SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

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28 CHAPTER 4. SIMULATION e 1 () (arrival of a job) B ← FB −1 (random()) S B,n+1 ← S B,n +1 if {S B,n+1 =1} then endif C 2+B ← T n+1 +70 C 1 ← T n+1 +40 e 2 () (finish at CPU A) S 0,n+1 ← S 0,n − 1 if {S 0,n+1 > 0} then C 2 ← T n+1 +70 endif e 3 () (finish at CPU B) S 1,n+1 ← S 1,n − 1 if {S 1,n+1 > 0} then C 3 ← T n+1 +70 endif 7. No answer provided. 8. (a)
29 (b) Ȳ 1 = {0(5 − 0) + 1(6 − 5) + 2(9 − 6) + 1(11 − 9) +2(13− 11) + 3(15 − 13) + 2(16 − 15) +1(18 − 16) + 0(19 − 18)}÷19 = 23 ≈ 1.2 customers waiting for service 19 Ȳ 2 = (c) N 19 = 5 arrivals An estimate of average delay is 0(3 − 0) + 1(19 − 3) 19 = 16 19 ≈ 0.84 utilization of the copier ∫ 19 0 Y 1,t dt = 23 =4.6 minutes N 19 5 9. Based on the available data, the sample-average interarrival-time gap was about 17.6 seconds, and might be modeled as exponentially distributed. Let F G be the cdf of the interarrival-time gaps, which we assume to be independent because students act independently. Let ∆t be the pick-up interval, in seconds. Let S n represent the number of students waiting after the nth state change. Define the following system events and associated clocks: e 0 () (start of day) S 0 ← 0 C 1 ← FG −1 (random()) C 2 ← ∆t e 1 () (student arrives) S n+1 ← S n +1 C 1 ← T n+1 + FG −1 (random()) e 2 () (pick up)
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: 27 S n+1 ← S n − FD −1 endif
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 and 48: 43 ⎧ ⎪⎨ λ(t) = ⎪⎩ 144, 0
Page 49 and 50: while {b
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
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81 8. See Exercise 7 for an analysi
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y conditioning on the first state v
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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
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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
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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
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101 ∑ 20 j=0 d j ≈ 453.388 ⎧
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103 (c) For the M/M/s/c with s ≤
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will be at the front of the queue o
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107 (b) For the M/M/1 For the M/D/1
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109 = (1− ρ 1 )(1 − ρ 2 ) ∞
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111 i a 0i µ (i) 1 20 30 2 0 30 (
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113 or or ⎛ Λ[(I − R) ′ ]
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115 i s i ρ i 1 19 0.88 2 4 0.75 3
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It appears that there will be very
Page 123 and 124:
Chapter 9 Topics in Simulation of S
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9. To obtain a rough-cut model, fir
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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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