Corrections to the Second Edition of Modeling and Analysis of Stochastic Systems

Transcription

1 Corrections to the Second Edition of Modeling and Analysis of Stochastic Systems Vidyadhar Kulkarni November, 200 Send additional corrections to the author at his address Chapter 2. Page, Equation (2.5): insert, after ). 2. Page 5, line 5 from below: the entry P 0, should be 2p d ( p d ), and P 0,2 should be ( p d ) Page 6, line 5: replace X n by Z n. 4. Page 6, line 7: replace Z n by Z k. 5. Page 25, line 0: replace exiting by existing. 6. Page 30, line 3 from below: replace serves by serve. 7. Page 38, line 3 from below: replace D = XAX by A = XDX. 8. Page 39, line 2 from below: replace x i by x i. 9. Page 46, line 6 of Exercise 2.20: replace i by k. Chapter 3. Page 56, line 7 from below: replace between between by between. 2. Page 62, line 2 of Example 3.7: replace form by from. 3. Page 68, line 2: replace v by v i. 4. Page 75, line 4 and 5: replace φ () by φ i (). 5. Page 70, Line 5 from below: insert is before tossed. 6. Page 74, line 5: replace by E(z T X 0 = i, X = 0) =. E(z T X 0 = i, X = 0) = z. 7. page 75, Equation 3.3: replace φ (k) by φ (k) (). 8. Page 90, line 2 of Definition 4.4: insert if after closed.

2 Chapter 4. Page 9, Figures 4.: The label on the arc (,2) should be α and on the arc (2,) should be β. 2. Page 9, Figures 4.2: The label on the arc (2,2) should be β. 3. Page 9, line 2 from below: replace C = {} by C = {0}. 4. Page 92, line 8 from below: replace the second n by m. 5. Page 92, Figure 4.3: the labels on the nodes should be 0,,2,i, i, i +, N and N. 6. Page 93, Example 4.8: Replace by P P = [ 0 0 ] [ 0 0 ] 7. Page 94, line 9: replace (n+)(n+2) by n(n+). 8. Page 94, line 3: replace (n+)(n+2) by n(n+). 9. Page 94, replace line 5 by 0. Page 94: replace lines 7, 8, 9 by n(n + ) = n n + m 0 = = = np( T 0 = n X 0 = 0) n=0 n= n=0 n n(n + ) n + =.. Page 97, Theorem 4.7 (ii): replace null transient by null recurrent. 2. Page 98, line 2 from below: replace finite by a communicating class. 3. Page 99, line : Replace The by Then. 4. Page 99, line 2: delete visits. 5. Page 02, line 5 from below: replace = by. 6. Page 03, Equations 4.6 and 4.7: replace ν(i) by d(i). 7. Page 03, line 3 from below: replace y i (r + ) by y (r+) i. 8. Page 04, line 3: replace by =. 9. Page 04, line 5: replace y (r+) i by y (n+) i. 20. Page 04, line 4 from below: insert positive before recurrent. 2. Page 05, line 4 of Theorem 4.2: replace > by < 2

3 22. Page 07, Equation 4.23: replace n = in the sum by n = Page 07, last line: replace n 0 by k Page 09, line 3: replace P( T j = m) by P( T j = m X 0 = j). 25. Page 09, line 4: replace ũ j (m) by u m. 26. Page 0, line 7: replace ν n by nu n. 27. Page 3, line 5: insert (by using the bounded convergence theorem) after get. 28. Page 4, Equation 4.3: replace p (nd+r) jj by p (nd+r) ij. 29. Page 4, Equation 4.32: replace M i j(n + ) by M (n+) ij. 30. Page 6: last line: replace two by three. 3. Page 7, line 5 from below: insert is after This. 32. Page 8: line 5: replace analytical examples by analytical solutions. 33. Page 8, line 4 in Example 4.2: replace positive recurrent by irreducible. 34. Page 9, the line above Example 4.22: replace is by has. 35. Page 9, line 6 in Example 4.22: replace recurrent by irreducible. 36. Page 20, line 4: replace positive recurrent by irreducible. 37. Page 22, line 0: insert unique before solution. 38. Page 22, line : replace an by and. 39. Page 23, line 2 of Theorem 4.22: insert non-negative after smallest. 40. Page 24, line 8 from below: replace α i (r) by u i (r). 4. Page 24, line 6 from below: replace m=n+ u m α i (r) ɛ/2 by u i (r) u m ɛ/2 m=n+ 42. Page 24, last line: replace the help of several examples by an example. 43. Page 27, last line: replace d( p u ) by dα( p u ). 44. Page 28, line 4: replace d( p u ) by dα( p u ). 45. Page 24, line 9: replace decision making by decisions. 46. Page 29, line 4 from below: replace π 2 by π B. 47. page 29, line 6 from below: replace DMTC by DTMC. 48. Page 34, line 4 of exercise 4.4: replace p o by p Page 40, line 6: replace east by least. 3

4 50. Page 4, line 2: insert ) after X n+. 5. Page 42, line 8 from below: insert recurrent after positive. Chapter 5. Page 47, line 8 from below: insert, after e λx. 2. Page 50, line 4: replace random t by random T. 3. Page 5, line 7 from below: replace exp by exp. 4. Page 52, line 2 of Theorem 5.5: replace Erl by Erl. Page 53, line 2 from above, and line 3 from below: replace X + 2 by X Page 52, line 4 from below: replace n! by (n )!. 6. page 54, line 2 of Theorem 5.7: replace G by G. 7. Page 57, line 3 from below: delete a. 8. Page 58, line 5 of Theorem 5.: replace k n in the superscript by k n. 9. Page 60: replace h i in the second sum by h i. 0. Page 63, lines 2,3 and 4 of the proof of Theorem 5.4: replace (t i + dt i ) by (t i, t i + dt i ).. Page 63, Page 63, line 5 of the proof of Theorem 5.4: replace (t i t i + dt i ) by (t i t i, t i t i + dt i ). 2. Page 67: figure 5.5: The label on the y-axis on the bottom graph should be N(t) = N (t) + N 2 (t). 3. Page 67, line from below: replace call by class. 4. Page 68, line 0: replace Z n by Z. 5. Page 68, line 6: replace in the by in this. 6. Page 69, line 8 from below: replace R n = i by R n = k. 7. Page 69, line 9 from below: replace iid be by iid. 8. Page 72, line 4 from below: insert has after 0}. 9. Page 73, line 3: replace G ) by G( ). 20. Page 77, Theorem 5.2: Replace on the right hand side by Λ n n! Λ n (n )! 2. Page 77, last line: replace call by called 22. Page 79, line 4: replace (sz in the middle term by s(z. 23. Page 82, line 3: replace lifetime by lifetimes. 24. Page 84, lines 6 and 7: replace minute by hour (two instances). 25. Page 84, line 9: insert a after is. 4

5 26. Page 87, line 9 from below: replace by. Chapter 6. Page 205: line 3 of Theorem 6.4: replace respect t by respect to t. 2. Page 209, line 5 from below: replace had by hand. 3. Page 20, line 3 in Theorem 6.7: replace M by N. 4. Page 20, line 2 from below: replace m by N. 5. Page 2, line 4 of Example 6.8: replace the (0,0)th entry of X by µ λ+µ. 6. Page 24, line 2 above Equation 6.27: delete. before Now. 7. Page 26, last line: replace computed by Equation 6.28 by defined in Theorem Page 227, line 2: remove all four large parentheses. 9. Page 230, line 4 from below: replace t 0 by t Y. 0. Page 230, line 2 from below: replace t 0 by t Y.. Page 230, line 0 from below: replace n 0 by n. 2. Page 233, line : replace p i = /(i + 2) by p i = (i + )/(i + 2). 3. Page 237, last line: replace n by t. 4. Page 242, line 5: insert death parameters before µ i. 5. Page 244, line 2: replace p i,n+ by p,n+. 6. Page 256, line 2: replace DTMC by CTMC. Chapter 7. Page 279, line 7 from below: replace sections by section. 2. Page 280, line 8 from below: replace 7.3 by Page 282, line 8: insert are after there. 4. Page 282, line 2 from below: replace i + by i and i + 2 by i Page 283, line 3: replace Once by One. 6. Page 284, line 6 from below: replace j by j. 7. Page 285, line 3 from below: replace p kj by b kj. 8. Page 286, line 5: replace 23 on page 6.25 by 6.25 on page Page 286, line 6 from below: delete we have,. 0. Page 288, line 2 from below: replace for a by for an.. Page 290, line : replace for an by for a. 2. Page 290, line 4: replace Little Law by Little s Law. 5

6 3. Page 29, line 7: replace Sn t<d n by {Sn t<d n}. 4. Page 296, line 5 from below: replace nµ by nµ,. 5. Page 297, line 9 in Section 7.3.5: replace (N n) by min(n n, s). 6. Page 297, line in Section 7.3.5: replace min(n, s) by n. 7. Page 30, line 0: insert given after is. 8. Page 303, Figure 7.4: the return arc label should be α. 9. page 303, line below Figure 7.4: insert ( before λ. 20. Page 309, lines 2 and 4 above Example 7.: replace G(K) by G N (K). 2. Page 3, Figure 7.7: the label on the arc from CPU tp Printer should be α and that on the arc from CPU to the Disc Drive should be α. 22. Page 35, line 7: insert a after in. 23. Page 36, line 5 from below: replace the n by the nth. 24. Page 39: line of Example 7.5: replace service by inter-arrival. 25. Page 32, Figure 7.8: The caption in the lower box should read Single Server No Waiting Space 26. Page 325, line 3 of Exercise 7.: replace essentially by an. 27. Page 328, line 3 of exercise 7.3: replace little s law by Little s Law. 28. Page 33, line 3 of exercise 7.29: insert is after server. 29. Page 33, Figure 7.9: the label on the horizontal arrow out of the last node should be p. 30. Page 332, Figure 7.9: the labels on the horizontal arrows should be p, p 2, p N, and p N. 3. Page 333, part 2 of exercise 7.4: replace j K by i N. 32. Page 336, line of Exercise 7.55: replace 7.2 by Page 336, line 2 of Exercise 7.56: replace 7.2 by 7.3. Chapter 8. Page 342, line 6: delete 0,. 2. Page 342, line 2: replace k n by k Page 342, line from below : replace the first k n by k Page 342, line 5 from below: replace S kn+ t n + X by S kn+ X t n. 5. Page 343, line : replace G(0 ) = by G(0 ) = Page 344, line 3 from below: replace which by whose probability is. 7. Page 347, line 6 from below: replace infty by infinity. 6

7 8. Page 348, equation 8.2: replace = x) by x) =. 9. Page 352, line 2 from below: replace Sn S n by S n Sn. 0. Page 355, line 2: replace p k by p r.. Page 355, line 2: replace G k by G r. 2. Page 370, line 7: insert in before the. 3. Page 370, line 9 from below: replace G by F. 4. Page 373, Figure 8.7: replace the label Z(t) by X(t) on the vertical axis. 5. Page 375, the lower part of Figure 8.8: replace the label Z(t) by X(t) on the vertical axis. 6. Page 377, last line: insert +P(U > t) before the, 7. Page 378, line 3: insert + lim t P(U > t) before the. 8. Page 378, line 5 from below: replace chapter by section. 9. Page 378, line 6 from below: insert = G ii ( ) after p ii. 20. Page 380, line 4 from below: replace S by Y. 2. Page 395, Figure 8.0: replace Z(t) by X(t) as the label for the vertical axis. 22. Page 400: replace Compute by Derive in the first line of Exercises 8.26, 8.27, 8.28 and Page 40, Exercise 8.39: replace An by A in parts 2 and Page 40, last line replace E(min(U + D, t) by E(min(U + D, t)). 25. Page 404, line 2 of Exercise 8.5: replace φ k (s) by µ/(s + µ). Chapter 9. Page 43, last line: replace min by sup. 2. Page 48, line 4: replace by =. 3. Page 424, line 2 from below: replace implicity by implicitly. 4. Page 425, lines 5 and 6 in Theorem 9.5: replace j S by j I. 5. Page 426, Equation 9.35: replace X(t) = k by Z(t) = k. 6. Page 427, line 3: replace τ rr by τ r. 7. Page 427, line from below: replace P(X(t) = j) by P(X(t) = j X(0) = i). 8. Page 428, lines 7 and 8: On each line insert to after S n+. 9. Page 430, line 7: replace s by u. 0. Page 430, line 5: replace α k by α kj.. Page 432, line 3: replace the right hand side by e 2µx, 2. Page 432, line 4: replace Ã(λ) by Ã(µ). 3. Page 433, line 8: insert death parameters before µ. 7

8 4. Page 433, Theorem 9.20: replace process by death. 5. Page 435, line 4 from below: insert be after Z(t). 6. Page 436, lines 4 and 5: replace Z(t)=j by {Z(t)=j}. 7. Page 44, line 3: replace (, 2) by (i =, 2). Chapter 0. Page 446, line 4: delete part of. 2. Page 446, line 5 from below: insert finite dimensional after the. 3. Page 447, lines 9 and : replace ] by ] T. 4. Page 447, line 5: insert = before Page 453, lines 0 and : replace a) by 0). 6. Page 462: the four lines above Theorem 0.7 should not be in italic. 7. Page 465, line : replace B(u) by X(u). 8. Page 466, line 6: the short should be longer. 9. page 467, line 3: the long should be shorter. 0. Page 47, line :6: replace is the by is.. Page 472, line 3 of Definition 0.9: replace X(u) by B(u). 2. Page 473, line 3: replace X(u) by B(u). 3. Page 483, line 7: delete and after American. 4. Page 483: line below Equation 0.69: replace Theorem 0.27 by Theorem Page 485, line : replace b(t) by b(t). Appendices. Page 49, last line: replace P (φ) = by P (φ) = Page 494, line 6 from below: replace function by functions. 3. Page 497, line 4 from below: delete for. 4. Page 499, line : replace differential by continuous. 5. Page 50, line 6 from below: replace by µ = [ µ µ2 ] µ = [µ µ2] 6. Page 50, line 3 from below: replace (a µ2) by (a µ2). 7. Page 503, line 6: replace in by are given in. 8. Page 505, line 6: replace in by are given in. 9. Page 53, line 6 from below: insert. after k 0. 8