Time switch and two-stage T-S network

What is a time (T) switch?

A time switch, or Time Slot Interchanger (TSI), is a digital switch that moves PCM samples from one time slot to another on a TDM highway. Since each subscriber has a fixed slot, interchanging slots connects two subscribers.

How does it work?

 TDM in                                TDM out
 ------->+------------------------+------->
         | Speech memory (SM)     |
         | c words x 8 bits       |
         +------------------------+
   write addr ^              ^ read addr
   +-----------+        +-----------+
   | slot      |------->| Control   |
   | counter   |        | memory CM |
   +-----------+        +-----------+
                              ^ processor

• Sequential write, random read: in input slot $i$ the sample is written into SM($i$). In output slot $j$, CM($j$) holds $i$, so SM($i$) is read out in slot $j$.
• Random write, sequential read: CM($i$) = $j$ gives the write address; SM is read in order, so the sample leaves in slot $j$.
• Example: to connect TS3 to TS7 (sequential write), set CM(7) = 3. Delay = 4 slots ≈ 15.6 µs (slot = 3.9 µs).
• Timing: one write and one read per slot, so $t_{access} \le 125\ \mu s / 2c$; for $c = 32$, $t \le 1.95$ µs.

Is a two-stage TS switch possible?

Partly, but not as a practical exchange network. A TS switch can be built and it will work at very light load, but it suffers from heavy blocking, so it cannot give an acceptable grade of service.

 HW1 -->[TSI 1]--+-----------+--> OHW1
 HW2 -->[TSI 2]--|  N x N    |--> OHW2
  ...            |  space S  |     ...
 HWN -->[TSI N]--+-----------+--> OHWN

Working of TS: to connect A (slot $i$, highway $a$) to B (slot $j$, highway $b$):

1. TSI $a$ moves A's sample from slot $i$ to slot $j$ (the slot B needs).
2. In slot $j$ the space stage closes crosspoint $(a, b)$, sending the sample to highway $b$.

Why the TS switch is not suitable

1. Single fixed path: the space stage must use slot $j$, the called party's slot. If another input highway is already sending a call to highway $b$ in slot $j$, the crosspoint column for $b$ is busy in slot $j$, and the new call is blocked, even though A and B are free.
2. Also, if slot $j$ on the output of TSI $a$ is already used by another call from highway $a$ (to a different output highway), the call is blocked.
3. The blocking probability is roughly the chance that one of these links is busy: with link occupancy $p$,

$$P_B = 1 - (1 - p)^2$$

For $p = 0.5$: $P_B = 0.75$; for $p = 0.2$: $P_B = 0.36$. Telephone networks need $P_B$ of about 0.002 to 0.01. 4. There is no alternative path, so path search cannot help. 5. The same problem exists in the ST switch, where the space stage must use the caller's slot $i$.

 Call 1: HW1 -> HW3 using slot 7   (S busy: col 3, slot 7)
 Call 2: HW2 -> HW3, B is in slot 7 -> BLOCKED

How the problem is solved

Add a third stage to create alternative paths: TST (or STS).

 HWa->[TSI-A: i->x]->[S: a->b in x]->[TSI-B: x->j]->HWb

1. Input T stage: moves the sample from slot $i$ to any free internal slot $x$, not just slot $j$.
2. Space stage: in slot $x$ connects highway $a$ to highway $b$.
3. Output T stage: moves the sample from slot $x$ to the wanted slot $j$.
4. The controller can choose among $l$ internal slots, so the call is blocked only if all $l$ are unusable.

Blocking with Lee graph ($l$ parallel paths, two links each, $p' = pc/l$):

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

Example: $c = l = 32$, $p = 0.5$:

$$P_B = (1 - 0.25)^{32} = 0.75^{32} \approx 1.0 \times 10^{-4}$$

compared with $0.75$ for the TS switch.

Non-blocking TST: with time expansion $l \ge 2c - 1$ (e.g. 64 internal slots for 32-slot links), no internal blocking at all.

Other solutions

• STS: space stages outside and $k$ TSIs in the middle; $k$ alternative paths; non-blocking if $k \ge 2N - 1$. It needs more hardware than TST.
• TSST / TSSST: for very large exchanges, the middle space stage is made multistage (Clos), keeping crosspoints low.

Hence digital exchanges use TST (time–space–time) networks rather than two-stage TS switches.