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

# Multistage switching and STS blocking

Telecommunications · Chapter 4: Digital Switching

- Asked in: 2072 Chaitra, 2+6 marks
- Video: [Telecommunication Systems (19 - Multi-Stage TST or STS Switch)](https://www.youtube.com/watch?v=ipZmBPa4SdM) — Dr. Samy S. Soliman
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/telecommunications/multistage-sts-blocking
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-4

## Question

What is multistage switching? Describe the STS switching with neat diagram and its blocking probabilities.

## Short answer

Multistage switching builds large switches from smaller stages to save crosspoints. An STS switch has input and output time-multiplexed space stages connected to k centre TSIs; blocking probability PB = [1 - (1 - p')^2]^k with p' = pN/k, and is non-blocking when k ≥ 2N-1.

## Answer

#### Multistage switching
**Multistage switching** builds a large switch from two or more stages of smaller switching arrays joined by internal links. It needs far fewer crosspoints than a single $N \times N$ matrix and gives alternative paths, but some blocking may occur unless the middle stage is large enough (e.g. $k \ge 2n - 1$).

#### STS switching
A **Space–Time–Space** switch has an input time-multiplexed space stage, a centre stage of $k$ TSIs, and an output space stage.

```text
 Link 1 -->+-------+  [TSI 1]  +-------+--> Link 1
 Link 2 -->| N x k |->[TSI 2]->| k x N |--> Link 2
   ...     | space |   ...     | space |      ...
 Link N -->+-------+  [TSI k]  +-------+--> Link N
   c slots  (in S)   centre T   (out S)
```

**Working:** to connect slot $i$ of link 1 to slot $j$ of link N:
1. The controller looks for a centre TSI $m$ that is free in input slot $i$ (link from input S) and in output slot $j$ (link to output S).
2. In slot $i$, the input space stage closes crosspoint (1, $m$), sending the sample to TSI $m$.
3. TSI $m$ stores the sample and reads it out in slot $j$.
4. In slot $j$, the output space stage closes crosspoint ($m$, N), sending it to outgoing link N.
5. The reverse direction is set up the same way.

There are $k$ alternative paths, one through each centre TSI.

#### Blocking probability
Using a Lee graph: each path has two internal links in series (input S → TSI and TSI → output S), and $k$ paths are in parallel.

- External link occupancy: $p$. With space expansion $\beta = k/N$, internal link occupancy is $p' = p/\beta$.
- Probability one path is free: $(1-p')^2$; busy: $1-(1-p')^2$.
- Blocking (all $k$ busy):
$$
P_B = \left[1 - (1 - p')^2\right]^{k}
$$
- Non-blocking when $k \ge 2N - 1$.

**Example:** $N = k = 16$, $p = 0.7$: $p' = 0.7$, $P_B = (1 - 0.09)^{16} = 0.91^{16} \approx 0.22$.

**Remark:** to reduce blocking, $k$ must be raised, which adds TSIs and crosspoints (space expansion). For large switches, TST is cheaper for the same blocking, so STS is used mainly in small switches.

## Check yourself

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

1. Why use multistage switching instead of one N x N matrix?
   A. It removes all blocking
   B. It needs far fewer crosspoints and gives alternative paths
   C. It needs no control
   D. It doubles the crosspoints
   Answer: B. Smaller arrays joined by links reduce crosspoints, though blocking may occur. (Key point: Multistage purpose)

2. In STS, which centre TSI can be used for a call from slot i to slot j?
   A. Any TSI
   B. The TSI numbered i
   C. One free in input slot i and output slot j
   D. Only TSI 1
   Answer: C. The controller needs a TSI free on both the link from input S in slot i and the link to output S in slot j. (Key point: STS path choice)

3. What is the STS blocking probability?
   A. [1-(1-p')^2]^k
   B. (1-p')^k
   C. p'^2
   D. 1-(1-p')^k
   Answer: A. Each path has two links in series (prob. free (1-p')^2); all k parallel paths must be busy. (Key point: Blocking formula)

4. For N = k = 16, p = 0.7, what is the approximate blocking?
   A. 0.91
   B. 0.22
   C. 0.09
   D. 0.7
   Answer: B. p' = 0.7, 1-(0.3)^2 = 0.91, and 0.91^16 is about 0.22. (Key point: Example value)

## Related in Telecommunications

- [Multistage switching; TST vs STS](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Amultistage-tst-sts-compare) — 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
- [Multistage switching and TST blocking](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Amultistage-tst-blocking) — 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
- [Multistage advantage and TST switching](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Amultistage-advantage-tst) — Telecommunications, chapter 4, asked 2×
- [Blocking and non-blocking switching](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Ablocking-nonblocking-switching) — Telecommunications, chapter 4

## Related in other subjects

- [Crossbar switch vs multistage network](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_organization_and_architecture%3Acrossbar-vs-multistage) — Computer Organization and Architecture, chapter 8