When we built up the previous model, we assume that the population only flow within the two closest shop. In fact the population flows everywhere. That how can we express the average population density as a average of a whole day? Putting them in equal proportion is obviously not fair. Our solution will be:

Assume that for a particular street, the closeset distance from the street to the Starbucks shops willbe a1,a2,..., am units while the distance to the Coffee Bean will be b1,b2,...,bn units. Then our market share will be:

*[Σ(1/bi)/Σ(1/bi)+Σ(1/ai)].*The idea is that we still have to show that the closer distance will have a higher chance of providing service to a (averagely) closer customers. It can also shows that a very close shop does have a very strong influence on that street (which is the real case).

"Distance tends to zero will results in an infinity-ly large influence to the streer?"

This problem makes no sense. The distance is assumes to be an integer, and the indfluence is measured in terms of the whole street. Even though the man living besides the shop goes there everyday, the shop can't influence the whole street at all.

Now note that consider the model when there's only 1 Starbucks and 1 Coffee Bean shop. The market share will be

*(1/b)/(1/a+1/b)=a/(a+b)*, which is just same ax the model of (i)! Then we are success in considering the extended model.

in this model, the locating strategy is nearly the same as (i) so we need not to discuss again. Just a point to note that the locating of n-junction will be much more important.

One important function of this model is that the share of a shop to a particular segment of a street is not as exact as 1 or 0 or 0.5, but they're equalizing to half-half. They are

**competing**. That's why we need some strategy to win the competition.

(Note that the share originally is 1, falled to less than one, higher than 0.5, if the share is half-half it's still the same, and if it is zero it will raise a bit.)

6)Population denstiy

In the above model we are considering the

**flow**of population. However one of the assumption still exist -- the population density is even. If it is not even, the proportion will be different.

In the real society, Coffee Shop starts their business in the business zone where people used to negotiate there. A rural area will have a higher potential population density than a urban area. It will be very complicated for us to compute those values. In order to show that the model is efficient to compute the real model.

consider the above map (2*4) again. Now assume that the segment (1,2) to (3,2) is a trade zone where people likes coffee so much there (they don't put a coffee machine at office right?), hence a double share to the coffee shop. The new share will be 5.5/16, where the rate of increasing will be 14% which compansated the deficit of locating in a bad location which fewer customers.

Combining thw two models, the new share will be about 7.2/16=0.45 which is nearly half! We see that the effect of trade zone is even larger than the density.

we will take care of the long-run considerationg in the next discussion.

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