---
title: "TST time slots for non-blocking operation"
subject: "Telecommunications"
chapter: 4
key: "semester_8/telecommunications:tst-time-slots-numerical"
source: https://www.bctengineer.com/q/semester_8/telecommunications/tst-time-slots-numerical
site: BCT Engineering
---

# TST time slots for non-blocking operation

Telecommunications · Chapter 4: Digital Switching

- Asked in: 2080 Baisakh, 6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/telecommunications/tst-time-slots-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-4

## Question

A TST network is used in a digital switch and the secondary multiplex contains 120 time slots. How many time slots would be included in the time multiplexed space stage for non-blocking operation? What would be the blocking probability if the time multiplexed space stage contained 120 time slots, 150 time slots, and 200 time slots? Assume channel occupancies of 0.6E and 0.9E.

## Short answer

Non‑blocking requires l ≥ 2c−1 = 239 slots. For c=120, P_B at p=0.6E: l=120 → 8.2×10^−10, l=150 → 2.9×10^−21, l=200 → 1.7×10^−46. At p=0.9E: 0.299, 4.8×10^−6, 2.2×10^−21. Even a small time expansion drastically reduces blocking.

## Answer

**Given:** time slots per frame on the external (secondary multiplex) link, $c = 120$; occupancy $p = 0.6$ E and $0.9$ E.

#### Non-blocking condition
A TST switch is strictly non-blocking when the number of space-stage time slots satisfies $l \ge 2c - 1$:
$$
l = 2(120) - 1 = 239 \text{ time slots}
$$

#### Blocking probability
Lee-graph formula for TST:
$$
P_B = \left[1 - (1 - p')^2\right]^{l}, \qquad p' = \frac{p\,c}{l}
$$

**Step 1: internal link occupancy $p'$**

| $l$ | $p'$ (p = 0.6) | $p'$ (p = 0.9) |
|---|---|---|
| 120 | 0.600 | 0.900 |
| 150 | 0.480 | 0.720 |
| 200 | 0.360 | 0.540 |

**Step 2: path-busy probability $q = 1 - (1-p')^2$**

| $l$ | $q$ (p = 0.6) | $q$ (p = 0.9) |
|---|---|---|
| 120 | 0.8400 | 0.9900 |
| 150 | 0.7296 | 0.9216 |
| 200 | 0.5904 | 0.7884 |

**Step 3: $P_B = q^{\,l}$**

For example, $l = 120$, $p = 0.9$:
$$
P_B = 0.99^{120} = 0.299
$$
and $l = 150$, $p = 0.6$:
$$
P_B = 0.7296^{150} = 2.90 \times 10^{-21}
$$

| $l$ | $P_B$ at 0.6 E | $P_B$ at 0.9 E |
|---|---|---|
| 120 | $8.19 \times 10^{-10}$ | $0.299$ |
| 150 | $2.90 \times 10^{-21}$ | $4.80 \times 10^{-6}$ |
| 200 | $1.70 \times 10^{-46}$ | $2.24 \times 10^{-21}$ |

**Comment:** even without time expansion ($l = 120$) the blocking is negligible at 0.6 E, but at 0.9 E it is about 30%. A modest time expansion (150 or 200 slots) brings it down to a negligible value, using far fewer slots than the 239 needed for strictly non-blocking operation.

**Answer:** $l = 239$ slots for non-blocking; $P_B$ = $8.19 \times 10^{-10}$, $2.90 \times 10^{-21}$, $1.70 \times 10^{-46}$ (0.6 E) and $0.299$, $4.80 \times 10^{-6}$, $2.24 \times 10^{-21}$ (0.9 E) for $l$ = 120, 150, 200.

## Check yourself

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

1. For a TST switch with c = 120, how many space-stage slots give non-blocking operation?
   A. 120
   B. 240
   C. 239
   D. 119
   Answer: C. l = 2c - 1 = 2(120) - 1 = 239. (Key point: Non-blocking slots)

2. For l = 150, c = 120 and p = 0.6 E, what is p'?
   A. 0.60
   B. 0.48
   C. 0.72
   D. 0.36
   Answer: B. p' = p c / l = 0.6 x 120 / 150 = 0.48. (Key point: Internal occupancy)

3. Which formula is used for P_B?
   A. P_B = [1-(1-p')^2]^l
   B. P_B = (1-p')^l
   C. P_B = p'^2
   D. P_B = 1-(1-p)^c
   Answer: A. Lee graph with l parallel paths of two links in series. (Key point: Blocking formula)

4. With l = 120 and p = 0.9 E, P_B is about
   A. 8.2e-10
   B. 0.299
   C. 0.99
   D. 4.8e-6
   Answer: B. q = 1 - 0.1^2 = 0.99 and 0.99^120 = 0.299, about 30%. (Key point: Result at 0.9 E)

## Related in Telecommunications

- [Space switch and why TST is used](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aspace-switch-why-tst) — 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
- [Digital exchange principles; non-blocking 3-stage](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Adigital-exchange-nonblocking-3-stage) — 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
- [Strictly non-blocking 3-stage condition](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Astrictly-nonblocking-3-stage) — Telecommunications, chapter 4
- [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

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