---
title: "S and T switches; 3-stage vs 2-stage"
subject: "Telecommunications"
chapter: 4
key: "semester_8/telecommunications:s-t-switches-3-stage"
source: https://www.bctengineer.com/q/semester_8/telecommunications/s-t-switches-3-stage
site: BCT Engineering
---

# S and T switches; 3-stage vs 2-stage

Telecommunications · Chapter 4: Digital Switching

- Asked in: 2069 Chaitra, 2+6 marks
- Video: [Lec-11_Two stage switching network | Telecommunication Engineering | ICT Engineering](https://www.youtube.com/watch?v=c5Pff6spVbg) — Computer_IT_ICT Engineering Department : LJIET
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/telecommunications/s-t-switches-3-stage
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-4

## Question

What do you mean by S (space) and T (time) switches? Show that 3-stage STS or TST network can minimize the switching problems associated with 2-stage ST or TS network with their working models.

## Short answer

Space switches change highway, time switches change time slot. Two-stage TS/ST networks offer a single path, so blocking is high. TST uses an internal slot x: input TSI moves to x, space connects highways in x, output TSI to destination slot, giving l paths. STS uses k centre TSIs, providing k paths. Blocking = [1-(1-p')^2]^k or ^l, vastly reduced.

## Answer

#### S and T switches
- **Space (S) switch:** a time-multiplexed crosspoint matrix that connects an input TDM highway to an output highway in a given time slot. It changes the **highway** but not the **slot**.
- **Time (T) switch / TSI:** a speech memory with a control memory that writes samples and reads them in a different order. It changes the **slot** but not the **highway**.

#### Problem with two-stage ST and TS networks
To connect slot $i$ on highway $a$ to slot $j$ on highway $b$, both the slot and the highway must change.

```text
 TS: HWa --[T: i->j]--[S: a->b in slot j]-- HWb
 ST: HWa --[S: a->b in slot i]--[T: i->j]-- HWb
```
- In **TS**, the space stage must use slot $j$; if slot $j$ is already used by another input highway to reach $b$, the call is blocked.
- In **ST**, the space stage must use slot $i$; if slot $i$ is busy into highway $b$, the call is blocked.
- There is only **one path**, so the blocking probability equals the chance that this particular link is busy (about $p$). This is too high for an exchange.

#### Three-stage TST network
```text
 HWa->[TSI-A: i->x]->[S: a->b in x]->[TSI-B: x->j]->HWb
```
1. Input TSI moves the sample from slot $i$ to **any free internal slot $x$**.
2. Space stage connects $a$ to $b$ in slot $x$.
3. Output TSI moves it from $x$ to $j$.

The controller can pick any of $l$ internal slots, so there are $l$ alternative paths.
$$
P_{B,TST} = \left[1 - (1 - p')^2\right]^{l}, \quad p' = \frac{pc}{l}
$$

#### Three-stage STS network
```text
 HWa->[S: a->m in i]->[TSI m: i->j]->[S: m->b in j]->HWb
```
1. Input space stage sends the sample in slot $i$ to **any free centre TSI $m$**.
2. TSI $m$ changes the slot from $i$ to $j$.
3. Output space stage connects TSI $m$ to highway $b$ in slot $j$.

There are $k$ alternative paths (centre TSIs):
$$
P_{B,STS} = \left[1 - (1 - p')^2\right]^{k}, \quad p' = \frac{pN}{k}
$$

#### Comparison (example)
For $p' = 0.3$: a two-stage network blocks with probability about 0.3 (one link busy). A three-stage network with 8 paths gives $(1 - 0.7^2)^8 = 0.51^8 \approx 4.6 \times 10^{-3}$, and with 32 paths about $4.4 \times 10^{-10}$.

| Network | Paths | Blocking |
|---|---|---|
| TS / ST | 1 | High (≈ $p$) |
| STS | $k$ | Low, non-blocking if $k \ge 2N-1$ |
| TST | $l$ | Very low, non-blocking if $l \ge 2c-1$ |

So 3-stage STS or TST networks remove the single-path problem of 2-stage networks; TST is preferred because time expansion is cheaper than space expansion.

## Check yourself

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

1. Which statement is correct?
   A. S changes slot, T changes highway
   B. S changes highway, T changes slot
   C. Both change slot
   D. Both change highway
   Answer: B. Space switch changes highway; time switch (TSI) changes slot. (Key point: S and T definitions)

2. Why is blocking in TS or ST high?
   A. Only one path exists, so blocking is about p
   B. Too many paths
   C. Memory is too slow
   D. No control memory
   Answer: A. With one path, blocking equals the chance that link is busy. (Key point: Two-stage problem)

3. How does TST give alternative paths?
   A. Any of l internal slots can be chosen
   B. Any of k TSIs
   C. Any highway
   D. Any external slot
   Answer: A. The input TSI can move the sample to any free internal slot x. (Key point: TST paths)

4. For p' = 0.3, how do 8 paths compare with 1 path?
   A. About 0.0046 vs about 0.3
   B. About 0.3 vs 0.0046
   C. Same
   D. About 0.51 vs 0.3
   Answer: A. (1-0.7^2)^8 = 0.51^8 is about 4.6e-3, versus about 0.3 for one path. (Key point: Comparison)

## Related in Telecommunications

- [Space and time switches; STS vs TST](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aspace-time-switch-sts-tst) — Telecommunications, chapter 4
- [TST vs STS and three-stage design](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atst-sts-design-crosspoints) — Telecommunications, chapter 4
- [Multistage switching; TST vs STS](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Amultistage-tst-sts-compare) — Telecommunications, chapter 4
- [ST and TS drawbacks solved by STS](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Ast-ts-drawbacks-sts) — 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
- [Time switch and two-stage T-S network](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atime-switch-two-stage-ts) — Telecommunications, chapter 4

## Related in other subjects

- [Context switching; distributed vs network OS](https://www.bctengineer.com/question.md?key=semester_7%2Fdistributed_system%3Acontext-switch-dos) — Distributed System, chapter 3