SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

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44 CHAPTER 5. ARRIVAL-COUNTING PROCESSES 22. The algorithms given here are direct consequences of the definitions, and not necessarily the most efficient possible. (a) Recall that the inverse cdf for the exponential distribution with parameter λ is algorithm Erlang a ← 0 for i ← 1ton do a ← a − ln (1-random())/λ enddo return X ← a (b) algorithm binomial a ← 0 for i ← 1ton do U ← random() if {U ≥ 1 − γ} then a ← a +1 endif enddo return X ← a (c) algorithm Poisson a ← 0 b ←−ln(1-random())/λ while {b
while {b
Page 1 and 2: SOLUTIONS MANUAL for Stochastic Mod
Page 3 and 4: ii CONTENTS
Page 5 and 6: Chapter 2 Sample Paths 1. The simul
Page 7 and 8: 3 7. Inputs: Number of hamburgers d
Page 9 and 10: Chapter 3 Basics 1. (a) Pr{X =4} =
Page 11 and 12: (b) 6. (a) F Y (a) = = ∫ a ⎧
Page 13 and 14: 9 (a) Pr{X 2 =1| X 1 =0} = Pr{X 2 =
Page 15 and 16: 11 16. Let U be a random variable h
Page 17 and 18: 13 Y = ⎧ ⎪⎨ ⎪⎩ 1, 0 ≤ U
Page 19 and 20: 15 (d) E[X] = ∑ all a ap X (a) =0
Page 21 and 22: 17 ( ) ∞∑ d 2 = γ a=1 dq 2 qa+
Page 23 and 24: 19 = ∫ ∞ −∞ ∫ ∞ a 2 f X
Page 25 and 26: 21 (b) Let T ≡ number of trials u
Page 27 and 28: (b) f is maximized at a = β giving
Page 29 and 30: Chapter 4 Simulation 1. An estimate
Page 31 and 32: 27 S n+1 ← S n − FD −1 endif
Page 33 and 34: 29 (b) Ȳ 1 = {0(5 − 0) + 1(6 −
Page 35 and 36: Chapter 5 Arrival-Counting Processe
Page 37 and 38: 33 Pr{Y 2,6 > 30} = 1− Pr{Y 2,6
Page 39 and 40: 8. (a) Restricted to periods of the
Page 41 and 42: 37 Pr{Y (A) 1.5 > 1000,Y (B) 1.5 >
Page 43 and 44: 39 Pr{Y (B) 2 − Y (B) 1 > 5,Y (B)
Page 45 and 46: 41 (b) 20. (a) Pr{Y 52 − Y 48 =20
Page 47: 43 ⎧ ⎪⎨ λ(t) = ⎪⎩ 144, 0
Page 51 and 52: 27. We suppose that the time to bur
Page 53 and 54: and the latter with rate ( λ ′
Page 55 and 56: Chapter 6 Discrete-Time Processes 1
Page 57 and 58: 53 n =5 ⎛ ⎜ ⎝ 0.00001 0.68645
Page 59 and 60: 55 with ŝe = √ ̂p(1 − ̂p) 45
Page 61 and 62: 57 14. Case 6.3 S n represents the
Page 63 and 64: 59 Pr{≤ 1 defective} = a + b + c
Page 65 and 66: 61 ⎛ P = ⎜ ⎝ 0.92 0.08 0 0 0
Page 67 and 68: 63 25. (a) We first argue that µ 1
Page 69 and 70: 65 The expected number of spins is
Page 71 and 72: 67 = 1− Pr{X 1 >n} Pr{X 2 >n} = 1
Page 73 and 74: 69 = ∑ h∈A Pr{S 1 = j | S 0 = h
Page 75 and 76: 71 Therefore, the long-run expected
Page 77 and 78: 73 31. We show (6.5) for k = 2. The
Page 79 and 80: Chapter 7 Continuous-Time Processes
Page 81 and 82: 77 dp i3 (t) dt dp i4 (t) dt dp i5
Page 83 and 84: 79 endif C 2 ← T n+1 + FH −1 (r
Page 85 and 86: 81 8. See Exercise 7 for an analysi
Page 87 and 88: y conditioning on the first state v
Page 89 and 90: 85 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Page 91 and 92: 87 or λµ 00 = 1 +λµ 10 λµ 01
Page 93 and 94: 89 e 2 () (order arrives from manuf
Page 95 and 96: 91 − λ M ∫ ∞ λ 1 + λ M t =
Page 97 and 98: Chapter 8 Queueing Processes 1. Let
Page 99 and 100:
95 λ i = µ i = { 20(1 − i/16),
Page 101 and 102:
97 (c) l q = ∞∑ π 2 + (j − 2
Page 103 and 104:
We have assumed a 24-hour day, but
Page 105 and 106:
101 ∑ 20 j=0 d j ≈ 453.388 ⎧
Page 107 and 108:
103 (c) For the M/M/s/c with s ≤
Page 109 and 110:
will be at the front of the queue o
Page 111 and 112:
107 (b) For the M/M/1 For the M/D/1
Page 113 and 114:
109 = (1− ρ 1 )(1 − ρ 2 ) ∞
Page 115 and 116:
111 i a 0i µ (i) 1 20 30 2 0 30 (
Page 117 and 118:
113 or or ⎛ Λ[(I − R) ′ ]
Page 119 and 120:
115 i s i ρ i 1 19 0.88 2 4 0.75 3
Page 121 and 122:
It appears that there will be very
Page 123 and 124:
Chapter 9 Topics in Simulation of S
Page 125 and 126:
9. To obtain a rough-cut model, fir
Page 127 and 128:
123 w q =0.49hours l q = 3.2 trucks
Page 129 and 130:
125 When we fix the initial state,
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