load vs delay)

4. Compared with M/M/1, the M/D/1 mean waiting time is:
   A. Double
   B. Half
   C. The same
   D. Zero
   Answer: B. Fixed service time halves Wq. (Key point: M/D/1)

## Related in Telecommunications

- [Short note: Queuing theory in delay system](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Ashort-note-queuing-delay) — Telecommunications, chapter 6
- [Evolution of telecommunication and switching systems](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aevolution-telecom-switching) — Telecommunications, chapter 1, asked 3×
- [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×

## Related in other subjects

- [Delays in packet switched network](https://www.bctengineer.com/question.md?key=semester_8%2Finternet_and_intranet%3Apacket-switched-delays) — Internet and Intranet, chapter 1
- [Queuing characteristics and application](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aqueuing-characteristics-application) — Simulation and Modeling, chapter 4
- [Short note: Application of queuing system](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Ashort-note-queuing-application) — Simulation and Modeling, chapter 4
- [How CM helps with delay, quality and cost](https://www.bctengineer.com/question.md?key=semester_8%2Fconstruction_management%3Acm-course-delay-cost) — Construction Management, chapter 1