SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

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98 CHAPTER 8. QUEUEING PROCESSES 8. (a) We approximate each system as an M/M/1 queue. One-person system λ = 24 cars/hour µ = 30 cars/hours w q =2/15 ≈ 0.13 hour w = w q +1/µ =1/6 ≈ 0.17 hour Two-person system µ = 48 cars/hour w q =1/48 ≈ 0.02 hour w = w q +1/µ =1/24 ≈ 0.04 hour (b) Need π 0 <strong>for</strong> each case One person π 0 =1/5 Two person π 0 =1/2 (c) The fraction of time that the intercom is blocked is ∞∑ η = π j = 1− π 0 − π 1 − π 2 j=3 2∑ = 1− (1 − ρ)ρ j j=0 One person η ≈ 0.51 Two person η ≈ 0.13 So in the one-person system the intercom is blocked over half the time. (d) Now we model the system as an M/M/1/6 queue. • One-person system π 6 ≈ 0.066 so the lost-customer rate is λπ 6 ≈ 1.6 customers/hour. • Two-person system π 6 ≈ 0.008 so λπ 6 ≈ 0.2 customer/hour. 9. This is one possible approach to the problem: Since the user population is relatively large, treat the arrival process as Poisson with rate λ = (10000)(0.7) = 7000 calls/day ≈ 292 calls/hour ≈ 4.86 calls/minute
We have assumed a 24-hour day, but perhaps a 12 or 18-hour day is better. 1/µ =2.5 minutessoµ =0.4 calls/minute. If we treat the connect times as exponentially distributed, then we have an M/M/s/s queue, where s is the number of ports. A worthwhile table is π s vs. s, sinceπ s is the probability a user cannot connect. π s s 5 0.63 6 0.56 7 0.49 8 0.43 9 0.37 10 0.31 11 0.25 12 0.20 13 0.16 14 0.12 15 0.09 16 0.06 17 0.04 18 0.03 19 0.02 20 0.01 10. (a) If we model the arrival of calls as Poisson, service times as exponentially distributed, <strong>and</strong> no reneging, then we have an M/M/2 queue. λ = 20 calls/hour 1/µ =3minutessoµ = 20 calls/hour (b) To keep up ρ = λ < 1orλ
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SOLUTIONS MANUAL for Stochastic Mod
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ii CONTENTS
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Chapter 2 Sample Paths 1. The simul
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3 7. Inputs: Number of hamburgers d
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Chapter 3 Basics 1. (a) Pr{X =4} =
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(b) 6. (a) F Y (a) = = ∫ a ⎧
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9 (a) Pr{X 2 =1| X 1 =0} = Pr{X 2 =
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11 16. Let U be a random variable h
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13 Y = ⎧ ⎪⎨ ⎪⎩ 1, 0 ≤ U
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15 (d) E[X] = ∑ all a ap X (a) =0
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17 ( ) ∞∑ d 2 = γ a=1 dq 2 qa+
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19 = ∫ ∞ −∞ ∫ ∞ a 2 f X
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21 (b) Let T ≡ number of trials u
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(b) f is maximized at a = β giving
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Chapter 4 Simulation 1. An estimate
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27 S n+1 ← S n − FD −1 endif
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29 (b) Ȳ 1 = {0(5 − 0) + 1(6 −
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Chapter 5 Arrival-Counting Processe
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33 Pr{Y 2,6 > 30} = 1− Pr{Y 2,6
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8. (a) Restricted to periods of the
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37 Pr{Y (A) 1.5 > 1000,Y (B) 1.5 >
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39 Pr{Y (B) 2 − Y (B) 1 > 5,Y (B)
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41 (b) 20. (a) Pr{Y 52 − Y 48 =20
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43 ⎧ ⎪⎨ λ(t) = ⎪⎩ 144, 0
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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
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: 97 (c) l q = ∞∑ π 2 + (j − 2
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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