0:03

So now I'm going to go through a small numerical example.

Â Clearly the scale or the scope does not illustrate the entire TD piece generality.

Â But it suffices to illustrate some of the key points in formulating one module in

Â the overall schematic that is the price optimization cuz I'm assuming all other

Â modules are already well done. And only focus on price or, equivalently,

Â price reduction or reward optimization in a very simple formulation.

Â Okay? So suppose we have, just two periods.

Â Okay? Night and, day.

Â And then, suppose that, on the particular training, we saw that there are two

Â classes of usage. Email and, file download.

Â I say, email and file download, which could be, say, a movie download from

Â iTunes. An email's, representation of, is a

Â representative of the delay intolerant traffic, and this is file download,

Â represent of delay tolerant traffic. And suppose, at night, the total, capacity

Â demand is a 300 megabyte from a small pool of, users in the neighborhood.

Â 1:43

And 200 out of that 300 belongs to e-mail. And the one remaining hundred megabyte

Â belongs to file download. And during the day, you also have 200

Â megabyte of e-mail, and then now however 200 instead of 100, megabyte of demand of

Â file download. So during night time, we have a total of

Â 300. During daytime, we have a total of 400.

Â And let's say the current maximum capacity supported is 350, represented by the blue

Â dotted line. Now clearly, at night time, it is

Â underutilized. And during daytime, it is over-utilized.

Â So we'll like to see if we can dump some of the daytime traffic to nighttime

Â traffic. And, ideally, stretch the line to a

Â straight line. In this particular, simple example's

Â numerical setup, such as straight line, ideally would be exactly 350, okay?

Â So the question is, you have to provide some incentives through whatever user

Â interface and pricing marketing plan so that some of these traffic will flow from

Â night, from daytime to nighttime. Let's say during nighttime, you offer a

Â certain reward. Okay, we're working now by P sub N.

Â And during day time we offer P sub D, N for night, D for day.

Â Now you would expect intuitively that a P sub N should be zero because you actually

Â don't exactly need incentive for traffic to move away from night into the day.

Â But for generality sake, let's give that a symbol.

Â And P sub D you would think would be strictly positive number, just don't know

Â exactly what that number is. So, let's assume that the ISP charge users

Â on a $ten per gigabyte basis, okay? Or one cents per megabyte.

Â And, the cost incurred for every gigabyte over capacity overshot, is, let's say,

Â $one. For example, representing the cost, you

Â need to roll out a track, a truck to, answer a customer complaint.

Â Or the consumer turn rate cost. Or the, Part of the cost for delivering,

Â 4:11

delivering higher capacities through, expansion plans.

Â What ever the cost Say this is $one per gigabyte for every gigabyte of, exceeding

Â the capacity. For example, during night time,

Â Before TDP, there is no cost of exceeding capacity during daytime.

Â However, there is, 50, megabyte of over [inaudible].

Â Okay, so that's the setup of this problem. Now, our optimization variables are

Â clearly P sub N and P sub T That's two simple scalar variables in this numerical

Â example. Now all we have to do is to understand

Â expected amounts that will be shifted, okay?

Â Let's say e-mail and file download are going to have a very simple weighting

Â function actually simpler, much simpler, than even this P over T + one to the beta.

Â Let's say, it's simply proportional to the price.

Â So, nighttime, let's say email, so probability of shifting a unit of traffic

Â simply the price p offered over four. And for daytime is P D / four.

Â For file transfer it is a more delayed tolerant and therefore the probability of

Â shifting unit traffic is P sub N / two during nighttime, and P sub D / two during

Â daytime. So, the fact is divided by four versus by

Â two is representative of the fact that file download is more delay tolerant than

Â e-mail. And this is a very simple linear

Â assumption here. So now we can write down the following,

Â that for e-mail, the amount that shift into night time,

Â 6:03

Okay, it is simply 200 units times P N / four.

Â And the amount shift into daytime is simply 200 times P D / four.

Â And for file transfer, the amount of shift into nighttime is 200 P N / two now, okay,

Â so this, and then the marsh shift into day time is, is simply, 100, P D / two, PD /

Â two because of this vector and 100 because it was 100 unit to start out with.

Â Okay? 100 unit during, night time, so, the amount that can be possibly shifted into

Â daytime, has a basis of 100 units. So now that we have this table, we can

Â easily calculate the expected amounts shifted into the nighttime and into the

Â daytime. Now since it's a binary, simple example-

Â two different time slots - of course, whatever goes into nighttime must have

Â come from daytime. Whatever goes into daytime must have come

Â from nighttime. This simplifies the derivation.

Â So now, we can express the following, okay.

Â 8:04

Okay, you can expand this and it is 150Pn^2 squared there's Pn times Pn +

Â 100PD squared. So there's pd times pd.

Â So it's a quadratic expression in the variables PN Pd for the cost of handing

Â out reward. So let's remember this term, and this will

Â be in the expression of the objective function, okay?

Â The other term in the expression of the objective function is the cause of

Â exceeding capacity. Which we said is $one for each gigabyte.

Â Now let's look at the amount of traffic here.

Â The amount of traffic shifted into night time, from the previous table is simply

Â the fall line. Equals 150 Pn okay, an amount of traffic

Â that shift into daytime, coming from nighttime, that is,

Â 9:10

Is simply the following. I'm just rewriting the expressions we just

Â found from that table two slides ago. Just 100 Pd.

Â So you add up these two terms. Well, I should say you subtract this term

Â from this term. Cuz this is the amount shifted into

Â nighttime. And this is the amount shifted into

Â daytime that is away. From nighttime.

Â This is into nighttime. Okay.

Â So the original nighttime traffic is 300 unit, plus 150 Pn is the amount you go

Â into. And then, minus the amount going out.

Â Minus 100 Pd And then you subtract the capacity you have, which is 350.

Â This entire expression therefore, is the cost of exceeding the capacity during the

Â nighttime. A more, strictly speaking we should say

Â this expression or zero whichever is larger so just in case this expression

Â becomes less than zero you just take zero, because you can never negative amount of

Â traffic. Now this is whole thing is for night time

Â you can, write down the similar expression for, the day time which turns out to be,

Â the max of 50 - 150 Pn + 100 Pd. The Pd, is, again, 300 minus, is actually

Â 400 - 350 for the daytime and, of course, the daytime nighttime shift in and out are

Â just exactly mirror image. Okay.

Â Whatever comes, from, gets into nighttime comes from daytime.

Â Whatever, gets out of nighttime goes into daytime.

Â And, you have to make sure that is positive.

Â So now you've got this term and this term. You can put them together, and that's the

Â total cost of exceeding capacity. So let's assume that this cost, the sum of

Â these two terms, and this cost, are equally weighted.

Â Then you just add up the whole thing together, and you have the following

Â objection, objective function: that's 150 Pn^2.

Â I'm simply arriving down. Well we are ready to write in the last two

Â slides. That's one term, okay, plus another term

Â which is the cost of exceeding capacity at nightt time.

Â 12:14

That's the entire objective function. And you would like to minimize this

Â objective, objective function with a following variables; simply two scalers,

Â Pn and Pd. As you can see this is not a smooth

Â function because of the max operation but it is a quadratic convex function and

Â you're minimizing over and therefore it's a convex optimization.

Â In any case it doesn't quite matter because it is only small problem with only

Â two degrees of freedom. Again, the price setting varies PNPD.

Â With a given Pn Pd, consumers will react differently And their reaction will, in

Â turn, drive the optimization. You can do it iteratively, or you can do

Â it one shot optimization. And, by incorporating the anticipated

Â reaction from the consumers. Where is that?

Â Well, that's all modeled in a simple weight here, wit these terms, okay?

Â That's the expected amount, probability of shifting for each unit of existing

Â traffic. So with that incorporated into the

Â objective function, we can solve this problem in one shot over these two

Â variables. And you see that the resulting pn is

Â actually 0.33 and pd is zero indeed. You should provide no incentive for

Â traffic to go into daytime. That's what PD denotes.

Â And optimized, that is, indeed, zero. Because daytime's already overcrowded.

Â Whereas, you should provide some strictly positive rewards while shifting traffic

Â into night time. That's what PN represents.

Â And the optimized value of PN star is 33. That is, you should discount the price

Â from $ten a gigabyte. Down to basically $6.67 per gigabyte for

Â nighttime. And if you induce, look at this, the

Â induced traffic is exactly 350 units for daytime and 350 units for nighttime.

Â 14:33

That is, We achieved the ideal target of shifting

Â just enough. 50, megabyte from daytime to night time.

Â And therefore straightening this line over a 24 hour period.

Â And exactly meeting the capacity. That is more like artifact of this

Â particular numerical example. In advanced material part of the lecture

Â meeting, we will talk about a more general, even though not the most general

Â steer, formulation of price optimization. And remember, this is only one out of

Â many, at least five, depending on how you count them, modules required for TDP to

Â work. In advanced material, I'll also be talking

Â about Paris Metro pricing, their smart idea to charge different prices, and then

Â induce different quality of service, as well as a brief introduction to two-sided

Â pricing, or this channelized 1-800 number for mobile data traffic.

Â So, before the advanced material part of the lecture, I would like to summarize

Â what we have covered. In this lecture, we highlight that SDP,

Â smart data pricing, may create win-wins across consumers, SUME, ISPs, and content

Â providers. In particular charging based on when is a

Â very powerful idea used in many different industries, in mobile data industry.

Â It may help time shift traffic through peak valley cyclic fluctuations.

Â And across the last two lectures, we have seen a suite of ideas, and models, and

Â design methodologies for mobile or, more generally, internet data, especially.

Â Data access pricing. And this is one big part of network

Â pricing. Together with what we already talked

Â about, such as auction. Methods, and other kind of ways to

Â internalize negative externalities. And together with future topics such as

Â fairness of allocation. These constitute our work, discussion on

Â the economics side of network behavior. Now we have talked about internet many

Â times including internet access pricing, the last two lectures and the overly

Â network, including Wikipedia, including Netflix, including Google, Facebook,

Â Twitter, on top of the internet. So what is the internet?

Â