SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

79
endif
C 2 ← T n+1 + FH
−1 (random())
e 3 () (end regular call)
S n+1 ← 1
e 4 () (end chair call)
if {S n =3} then
S n+1 ← 1
else
S n+1 ← 2
endif
5. (a) M = {0, 1, 2, 3} represents the number of vans in use at time t days.
Approximate the time between requests and the time in use as exponentially
distributed. Therefore, we have a Markov-process model with
(b)
⎛
G = ⎜
⎝
π = ⎜
⎝
− 8 8
7
1
− 23
2 14
7
0 0
8
0
7
8
7 7
0 1 − 15
3
0 0 − 3 2 2
⎛
0.1267709745
0.2897622274
0.3311568313
0.2523099667
The rate at which requests are denied is 8/7 π 3 ≈ 0.29 requests/day.
(c) 0π 0 +1π 1 +2π 2 +3π 3 ≈ 1.71 vans
6. (7.1) for a and n integer
⎞
⎟
⎠
⎞
⎟
⎠
Pr{X >n+ a | X>n} =
=
Pr{X >n+ a, X > n}
Pr{X >n}
Pr{X >n+ a}
Pr{X >n}
=
(1 − γ)n+a
(1 − γ) n
= (1− γ) a =Pr{X >a}