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

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.

 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

 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

 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 $4.4 \times 10^{-10}$.

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.