Let's try now to come up with a way to describe the throughput in an ALOHA based
WiFi network. So, let's consider the situation again
where we have three devices, A, B and C. And then, we have two access points, D
and E, all of which are WiFi stations. Again, and all of which can interfere
with one another. So, first, let's look at the different
probabilities and things what, so what has to happen for a successful
transmission? Well, two things has to happen at the
same time. The first is that one station has to
transmit. We looked at it before, if probability of
transmission is too low, then those stations might transmit.
And therefore, there's no successful transmission and no throughput in that
time slot. So, what's the probability of having one
station transmit? So, let's look at station B.
And we want to say, for station B, what's the probability that station B is going
to transmit in this time slot? Well, simple, it's just ProbTrans, which
we said before. So, we could just write down here,
ProbTrans for that probability. the second thing that must happen though
is that every other station has to not transmit.
So, while B will transmit in this time slot, we have to ensure that A, C, D and
E each will not transmit. Okay, so now, we're just taking the
logical negation of the probability that it transmits.
So, if there is a 10% chance that you transmit, then that means there's a 90%
chance that you don't transmit. If there is a 1% chance that you
transmit, that means there is a 99% chance that you don't.
Just like heads and tails. When you flip the coin, the chance to get
heads is 50% and the chance to get tails is 1 minus 0.5, or 100% minus 50% which
is 50%. So now, when we express probabilities and
that's another thing we should point out. In everyday speech we would say the
probability of something would be 10% or 20%, and we use a percent sign.
When we write them in mathematical equations we express them typically in
decimal form unless otherwise stated. So, 50% is the same thing as saying 0.5,
in probability. So, the probability must always range
between 0 and 1. A 0 means that it'll never happen, and
that 1 means that it will always happen. So, ProbTrans, for instance, if we made
ProbTrans 0, then that means that the stations would never transmit.
Clearly, that's very undesirable because then you're never going to get any
throughput. But then, at the same time if you've made
them all 1 or 100%, then they would all be constantly transmitting which is also
undesirable. Because then, you would also have no
throughput. So again, it needs to be somewhere in
between, but so we back to this again we are saying that every other station has
to not transmit in this time slot. Okay, so ProbTrans the probability that b
or any other station transmits. The probability that another station
doesn't transmit is 1 minus ProbTrans. And it's not just one station though.
Every station must not transmit. And so, when we say and which we're
saying A does not transmit and D does not transmit and C does not transmit and E
does not transmit, we multiply probabilities multiply as long as they're
independent. And we're talking about things happening
simulationously. And we won't get into what we mean by
independence. Thinks on those source.
Suffice it to say that these are independent events, therefore we multiply
the probability. So, we have 1 minus ProbTrans, then we
multiply that by 1 minus ProbTrans. And again, for the third station times 1
minus ProbTrans. And then again for the fourth station,
time 1 minus ProbTrans. So, we just want to do that
multiplication four times because it's four different devices that have to not
be transmitting at the same time. And this is that probability.
So, suppose, for instance, the probability of transmission was 50%, then
1 minus the probability of transmitting would also be 50%.
So, we'd have 0.5 times 0.5 times 0.5 times 0.5.
And that would give us 0.25 times 0.5 is 0.125 times 0.5 is 0.0625 is that.
and you can take other examples into that out.
But basically we're just multiplying this through, and you just have to subtract 1,
do 1 minus ProbTrans. So, if it was, for instance, 20%, then
this would be 1 minus that is 0.8. And then, multiplied by 0.8 again, will
give you 0.64. And then, keep doing that multiplication
four times to get the probability that every other station is not going to be
transmitting except for the one that we're looking at.
And you can more conveniently express this.
This is just a side note as 1 minus ProbTrans to the 4th power and that
denotes multiplying four times.
