Multistage advantage and TST switching

Advantages of multi-stage over single-stage switching

• Far fewer crosspoints: a single-stage $N \times N$ switch needs $N^2$ crosspoints (e.g. $10^6$ for 1000 lines); a three-stage switch needs only a small fraction, so cost and size fall.
• Better crosspoint utilisation: in a single stage at most $N$ of the $N^2$ crosspoints can be in use at once (each input and output takes part in only one connection), so utilisation is at most $1/N$; in multi-stage each crosspoint is shared by many possible connections.
• Alternative paths: a call can go through any of several middle-stage arrays, so failure of one crosspoint does not isolate a subscriber (better reliability).
• Modular growth: capacity can be expanded by adding arrays.
• Concentration and expansion can be built into the stages to suit traffic.
• Drawbacks: some blocking may occur, and path search (control) is more complex.

TST switching

A Time–Space–Time (TST) switch is a three-stage digital switching network: an incoming time stage, a space stage in the middle, and an outgoing time stage. It is the most common structure in large digital exchanges.

 incoming     T stage      S stage       T stage   outgoing
 highways   (time slot   (crosspoint   (time slot  highways
            interchange)   matrix)    interchange)
 PCM 1 --->[ TSI ]--->+-------------+--->[ TSI ]---> PCM 1
 PCM 2 --->[ TSI ]--->|  N x N      |--->[ TSI ]---> PCM 2
  ...                 |  space sw.  |
 PCM N --->[ TSI ]--->+-------------+--->[ TSI ]---> PCM N
 slot i  -> slot j (internal) -> slot j -> slot m

Operation (connect slot $i$ of PCM 1 to slot $m$ of PCM N):

1. First time stage: the time slot interchanger (TSI) on PCM 1 writes incoming samples into its speech memory in order and reads the sample of slot $i$ out in a free internal slot $j$ (chosen by the control).
2. Space stage: in internal slot $j$ the crosspoint connecting row 1 to column N is closed (set by the space-switch control memory), passing the sample to the outgoing side. The same crosspoint is reused by other connections in other slots.
3. Second time stage: the TSI on output highway N stores the sample and reads it out in the required outgoing slot $m$.
4. The reverse direction uses a related internal slot (often half a frame away, $j + l/2$, where $l$ is the number of internal slots) so both directions are set up together.

Because the space stage only needs to find one free internal slot common to both links, the many internal time slots act like many middle-stage arrays, giving very low blocking at low cost. The time stages can also expand ($l > c$ internal slots) to reduce blocking further.

Blocking probability of TST (Lee graph)

           slot 1
       +--- ... ---+
  A ---+--- slot j-+--- B     l parallel paths
       +--- ... ---+          (internal slots)
           slot l

Let $p$ = occupancy of an incoming channel, $c$ = channels per highway, $l$ = internal slots, time expansion $\alpha = l/c$. Each internal link is busy with probability $p' = p/\alpha$. A path is blocked if either of its two links is busy: $1-(1-p')^2$. The call is blocked only if all $l$ paths are blocked:

$$\begin{aligned} p' &= \frac{p}{\alpha},\quad \alpha = \frac{l}{c} \\ P_B(\text{TST}) &= \left[1 - (1-p')^2\right]^{l} \end{aligned}$$

Example: $p = 0.2$, $l = c = 32$: $P_B = (1 - 0.8^2)^{32} = 0.36^{32} \approx 6.3 \times 10^{-15}$, practically non-blocking.