---
title: "Decision tree; infinite source blocking formulas"
subject: "Telecommunications"
chapter: 6
key: "semester_8/telecommunications:decision-tree-infinite-sources"
source: https://www.bctengineer.com/q/semester_8/telecommunications/decision-tree-infinite-sources
site: BCT Engineering
---

# Decision tree; infinite source blocking formulas

Telecommunications · Chapter 6: Telephone Traffic

- Asked in: 2073 Shrawan, 4+4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/telecommunications/decision-tree-infinite-sources
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-6

## Question

What are the formulas used in telecommunication traffic engineering decision tree? Describe the blocking formulas used in infinite sources.

## Short answer

The decision tree uses source type (infinite/finite) and blocked call handling (cleared/held/delayed). For infinite sources: Erlang B for cleared calls B = A^N/N! / Σ_{k=0}^N A^k/k!; Poisson for held calls P = e^{-A} Σ_{k=N}^∞ A^k/k!; Erlang C for delayed calls C = (A^N/N! · N/(N-A)) / (Σ_{k=0}^{N-1} A^k/k! + (A^N/N! · N/(N-A))).

## Answer

#### Formulas in the traffic engineering decision tree
The correct formula is chosen by two questions: are the **sources** infinite or finite, and what happens to **blocked calls** — cleared (LCC), held (LCH) or delayed (LCD)?

```text
                 Traffic sources
           +-------------+-------------+
       Infinite                     Finite
   +-------+-------+          +-------+-------+
  LCC     LCH     LCD        LCC     LCH     LCD
Erlang B Poisson Erlang C   Engset  Binomial Finite
                                            queue
```

| Sources | Blocked calls | Formula |
|---|---|---|
| Infinite | Cleared | Erlang B |
| Infinite | Held | Poisson |
| Infinite | Delayed | Erlang C |
| Finite | Cleared | Engset |
| Finite | Held | Binomial |
| Finite | Delayed | Finite queue |

#### Blocking formulas for infinite sources
Assumptions: Poisson arrivals, exponential holding times, N servers, full availability, offered traffic $A$ in Erlangs.

**1. Erlang B (lost calls cleared)**
$$
B = \frac{A^N/N!}{\sum_{k=0}^{N}A^k/k!}
$$
Blocked calls vanish. Most widely used for trunk dimensioning (Erlang B tables).

**2. Poisson (lost calls held)**
$$
P = e^{-A}\sum_{k=N}^{\infty}\frac{A^k}{k!}
$$
Blocked calls are assumed to stay in the system (keep retrying) for a holding time; gives slightly higher blocking.

**3. Erlang C (lost calls delayed)**
$$
C = \frac{\dfrac{A^N}{N!}\dfrac{N}{N-A}}{\sum_{k=0}^{N-1}\dfrac{A^k}{k!} + \dfrac{A^N}{N!}\dfrac{N}{N-A}}
$$
Probability that a call has to wait; mean wait $= C\,h/(N-A)$. Used for call centres and common-control equipment.

## Check yourself

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

1. Which two questions select the formula?
   A. Number of trunks and call duration
   B. Time of day and GoS
   C. Holding time and arrival rate only
   D. Infinite or finite sources, and whether blocked calls are cleared, held or delayed
   Answer: D. The tree branches on source type and blocked-call handling. (Key point: Decision tree)

2. Infinite sources with blocked calls held uses?
   A. Poisson
   B. Erlang B
   C. Engset
   D. Binomial
   Answer: A. Infinite LCH is Poisson. (Key point: Formula mapping)

3. Which formula describes infinite sources with delayed calls?
   A. Erlang B
   B. Erlang C
   C. Engset
   D. Poisson
   Answer: B. Erlang C gives probability of waiting. (Key point: Erlang C use)

4. What does Erlang B assume for blocked calls?
   A. They wait
   B. They retry for one holding time
   C. They vanish
   D. They overflow to a queue of infinite size
   Answer: C. Lost calls cleared. (Key point: Erlang B)

## Close variants set in other exams

- [Decision tree; finite source blocking formulas](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Adecision-tree-finite-sources) — Telecommunications, chapter 6, asked 2×

## Related in Telecommunications

- [Decision tree formulas; 20 servers, 10 E](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Adecision-tree-20-servers) — Telecommunications, chapter 6
- [Infinite-source blocking; 40 subscribers](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Ainfinite-source-blocking-40-subscribers) — Telecommunications, chapter 6