SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

42 CHAPTER 5. ARRIVAL-COUNTING PROCESSES
Pr{Y 8 −Y 2 > 12} =
∞∑ e −8 8 m
m=13
m!
= 1−
12 ∑
m=0
e −8 8 m
m!
≈ 0.06
E[Y 8 −Y 2 ]=8patients
(c) 10 a.m. → hour 4
Λ(4) − Λ(2) = 2
(d)
Pr{Y 4 =9|Y 2 =6} = Pr{Y 4 −Y 2 =3|Y 2 =6}
= Pr{Y 4 −Y 2 =3}
= e−2 2 3
≈ 0.18
3!
Pr{Y 1/4 > 0} = 1− Pr{Y 1/4 =0}
=
( e −1/4 (1/4) 0 )
1−
0!
= 1− e −1/4 ≈ 0.22
(since Λ(1/4) = 1/4)
(e) 1 p.m. → hour 7
Λ(7) = 2(7) − 6=8
(from part (b))
∞∑
Pr{Y 7 ≥ 13} =
m=13
e −8 8 m
m!
≈ 0.06
21. (a) Compute the average number of arrivals (and its se) for each hour
hour average (se) estimates
1 144.2 (12.3) λ 1
2 229.4 (7.2) λ 2
3 382.6 (7.9) λ 3
4 96.0 (2.7) λ 4