So just taken again the equation that we just found for the per-station throughput. We said that a per-station throughput was going to be the probability of transmitting times 1 minus the probability of transmitting. And then we would do that multupliatin. However, many stations we had minus 1, because the one that's transmitting we don't have to include in this. So this is numstat minus 1 times. So we see that This throughput is dependent on two things. The first is the probability of transmitting, which we expected, and then the seconds also the number of stations, okay, and so we don't really have control over the number of stations that are under our access point because they will come and go especially if it's a free access point, that's one of the advantages of having a locked... [INAUDIBLE] besides the fact that people can't steal your WiFi, then you don't have people actually disrupting your data rate, as well, but what we can control is this probability of transmitting. So, let's see what happens, and let's see if we can figure out what the throughput will be as we change The probability of transmitting. So in this graph right here, we're showing, for example five stations. So we're saying that Numstat is equal to five in that equation. So this really gets, this multiplication happens four times. And then we're seeing what happens when we change the probability of a station actually transmitting. And so, we can look at either the total or the per-station throughput here it doesn't really make much of a difference because it's just a multiplication factor as we said. We just, to get the total throughput we just multiply this again by the number of stations we have and that's how we translate from, from this graph up to the top. And the higher one, total throughput will always be greater than the per-station throughput. Obviously, because we're just considering more stations. But you see this trend here that first it's going to increase. Right? And then it's going to hit some maximum value before then it starts to taper off and decrease. And this fits our intuition that we don't want it to be zero, because then the throughput will be zero, and we also don't want it to be one because then we'll always have collisions, because that's 100% transmission. It's somewhere in between but it's skewed now towards the lower side. And that maximum we hit we can see right here graphically, is when the probability of transmission is at 20%. So we can say the best ProbTrans is 20%, so then we might say from this graph, well okay then let's set probability of transmission to be 20%. So then one in every five frames, on average, every station will transmit. Notice how there are five stations. So the next question is, well, are we done? Are we satisfied with this 20% probability of transmission. Clearly we can't do any better if we have five stations. We see here it's, we're getting our maximum. We don't have anything else to play with. There's no other knobs that we can turn or anything to change any other parameters. And, so, but it turns out that as we change the number of stations we're going to see a little bit of a different story. Okay? So, that what we just saw was for NumStat which we're showing here equal to five. And now, this is a graph That's showing what happens as NumStat varies. So it's just a bunch of superimposed plots. The first one in the blue over here, this straight line is for NumStat equal to one and that should make sense that if we had one station, we would always want that station to transmit. Because then we get 100% throughput, because we never have to worry about interfering, so if you're the only station, go ahead. You can get your maximum possible throughput and your really impressive data rate, but once we have even two stations, that's when these things start to taper off. Right, and that's when the throughput starts to decrease. So if you notice a few trends about this graph. The first thing is that the maximum that we can achieve goes down. And it goes down pretty quickly as the number of stations increases. So we have one station we can get 100%. Two stations, the maximum we could get was 0.5. Three stations, the maximum we can get is up over here and then with five stations, it's down here. With ten stations, it's down here. And so, as you can see, [INAUDIBLE] as soon as we add a second station in, we already have had their total throughput that we can get. So, and that's not even saying that pre-station throughput is half, that each one gets half. It's the total throughput. So when you add the two throughputs up, they're less than what one station can get if the one station was there just by itself. So that's a pretty unfortunate situation, clearly. The second thing that you'll notice is the probability of transmitting. The ideal, the optimal probability of transmitting, or the best one that we can select goes down as the number of stations goes up. Which should make sense because it's kind of along the lines that you have to wait more. As there's more stations there because there's more of a chance that someone else Has something to send. Or someone else is going to send in that time slot. So, prob-trans here, for two stations, the ideal is 50%. So, half, and half, which should make sense. For three stations it's 0.333. It's actually one third. This is one half. For five station its actually 1 5th and for 10 stations its 1 10th. You should know there is the trend there and indeed that trend is the case that the [UNKNOWN] probability of transmitting is simply going to be equal to one over the number of station [SOUND] For this version of Aloha that we're looking at. So, if you could find the total number of stations and you took one over that, and every device was following that, that would be giving you the maximum total throughput that you could possibly get. But the key idea again is that the best probability of transmitting is going to depend upon the number of stations. So we can't just set it at 20%, or 50%, or anything. We have to know the total number of stations in order to get the best total throughput. So in this graph right here I'm plotting a maximum total throughput So I am basically taking this graph and I am just plotting these values here and seeing what happens as the change number of stations, this is for one station, this is for two, this is for three stations, this is for This is for five, this is for ten. So, this is what happens as the number of stations change. So, we start at one station we saw before it drops down to 50%. This is just showing there's a simpler form then we dropping it down again and again and it turns out that the, the maximum or the sorry, the minimum that we can get as we increase the number of stations to infinity we're going to approach Some limit here, that it won't drop below. And at least that's something good, I suppose, in that we do hit some limit, so it doesn't go down to zero. And that nobody's ever going to get through. And that limit is about about 37%. [SOUND] So it's going to go down, and keep going down and crawl around 37%. But this is again, this is maximum total throughput. So, the per-station throughput, you have to take this and divide this by the number of stations. So, as the number of stations goes to infinite, this would go down to zero for the per-station throughput. So, the total throughput, the limit and the lowest I can go is 37%. [SOUND] But for the per-station [SOUND] that would go all the way down theoretically to 0%. I mean, you'll never actually hit 0% because obviously someone has to be getting something in order for to be a total of 37%, but in the [UNKNOWN] you're going to be getting such a small amount of throughput. And that's really not a very desirable scenario to be in.