So now I'm going to go through a small numerical example. Clearly the scale or the scope does not illustrate the entire TD piece generality. But it suffices to illustrate some of the key points in formulating one module in the overall schematic that is the price optimization cuz I'm assuming all other modules are already well done. And only focus on price or, equivalently, price reduction or reward optimization in a very simple formulation. Okay? So suppose we have, just two periods. Okay? Night and, day. And then, suppose that, on the particular training, we saw that there are two classes of usage. Email and, file download. I say, email and file download, which could be, say, a movie download from iTunes. An email's, representation of, is a representative of the delay intolerant traffic, and this is file download, represent of delay tolerant traffic. And suppose, at night, the total, capacity demand is a 300 megabyte from a small pool of, users in the neighborhood. And 200 out of that 300 belongs to e-mail. And the one remaining hundred megabyte belongs to file download. And during the day, you also have 200 megabyte of e-mail, and then now however 200 instead of 100, megabyte of demand of file download. So during night time, we have a total of 300. During daytime, we have a total of 400. And let's say the current maximum capacity supported is 350, represented by the blue dotted line. Now clearly, at night time, it is underutilized. And during daytime, it is over-utilized. So we'll like to see if we can dump some of the daytime traffic to nighttime traffic. And, ideally, stretch the line to a straight line. In this particular, simple example's numerical setup, such as straight line, ideally would be exactly 350, okay? So the question is, you have to provide some incentives through whatever user interface and pricing marketing plan so that some of these traffic will flow from night, from daytime to nighttime. Let's say during nighttime, you offer a certain reward. Okay, we're working now by P sub N. And during day time we offer P sub D, N for night, D for day. Now you would expect intuitively that a P sub N should be zero because you actually don't exactly need incentive for traffic to move away from night into the day. But for generality sake, let's give that a symbol. And P sub D you would think would be strictly positive number, just don't know exactly what that number is. So, let's assume that the ISP charge users on a $ten per gigabyte basis, okay? Or one cents per megabyte. And, the cost incurred for every gigabyte over capacity overshot, is, let's say, $one. For example, representing the cost, you need to roll out a track, a truck to, answer a customer complaint. Or the consumer turn rate cost. Or the, Part of the cost for delivering, delivering higher capacities through, expansion plans. What ever the cost Say this is $one per gigabyte for every gigabyte of, exceeding the capacity. For example, during night time, Before TDP, there is no cost of exceeding capacity during daytime. However, there is, 50, megabyte of over [inaudible]. Okay, so that's the setup of this problem. Now, our optimization variables are clearly P sub N and P sub T That's two simple scalar variables in this numerical example. Now all we have to do is to understand expected amounts that will be shifted, okay? Let's say e-mail and file download are going to have a very simple weighting function actually simpler, much simpler, than even this P over T + one to the beta. Let's say, it's simply proportional to the price. So, nighttime, let's say email, so probability of shifting a unit of traffic simply the price p offered over four. And for daytime is P D / four. For file transfer it is a more delayed tolerant and therefore the probability of shifting unit traffic is P sub N / two during nighttime, and P sub D / two during daytime. So, the fact is divided by four versus by two is representative of the fact that file download is more delay tolerant than e-mail. And this is a very simple linear assumption here. So now we can write down the following, that for e-mail, the amount that shift into night time, Okay, it is simply 200 units times P N / four. And the amount shift into daytime is simply 200 times P D / four. And for file transfer, the amount of shift into nighttime is 200 P N / two now, okay, so this, and then the marsh shift into day time is, is simply, 100, P D / two, PD / two because of this vector and 100 because it was 100 unit to start out with. Okay? 100 unit during, night time, so, the amount that can be possibly shifted into daytime, has a basis of 100 units. So now that we have this table, we can easily calculate the expected amounts shifted into the nighttime and into the daytime. Now since it's a binary, simple example- two different time slots - of course, whatever goes into nighttime must have come from daytime. Whatever goes into daytime must have come from nighttime. This simplifies the derivation. So now, we can express the following, okay. The cost of reward, is simply the following. For nighttime is unit reward P N times, the, a month of reward. Amount of traffic expecting reward. What is that? I'm simply, okay adding these two terms here. Similarly, I'm going to add up these two terms, and multiply by the daytime reward. Okay, you can expand this and it is 150Pn^2 squared there's Pn times Pn + 100PD squared. So there's pd times pd. So it's a quadratic expression in the variables PN Pd for the cost of handing out reward. So let's remember this term, and this will be in the expression of the objective function, okay? The other term in the expression of the objective function is the cause of exceeding capacity. Which we said is $one for each gigabyte. Now let's look at the amount of traffic here. The amount of traffic shifted into night time, from the previous table is simply the fall line. Equals 150 Pn okay, an amount of traffic that shift into daytime, coming from nighttime, that is, Is simply the following. I'm just rewriting the expressions we just found from that table two slides ago. Just 100 Pd. So you add up these two terms. Well, I should say you subtract this term from this term. Cuz this is the amount shifted into nighttime. And this is the amount shifted into daytime that is away. From nighttime. This is into nighttime. Okay. So the original nighttime traffic is 300 unit, plus 150 Pn is the amount you go into. And then, minus the amount going out. Minus 100 Pd And then you subtract the capacity you have, which is 350. This entire expression therefore, is the cost of exceeding the capacity during the nighttime. A more, strictly speaking we should say this expression or zero whichever is larger so just in case this expression becomes less than zero you just take zero, because you can never negative amount of traffic. Now this is whole thing is for night time you can, write down the similar expression for, the day time which turns out to be, the max of 50 - 150 Pn + 100 Pd. The Pd, is, again, 300 minus, is actually 400 - 350 for the daytime and, of course, the daytime nighttime shift in and out are just exactly mirror image. Okay. Whatever comes, from, gets into nighttime comes from daytime. Whatever, gets out of nighttime goes into daytime. And, you have to make sure that is positive. So now you've got this term and this term. You can put them together, and that's the total cost of exceeding capacity. So let's assume that this cost, the sum of these two terms, and this cost, are equally weighted. Then you just add up the whole thing together, and you have the following objection, objective function: that's 150 Pn^2. I'm simply arriving down. Well we are ready to write in the last two slides. That's one term, okay, plus another term which is the cost of exceeding capacity at nightt time. And the cost of exceeding capacity in daytime. That's the entire objective function. And you would like to minimize this objective, objective function with a following variables; simply two scalers, Pn and Pd. As you can see this is not a smooth function because of the max operation but it is a quadratic convex function and you're minimizing over and therefore it's a convex optimization. In any case it doesn't quite matter because it is only small problem with only two degrees of freedom. Again, the price setting varies PNPD. With a given Pn Pd, consumers will react differently And their reaction will, in turn, drive the optimization. You can do it iteratively, or you can do it one shot optimization. And, by incorporating the anticipated reaction from the consumers. Where is that? Well, that's all modeled in a simple weight here, wit these terms, okay? That's the expected amount, probability of shifting for each unit of existing traffic. So with that incorporated into the objective function, we can solve this problem in one shot over these two variables. And you see that the resulting pn is actually 0.33 and pd is zero indeed. You should provide no incentive for traffic to go into daytime. That's what PD denotes. And optimized, that is, indeed, zero. Because daytime's already overcrowded. Whereas, you should provide some strictly positive rewards while shifting traffic into night time. That's what PN represents. And the optimized value of PN star is 33. That is, you should discount the price from $ten a gigabyte. Down to basically $6.67 per gigabyte for nighttime. And if you induce, look at this, the induced traffic is exactly 350 units for daytime and 350 units for nighttime. That is, We achieved the ideal target of shifting just enough. 50, megabyte from daytime to night time. And therefore straightening this line over a 24 hour period. And exactly meeting the capacity. That is more like artifact of this particular numerical example. In advanced material part of the lecture meeting, we will talk about a more general, even though not the most general steer, formulation of price optimization. And remember, this is only one out of many, at least five, depending on how you count them, modules required for TDP to work. In advanced material, I'll also be talking about Paris Metro pricing, their smart idea to charge different prices, and then induce different quality of service, as well as a brief introduction to two-sided pricing, or this channelized 1-800 number for mobile data traffic. So, before the advanced material part of the lecture, I would like to summarize what we have covered. In this lecture, we highlight that SDP, smart data pricing, may create win-wins across consumers, SUME, ISPs, and content providers. In particular charging based on when is a very powerful idea used in many different industries, in mobile data industry. It may help time shift traffic through peak valley cyclic fluctuations. And across the last two lectures, we have seen a suite of ideas, and models, and design methodologies for mobile or, more generally, internet data, especially. Data access pricing. And this is one big part of network pricing. Together with what we already talked about, such as auction. Methods, and other kind of ways to internalize negative externalities. And together with future topics such as fairness of allocation. These constitute our work, discussion on the economics side of network behavior. Now we have talked about internet many times including internet access pricing, the last two lectures and the overly network, including Wikipedia, including Netflix, including Google, Facebook, Twitter, on top of the internet. So what is the internet? Okay. It is a network of networks. And what kind of protocols control this network of networks? That would be the subject matter in the next sequence of lectures, starting with the basic foundational ideas behind the design of Internet, for our next lecture. So see you at the next lecture.