SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

46 CHAPTER 5. ARRIVAL-COUNTING PROCESSES
25.
Notice that lim t→∞ Pr{H t = t} = lim t→∞ e −λt = 0. Therefore, lim t→∞ Pr{H t ≤ a} =
1 − e −λa ,a≥ 0.
E[L t ] = E[H t + R t ]
= E[H t ]+E[R t ]
= E[H t ]+E[G] (memoryless)
> E[G]
since certainly E[H t ] > 0.
26. Now p G (a) =1/60 for a =1, 2,...,60. Therefore, δ =E[G] =61/2 =30 1 2 months.
η =E[R] =E[d(e iG − 1)]
=
60 ∑
a=1
d(e ia − 1)p G (a)
= d
( 1
60
For the specific case i =0.06/12 = 0.005
∑60
a=1
e ia − 1
)
( ) 70.14
≈ d
60 − 1 = d(0.169)
=50, 700 for d = 300, 000
Therefore the long-run rate of return is
η
δ
≈ $1, 662.29 per month
≈ $19, 947.54 per year
This is about $52 per year less than the other model.