SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

36 CHAPTER 5. ARRIVAL-COUNTING PROCESSES
10. Let
λ 0 = 8/hour
λ 1 = 1/18/hour p 1 =0.005
λ 2 = 1/46/hour p 2 =0.08
(a) λ = λ 0 + λ 1 + λ 2 =8 32 surges/hour
414
By the superposition property, the arrival of all surges is also a Poisson process.
Therefore, E[Y 8 ]=λ8 ≈ 64.6 surges.
(b) By the decomposition property, the “small” and “moderate” surges can be decomposed
into Poisson processes of computer-damaging surges.
λ 10 = p 1 λ 1 =1/3600/hour
λ 20 = p 2 λ 2 =1/575/hour
These processes can be superposed to give λ 3 = λ 10 + λ 20 ≈ 0.0020. Therefore,
E[Y 8 ]=8λ 3 ≈ 0.016 computer-damaging surge.
(c) Using the arrival rate from part (b)
11. No answer provided.
Pr{Y 8 =0} = e −λ 38 ≈ 0.98
12. Let {Y (0)
t ; t ≥ 0} be a Poisson arrival process representing the arrival of requests that
require printing; λ (0) = 400/hour.
{Y (1)
t ; t ≥ 0} similarly represents the arrival of requests that do not require printing;
λ (1) = 1000/hour.
(a) Y t = Y (0)
t + Y (1)
t is Poisson with rate λ = λ (0) + λ (1) = 1400/hour.
∞∑
Pr{Y 1.5 > 2000} =
m=2001
e −1400(1.5) (1400(1.5)) m
m!
≈ 0.985
(b) Let Y (A)
t
and Y (B)
t
represent the arrivals to computers A and B, respectively.
Y (A)
t is Poisson with λ (A) =(1/2)λ = 700
and they are independent, therefore,
Y (B)
t is Poisson with λ (B) =(1/2)λ = 700