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Arcane, or just good manners? (Not that being late is good manners anyway, but then disturbing the debate whilst making your way to a seat compounds the insult).
There is not enough space for everyone (around 427 seats for 650 MPs, and it was even rebuilt this way after war damage), but:
1. It's pretty rare for everyone to be present anyway, except maybe for state occasions or really important and close-run debates
2. There are more MPs than when the Chamber was originally designed...
3. ... and people have got bigger since then, too. (For one thing, there are now many many female MPs who tend to have wider hips)
However it's not unusual to see them really crammed in, especially on the front benches; one can't help but think that the crush must both distract from concentrating on the issues at hand, and also hurry things along to some extent.
It's not unusual for MPs to spend debate time in other locations, and only attend when the division bell is rung. This is why there is a "repeater" bell in several nearby pubs - so long as they're within an 8-minute dash.
I did laugh at the OP's terminology of "deputies". Technically of course they are deputised to represent their constituents, but most of them would never consider themselves to be deputy to anyone!!
Deputy is the Irish coming out in me. It's an alternative term for our members of parliament, officially a Teachta Daila or TD. We would use it where the British might use Right Honourable or Honourable Gentleman - not sure which, perhaps both