Multistage switching and TST blocking

Multistage switching

Multistage switching builds a large switch from several stages of smaller switches connected by internal links, instead of one big $N \times N$ matrix. It cuts the number of crosspoints (or memory) needed, and gives several alternative paths for each call. The cost is a small chance of internal blocking unless enough paths are provided. Examples: three-stage space networks, STS, TST, TSST.

TST switching

A Time–Space–Time switch has an input time stage, a time-multiplexed space stage and an output time stage.

 Link 1 -->[TSI-A1]--+-----------+--[TSI-B1]--> Link 1
 Link 2 -->[TSI-A2]--| N x N     |--[TSI-B2]--> Link 2
   ...        ...    | space     |     ...
 Link N -->[TSI-AN]--| (TMS)     |--[TSI-BN]--> Link N
                     +-----------+
  c slots   in T      l internal    out T    c slots
                       slots

Working (example): connect slot 5 of link 1 to slot 20 of link N.

1. The control processor searches for an internal slot $x$ that is free both on the output of TSI-A1 and on the input of TSI-BN (say $x = 12$).
2. Input T stage: TSI-A1 moves the sample from slot 5 to slot 12.
3. Space stage: in slot 12 its control memory closes crosspoint (1, N), so the sample passes from highway 1 to highway N.
4. Output T stage: TSI-BN moves the sample from slot 12 to slot 20.
5. The return direction uses another slot, often $x + l/2$ (mod $l$), so one search serves both directions.

Since any internal slot $x$ can be used, there are $l$ alternative paths.

Blocking probability

Using a Lee graph: each path has two internal links in series (TSI-A → S and S → TSI-B), and there are $l$ paths in parallel. With external occupancy $p$ and time expansion $\alpha = l/c$, the internal link occupancy is $p' = p/\alpha$:

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

• Non-blocking when $l \ge 2c - 1$.
• Example: $c = l = 32$, $p = 0.6$: $P_B = [1 - 0.4^2]^{32} = 0.84^{32} \approx 3.8 \times 10^{-3}$.

Why it is preferred: time expansion is cheap (faster memory), the space stage is small, and the blocking is very low, so TST (and TSST) is used in most large digital exchanges.