7/23/2019 Markov Chain Example Problem

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Markov Chain Problem (MAT-208)

Joseph Heavner 

December 11, 2015

PROBLEM:

UPS has a fleet of 270 planes that shuttle between three distribution hubs: Seattle, Chicago, and Atlanta. Each

day, most of the planes fly round trips, returning to their starting location; however, some remain at the other

two airports. Here are the percentages that move from each location:

Each day 2% of the planes from Seattle remain at Chicago and 8% remain at Atlanta; the rest return to Seattle.

Each day 3% of the planes from Chicago remain at Seattle and 6% remain at Atlanta; the rest return to Chicago.

Each day 9% of the planes from Atlanta remain at Seattle and 4% remain at Chicago; the rest return to Atlanta.

1.   Find the stochastic matrix that represents this situation.

2.   Assuming that the distribution of planes has reached a steady state, how many planes are at each location? 

SOLUTION:

1. The stochastic matrix can be found by simply making the entry in the  i, j   position the percent of planes coming from location i  and going to location  j , where i  = 1, 2, 3 corresponds to Seattle, Chicago,and Atlanta, respectively, and similarly for  j  = 1, 2, 3. Let  M  be the stochastic matrix, then

M =

.90   .03   .09.02   .91   .04.08   .06   .87

2. A steady-state vector is an eigenvector with eigenvalue 1. As such, we compute  x  as follows.

(M  − I )x =  0   =⇒

−.10   .03   .09 0

.02   −.09   .04 0

.08   .06   −.23 0

∼ · · · ∼

−.623656 1 0 0−.903226 0 1 0

0 0 0 0

  =⇒   x =  x1

1.623656.903226

We would usually try to choose a nice value for   x1, but because   x   has no simple fraction entries oranything of the like, there is no obvious choice for  x1   that would simplify the vector. So, we simplychoose x1 = 1. This gives us a basis for the solution space, which we shall denote  q.

Now, we want a  probability vector , call it   p, so we need to normalize   q. To do this we simply sum thecomponents: 1 + .623656 + .903226 = 2.526882, and divide each component by this sum,  i.e.

1 → 1/2.526882 = .3957,  et cetera . It is worth noting at this point that almost all of our values hithertohave been approximate. Henceforth we may truncate decimals, as few additional calculations are neededand, after all, we do not have an arbitrarily high degree of certainty in the real world, anyhow. After

normalizing, we get:

p =

.40.25.35

We can check this by computing  M p. We will not show this here, but indeed,  M p =  p, as expected. Toanswer this question, we note that  p  provides the percentage of the total of 270 planes that are at eachlocation. So, we simply multiply each pi   (using the precise values) by 270. This yields, multiplying inthe obvious order, 106.85, 66.64, and 96.51. We choose to round the last value down to keep the totalnumber of planes to 270. Therefore, the number of planes at each location (our solution) is as follows:

Seattle :  107 Chicago :  67 Atlanta :  96

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