---
title: "Multistage switching and TST blocking"
subject: "Telecommunications"
chapter: 4
key: "semester_8/telecommunications:multistage-tst-blocking"
source: https://www.bctengineer.com/q/semester_8/telecommunications/multistage-tst-blocking
site: BCT Engineering
---

# Multistage switching and TST blocking

Telecommunications · Chapter 4: Digital Switching

- Asked in: 2078 Bhadra, 2+5 marks
- Video: [Three stage Networks](https://www.youtube.com/watch?v=1RnBCgqJkjw) — @Ahmad Sir 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/telecommunications/multistage-tst-blocking
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-4

## Question

What is multi stage switching? Explain TST switching with neat diagram and its blocking probability.

## Short answer

Multistage switching uses several stages of smaller switches to reduce crosspoints and provide alternative paths. A TST switch has input TSI, a time-multiplexed space stage, and output TSI. Its blocking probability is P_B = [1-(1-p')^2]^l with p'=p/alpha; non-blocking when l >= 2c-1.

## Answer

#### Multistage switching
**Multistage switching** builds a large switch from several stages of smaller switches connected by internal links, instead of one big $N \times N$ matrix. It cuts the number of crosspoints (or memory) needed, and gives several alternative paths for each call. The cost is a small chance of internal blocking unless enough paths are provided. Examples: three-stage space networks, STS, TST, TSST.

#### TST switching
A **Time–Space–Time** switch has an input time stage, a time-multiplexed space stage and an output time stage.

```text
 Link 1 -->[TSI-A1]--+-----------+--[TSI-B1]--> Link 1
 Link 2 -->[TSI-A2]--| N x N     |--[TSI-B2]--> Link 2
   ...        ...    | space     |     ...
 Link N -->[TSI-AN]--| (TMS)     |--[TSI-BN]--> Link N
                     +-----------+
  c slots   in T      l internal    out T    c slots
                       slots
```

**Working (example):** connect slot 5 of link 1 to slot 20 of link N.
1. The control processor searches for an internal slot $x$ that is free both on the output of TSI-A1 and on the input of TSI-BN (say $x = 12$).
2. **Input T stage:** TSI-A1 moves the sample from slot 5 to slot 12.
3. **Space stage:** in slot 12 its control memory closes crosspoint (1, N), so the sample passes from highway 1 to highway N.
4. **Output T stage:** TSI-BN moves the sample from slot 12 to slot 20.
5. The return direction uses another slot, often $x + l/2$ (mod $l$), so one search serves both directions.

Since any internal slot $x$ can be used, there are $l$ alternative paths.

#### Blocking probability
Using a Lee graph: each path has two internal links in series (TSI-A → S and S → TSI-B), and there are $l$ paths in parallel. With external occupancy $p$ and time expansion $\alpha = l/c$, the internal link occupancy is $p' = p/\alpha$:
$$
P_B = \left[1 - (1 - p')^2\right]^{l}
$$
- Non-blocking when $l \ge 2c - 1$.
- Example: $c = l = 32$, $p = 0.6$: $P_B = [1 - 0.4^2]^{32} = 0.84^{32} \approx 3.8 \times 10^{-3}$.

**Why it is preferred:** time expansion is cheap (faster memory), the space stage is small, and the blocking is very low, so TST (and TSST) is used in most large digital exchanges.

## Check yourself

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

1. What is the main benefit of multistage switching over a single N x N matrix?
   A. Fewer crosspoints and alternative paths
   B. No chance of blocking ever
   C. Fewer links
   D. Single-path routing
   Answer: A. Small switches in stages reduce crosspoints and give several paths, at the cost of possible internal blocking. (Key point: Multistage purpose)

2. In a TST switch connecting slot 5 of link 1 to slot 20 of link N via internal slot x, what does the input TSI do?
   A. Moves the sample from slot 5 to slot x
   B. Moves the sample from slot x to slot 20
   C. Closes crosspoint (1, N)
   D. Searches for slot x
   Answer: A. Input T stage: slot 5 to x; space stage passes it in slot x; output T stage moves x to 20. (Key point: TST working)

3. What must the processor ensure when choosing internal slot x?
   A. x is free on both the output of the input TSI and the input of the output TSI
   B. x equals the input slot number
   C. x equals the output slot number
   D. x is the highest slot
   Answer: A. The same slot x is used on both links of the path, so it must be free on both. (Key point: Slot search)

4. For c = l = 32 and p = 0.6 E, P_B is about
   A. 0.84
   B. 3.8e-3
   C. 0.6
   D. 1e-10
   Answer: B. [1 - 0.4^2]^32 = 0.84^32, about 3.8e-3. (Key point: Blocking example)

## Related in Telecommunications

- [Multistage advantage and TST switching](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Amultistage-advantage-tst) — Telecommunications, chapter 4, asked 2×
- [Multistage switching; TST vs STS](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Amultistage-tst-sts-compare) — Telecommunications, chapter 4
- [Multistage switching and STS blocking](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Amultistage-sts-blocking) — Telecommunications, chapter 4
- [TST time slots for non-blocking operation](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atst-time-slots-numerical) — Telecommunications, chapter 4
- [Clos network and STS/TST blocking](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aclos-network-sts-tst-blocking) — Telecommunications, chapter 4
- [Derive STS and TST blocking probabilities](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Asts-tst-blocking-derivation) — Telecommunications, chapter 4

## Related in other subjects

- [Pipelining and multi-stage pipeline](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Apipelining-stages) — Embedded System, chapter 2, asked 3×