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601-jra-p345 - Solution Markov Chains(J WA A special type...

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Unformatted text preview: Solution Markov Chains (J WA A special type of difference equation arises in the study of arkov chains or Markov processes. We cannot go into the interesting theory of Markov chains, but we will give an example that illustrates some of the ideas. An automobile rental company has three locations, which we designate as P, Q, and R. When an automobile is rented at one of the locations, it may be returned to any of the three locations. Suppose, at some specific time, that there are p cars at location P, q cars at Q, and r cars at R. Experience has shown, in any given week, that the p cars at location P are distributed as follows: 10% are rented and returned to Q, 30% are rented and returned to R, and 60% remain at P (these either are not rented or are rented and returned to P). Similar rental histories are known for locations Q and R, as summarized below. Weekly Distribution History Location P: 60% stay at P, 10% go to Q, 30% go to R. Location Q: 10% go to P, 80% stay at Q, 10% go to R. Location R: 10% go to P, 20% go to Q, 70% stay at R. Let x], represent the state of the rental fleet at the beginning of week k: p(k) q(k) r(k) Xk For the state vector xk, p(k) denotes the number of cars at location P, q(k) the number at Q, and r(k) the number at R. From the weekly distribution history, we see that p(k +1) =.6p(k)+.lq(k)+.lr(k) q(k +1) =.lp(k)+.8q(k)+.2r(k) r(k + l) = .3p(k) + .lq(k) + .7r(k). (For instance, the number of cars at P when week k + 1 begins is determined by the 60% that remain at P, the 10% that arrive from Q, and the 10% that arrive from R.) To the extent that the weekly distribution percentages do not change, the rental fleet is rearranged among locations P, Q, and R according to the rule xk.“ = Axk, k = 0, 1,..., where A is the (3 x 3) matrix .6 .1 .1 A = .1 .8 .2 . .fl .3 .1 .7 Example 5 represents a situation in which a fixed p0pulation (the rental fleet) is rearranged in stages (week by week) among a fixed number of categories (the locations P, Q, and R). Moreover, in Example 5 the rules governing the rearrangement remain EXAMPLE (5 Solution fixed from stage to stage (the weekly distribution percentages stay constant). In general, such problems can be modeled by a difference equation of the form ‘ xk+x=Axk, k=0,l,.... For such problems the matrix A is often called a transition matrix. Such a matrix has two special properties: The entries of A are all nonnegative. (8a) In each column of A, the sum of the entries has the value I. (8h) It turns out that a matrix having properties (8a) and (8b) always has an eigenvalue of A = 1. This fact is established in Exercise 26 and illustrated in the next example. Suppose the automobile rental company described in Example 5 has a fleet of 600 cars. Initially an equal number of cars is based at each location, so that p(0) = 200, q(0) = 200, and r(0) = 200. As in Example 5, let the week-by-week distribution of cars be governed by xk+1 = Axk, k = 0, I, . . . , where p(k) .6 .l .1 200 xk = q(k) , A = .l .8 .2 , and x0 = 200 r(k) .3 .l .7 200 Find lim;Moo xk. Determine the number of cars at each location in the kth week, for k = 1,5, and 10. If A is not defective, we can use Eq. (6) to express X], as Xk = 01(A1)kui + €120»sz + 03(As)kl13. where {u}, uz, ml is a basis for R3, consisting of eigenvectors of A. It can be shown that A has eigenvalues A, = 1, A2 = .6, and A3 = .5. Thus A has three linearly independent eigenvectors: 4 0 l1=1, 111: 9 , A2=6 u2= 1 , 7 —1 —1 A3=.5, u3= ~l 2 The initial vector, x0 = [200, 200, 200]T, can be written as x0 = 30u1 — 150u2 — 80u3. Thus the vector xk = [p(k), q(k), r(k)]T is given by xk = Akxo = A"(30u, — 150u2 — 80u3) = 30(A])kul — 150(x2)’<u2 — 80(A3)ku3 = 30u, — 150(.6)"u2 — 80(.5)"u3. (9) From the expression above, we see that 120 ‘lim X; = 30u| = 270 k—)oo 210 Therefore, as the weeks proceed, the rental fleet will tend to an equilibrium state with 120 cars at P, 270 cars at Q, and 210 cars at R. To the extent that the model is valid, location Q will require the largest facility for maintenance, parking, and the like. Finally, using Eq. (9), we can calculate the state of the fleet for the kth week: I60 122.500 120.078 x]: 220 , x5 = 260.836 , and x10: 269.171 . fl 220 216.664 210.751 : i'T‘ii'IT": EXERCISES In Exercises 1—6, consider the vector sequence {xk}, 12. A and x0 in Exercise 6 where xk = AXk_], k = 1,2, . . . . For the given starting 3 _1 -1 3 vector x0, calculate x1, x2, X3, and x4 by using direct _ _ _ multiplication, as in Example 1. 13' A _ 12 0 5 ’ x0 _ —l4 1 A 0 l 2 4 ——2 —-1 8 . — 1 0 . X0 — 4 —6 l 3 l 14. A = —3 0 2 , x0 = 1 2 A ’5 '5 16 20 2 IO 1 ' " .5 .5 ’ x° ’ 8 5 25 128 In Exercises 15—18, solve the initial-value problem. 3, A = [ ' ' ], x0 = [ ] 15. u’(l) = 5u(t) —- 6v(t), u(0) = 4 .5 .75 64 v’(t) = 314(2) —4v(t). W) = 1 2 —1 3 16. u’(t) = 140) +2v(r), u(0) = l 4- A = -1 2 . x0 = 1 v’(t) = 2u(t) + v(t), v(0) = 5 l7. u’(t) = u(!) + v(t) + w(t), 14(0) = 3 5 A = ‘ 4 x = " v'(:) = ave) +3wm. um) = 3 ' 1 1 ’ ° 2 mm = —2u(:) + v(:) + wrr). mm) = 1 3 1 2 18. u’(t) = —2u(t) + 2v(t) —- 3w(t), u(0) = 3 6. A = , x0 = [ ] v'(t) = 2u(t) + v(t) — 6w(t), 12(0) = -—l 4 3 0 w’(t) = —u(t) — 2w), w(0) = 3 In Exercises 7-14, let xk = Axk_;, k = l, 2, . . . , for 19- Consider the matrix A given by the given A and x0. Find an expression for xk by using Eq. (6), as in Example 3. With a calculator, compute x; A = 1 2 . and x10 from the expression. Comment on limb“, xk. 0 1 7. . .‘ A and x0 1n Exercrse 1 Note that A = 1 is the only eigenvalue of A. 8. A and x ' Exerc'se 2 0 m l a) Verify that A is defective. . _ b) Consider the sequence {39,} determined by 10. A and x0 m Exercrse 4 Xi = Axk—hk = 1’ 2’ _ ‘ _, where 11. A and x0 in Exercise 5 x0 = [1, HT. Use induction to show that 9. A and x0 in Exercise 3 xk = [2k + 1, HT. (This exercise gives an example of a sequence xk = Axk_1, where lim,H00 lekll = 00, even though A has no eigenvalue larger than 1 in magnitude.) In Exercises 20 and 21, choose a value a so that the matrix A has an eignevalue of A = 1. Then, for x0 = [1, HT, calculate limkam xk, where xk = Axk_,, k = l, 2, . . . . .5 .5 20. A =[ :l .5 1+0: [0 .3 ] 21.A= .6 1+0: 22. Suppose that {uk} and {vk} are sequences satisfyv ing uk = Auk-“ k = 1,2,..., and vk = Avk_., k =1,2,....Showthatifuo = v0, then u,- = v,- foralli. 23. Let B = (b;,-) be an (n x n) matrix. Matrix B is called a stochastic matrix if B contains only non- ncgative entries and if b” + big + - -- + b;,, = 1, l 5 i 5 n. (That is, B is a stochastic matrix if 87 satisfies conditions 8a and 8b.) Show that A = 1 is an eigenvalue of B. [Him: Consider the vector w=[1,l,...,l]T.] 24. Suppose that B is a stochastic matrix whose entries are all positive. By Exercise 23, A = l is an eigen- value of B. Show that if Bu = u, u are 0, then u is a multiple of the vector w defined in Exercise 23. [Hint Define v = all so that v.- = l and [va 5 1, l 5 j 5 n. Consider the ith equations in Bw = w and Bv = v.] 25. Let B be a stochastic matrix, and let A by any eigen- value of B. Show that lAl 5 1. For simplicity, as- sume that A is real. [Hinn Suppose that Bu 2 Au, u ,4: 0. Define a vector v as in Exercise 24.] 26. Let A be an (n x n) matrix satisfying conditions (8a) and (8b). Show that A = l is an eigenvalue of A and that ifAu = Bu, 1! ye 0, then lfil 5 l. [Hint Matrix AT is stochastic.] 27. Suppose that (A — Al)u = 0, 11 9E 0, and there is a vector v such that (A — A1)v = n. Then v is called a generalized eigenvector. Show that {u, v} is a linearly independent set. [Hint Note that Av = Av + u. Suppose that an + bv = 0, and multiply this equation by A.] 28. Let A, u, and v be as in Exercise 27. Show that Akv = A"v+kA""u, k = 1, 2. 29. Consider matrix A in Exercise 19. a) Find an eigenvector u and a generalized eigenvector v for A. b) Express x0 = [1, HT as x0 = au + bv. c) Using the result of Exercise 28, find an expression for Akxo = A"(au + bv). d) Verify that Akxo = [2k + 1, HT as was shown by other means in Exercise 19. SUPPLEMENTARY EXERCISES 1. Find all values x such that A is singular, where x 12 A: 3 x 0 0-11 2. For what values x does A have only real eigenvalues, where 3. Let A = , c d wherea+b = 2andc+d = 2. ShowthatA = 2i5 an eigenvalue for A. [Hint: Guess an eigenvector.] 4. Let A and B be (3 x 3) matrices such that det(A) = 2 and det(B) = 9. Find the values of each of the following. a) det(A“Bz) b) det(3A) c) det(ABzA“) ...
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