SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

8. (a) Restricted to periods of the day when the arrival rate is roughly constant, a
Poisson process is appropriate to represent a large number of customers acting
independently.
(b) Not a good approximation, since most arrivals occur during a brief period just
prior to the start, and only a few before or after this period. Therefore, arrivals
do not act independently.
(c) Not a good approximation if patients are scheduled. We do not have independent
increments because patients are anticipated. (May be a good approximation for
a walk-in clinic, however.)
(d) Not a good approximation because the rate of finding bugs will decrease over
time.
(e) Probably a good approximation since fires happen (largely) independently, and
there are a large number of potential customers (buildings).
9. (a)
c = 60 + 72 + 68 = 200 total arrivals
d = 3 + 3 + 3 = 9 total hours observed
̂λ = c/d ≈ 22 customers/hour
35
ŝe =
√ ̂λ
d
≈ 1.6 customers/hour
(b) Let Y t ≡ number of arrivals by time t, and suppose λ = 22/hour.
Pr{Y 3 ≤ 56} =
56 ∑
j=0
e −22(3) (22(3)) j
j!
≈ 0.12
Since this probability is rather small, we might conclude that Fridays are different
(have a lower arrival rate).
At λ =22+ŝe = 23.6
and at λ =22− ŝe = 20.4
Pr{Y 3 ≤ 56} ≈0.04
Pr{Y 3 ≤ 56} ≈0.29
So less than or equal to 56 could be quite rare if λ =23.6. This is further evidence
that Fridays could be different.