Multistage switching; TST vs STS

Advantages of multi-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 switch

 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

An incoming time slot $i$ is moved by the first TSI to a free internal slot $j$, passed through the space matrix in slot $j$, and moved by the outgoing TSI to the wanted slot $m$. Blocking happens only if no internal slot is free on both the input and output highways. With $l$ internal slots and $p' = p\,c/l$:

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

STS switch

 incoming    S stage      T stage      S stage   outgoing
 PCM 1 --->+--------+--->[ TSI 1 ]--->+--------+---> PCM 1
 PCM 2 --->| N x k  |--->[ TSI 2 ]--->| k x N  |---> PCM 2
  ...      | space  |     ...         | space  |
 PCM N --->+--------+--->[ TSI k ]--->+--------+---> PCM N

The first space stage connects the incoming highway (in its own slot) to one of $k$ time switches; the TSI moves the sample to the outgoing slot; the second space stage connects it to the outgoing highway. Blocking happens if none of the $k$ modules has both links free. With $p' = p/\beta$:

$$\begin{aligned} p' &= \frac{p}{\beta},\quad \beta = \frac{k}{N} \\ P_B(\text{STS}) &= \left[1 - (1-p')^2\right]^{k} \end{aligned}$$

Comparison of TST and STS

Example ($p' = 0.2$): STS with $k = 8$ gives $P_B = 0.36^8 \approx 2.8 \times 10^{-4}$, while TST with $l = 32$ gives $0.36^{32} \approx 6.3 \times 10^{-15}$. Hence TST is preferred for large exchanges.