SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

40 CHAPTER 5. ARRIVAL-COUNTING PROCESSES
(d)
Pr{Y (D)
1 > 300 | Y 1 = 1200}
=
1200 ∑
n=301
( ) 1200
(0.13) n (0.87) 1200−n ≈ 0
n
18. Let {Y (T )
t ; t ≥ 0} represent arrival of trucks to the restaurant.
Poisson with rate λ (T ) = 10(0.1) = 1/hour
Let {Y (C)
t ; t ≥ 0} represent arrival of cars to the restaurant.
Poisson with rate λ (C) = 20(0.1) = 2/hour
(a) Y t = Y (T )
t
+ Y (C)
t
is Poisson with rate λ = λ (T ) + λ (C) =3/hour
E[Y 1 ]=λ(1) = 3 cars and trucks
(b) Pr{Y 1 =0} = e −3(1) ≈ 0.05
(c) Let C ≡ number of passengers in a car.
E[C] =(0.3)(1) + (0.5)(2) + (0.2)(3) = 1.9
Let P ≡ number of passengers arriving in 1 hour.
E[P ] = E [ Y (T )
1 +E[C]Y (C) ]
1
= 1+(1.9)2 = 4.8 passengers
19. Traffic engineer’s model: {Y t ; t ≥ 0} models the number of accidents and is Poisson
with rate λ = 2/week.
(a)
Pr{Y 2 ≥ 20} =
∞∑ e −2(2) (2(2)) m
m=20
m!
= 1−
19 ∑
m=0
e −4 4 m
≈ 1.01 × 10 −8
m!