ST and TS drawbacks solved by STS

Drawbacks of ST and TS switches

A TS switch has a time stage followed by a space stage; an ST switch has a space stage followed by a time stage. Both are two-stage and give only one possible path for each connection, so they block heavily.

• TS drawback: the input TSI moves the sample from slot $i$ to the wanted output slot $j$ on its own highway, and then the space stage must connect the highway to the output highway in slot $j$. If another input highway already uses slot $j$ to reach the same output highway, the call is blocked, even though both A and B are free.
• ST drawback: the space stage must connect input highway $a$ to output highway $b$ in the input slot $i$. If slot $i$ is already used on output highway $b$ by a call from another input, the call is blocked. The output TSI could change the slot afterwards, but it never receives the sample.

 TS: [T] -> [S]   S must use slot j  (fixed)
 ST: [S] -> [T]   S must use slot i  (fixed)
 => one path only, blocking = P(that slot busy)

How STS solves this

An STS switch has an input space stage, a middle stage of $k$ TSIs and an output space stage.

 HW1 -->+------+  [TSI 1]  +------+--> HW1
 HW2 -->|N x k |->[TSI 2]->|k x N |--> HW2
 HWN -->+------+  [TSI k]  +------+--> HWN

1. In input slot $i$, the input S stage can send the sample to any free centre TSI $m$ (one with a free link in slot $i$).
2. TSI $m$ changes the slot from $i$ to $j$.
3. In slot $j$, the output S stage connects TSI $m$ to output highway $b$.
4. The call is blocked only if all $k$ centre TSIs are busy in slot $i$ or slot $j$.

So there are now $k$ alternative paths instead of one. Using a Lee graph with internal link occupancy $p' = pN/k$:

$$P_B = \left[1 - (1 - p')^2\right]^{k}$$

For $p' = 0.3$ and $k = 8$: $P_B = 0.51^8 \approx 4.6 \times 10^{-3}$, compared with about $0.3$ for a single path. With $k \ge 2N - 1$ the STS switch becomes strictly non-blocking.

(TST solves the same problem using $l$ internal time slots as alternative paths, and is usually cheaper.)