Multistage switching and STS blocking

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.

 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.