Imagine a collection of nine rooms, R1, R2.. Rs connected as in the figure below
ID: 3194992 • Letter: I
Question
Imagine a collection of nine rooms, R1, R2.. Rs connected as in the figure below. A social gathering involving N people takes place in these rooms, and the percentage of persons in each room is x1, V2,..., rg As time progresses it is observed that in a time span of fifteen minutes people mingle and either stay where they are or migrate to an adjacent room with equal probability (a) Suppose that 100 people are distributed through out the rooms so that R1 has 30 people, R2 has 40 people, R has 10 people and Rs has 20 people. After fifteen minutes how many people do we expect to have in each room (b) The number of people observed at a given time in rooms R, R2,., Ro respec- tively is (125, 50, 150, 75, 275, 100, 125, 50, 150). What was the number of persons in each room fifteen minutes earlier? (c) If everybody begins in R, how long until there are people in Rg? At that time, what is the distribution of people throughout the rooms!? (d) If you were to cater an event at this location, how would you divide your resources through out the nine rooms?Explanation / Answer
A) Given that people stay in the same room or move to an adjacent room with equal probability. By that logic ---->
The 30 people in R1 have 3 equally likely options - Stay in R1 / go to R2 / go to R4.
That would mean after 15 minutes, R1=10. R2=10, R4=10, Rest = 0 ---------(1)
For the 40 people in R2 have 4 equally likely options.
Stay in R2 / Go to R1 / Go to R3 / Go to R5
Therefore, combining with equation (1) ---->
R1 = 20, R2 = 20, R3 = R4 = R5 = 10, Rest = 0.
For the 10 people in R5, they have 5 equally likely options.
Stay in R5 / Go to R2 / Go to R4 / Go to R6 / go to R8
After 15 mins,
R1 = 20, R2 = 22, R3 = 10, R4 = 12, R5 = 12, R6 = 2, R7 = 0, R8 = 2, R9 = 0
For the 20 people in Room 8 have 4 equally likely options.
Stay in R8 / Go to R7 / Go to R9 / Go to R5.
After 15 mins,
FINAL ANSWER to A) : 1 = 20, R2 = 22, R3 = 10, R4 = 12, R5 = 17, R6 = 2, R7 = 5, R8 = 7, R9 = 5
B) We need to assume that 15 minutes earlier, there were n1 number of people in R1, n2 number of people in R2, ... till n9 number of people in R9.
We need to trace developments for each room through the entire 15 minute span keeping in mind the movement associated with its adjacent rooms as well.
For example. at the start of the time period, if R1 has n1 people.
Splitting in R1 -> n1/3 in R1
Splitting in R2 -> n2/4 in R1
Splitting in R4 -> n4/4 in R1
Splitting associated with the other rooms will not affect the number of people in R1 anyway. So we calculate the total number of people in R1 = n1/3 + n2/4 + n4/4 = 125
Similarly we can find the total number of people in each room at the end of 15 minutes.
Room R2 = n2/4 + n1/3 + n3/3 + n5/5 = 50
Room R3 = n3/3 + n2/4 + n6/4 = 150
Room R4 = n4/4 + n1/3 + n7/3 + n5/5 = 75
Room R5 = n5/5 + n2/4 + n4/4 + n6/4 + n8/4 = 275
Room R6 = n6/4 + n3/3 + n9/3 + n5/5 = 100
Room R7 = n7/3 + n8/4 + n4/4 = 125
Room R8 = n8/4 + n7/3 + n9/3 + n5/5 = 50
Room R9 = n9/3 + n8/4 + n6/4 = 150
Solving these equations, we get the following values for n1, n2 ,... n9 ----->
FINAL ANSWER :
n1 = 0
n2 = 200
n3 = 0
n4 = 300
n5 = 0
n6 = 400
n7 = 0
n8 = 200
n9 = 0
C) At time T = 0, let us assume there are N people in Room R1 and 0 people in other rooms (R2-R9).
At time T = 15, people would have moved from R1 to R2, R4. So there are people in R1, R2 and R4.
At time T = 30, People would move from R2 to R1,R5,R3 and from R4 to R5, R7, R1. So we have people in R1,R2,R3,R4,R5,R7.
At time T = 45, People would move from R3 to R2, R6 and from R7 to R8, R4. So we now have people in R1,R2,R3,R4,R5,R6,R7,R8.
At time T = 60, People would have moved from R8 or R6 to R9.
FINAL ANSWER: Hence, after 60 minutes, there will be people in R9.
The distribution of the rooms at T = 60 ---->
Again, assuming there were N people in R1 and 0 people in other rooms at Time T = 0.
At time T = 15 --->
At time T = 30 ----->
At time T = 45 ---->
At time T = 60 ----->
D) As an Event Co-ordinator at this location, the strategy would be subjective and would be catered according to requirements. But in my opinion,
R1, R3, R7, R9 are Category 1.
R2,R4,R6,R8 are Category 2.
R5 is Category 3.
To allocate resources, Category 3 i.e R5 has the most adjacent neighbours so the number in R5 goes down over time. I would allocate my most resources to R5 hence to get an equitable number over time.
Next, I would give lesser resources to Category 2 and the least resources to Category 1 as Category 1 has lesser neighbours than Category 2. So the number would stay more stable in Category 1 Rooms than Category 2 rooms. Hence, the order of allocation of resources would be :
R5 > R2,R4,R6,R8 > R1,R3,R7,R9
Cheers!
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