SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

9. To obtain a rough-cut model, first note that the registration function is not really
relevant to the issue of additional bed versus additional doctor, so we will ignore it.
We also (a) ignore the interaction of beds and doctors, (b) treat treatment time as
exponentially distributed, and (c) develop a composite patient and ignore patient type
and priority.
• Additional doctor.
With an additional doctor there is a one-to-one correspondence between bed and doctor.
λ = 1/20 patient/minute
1/µ = 72(0.15) + 25(0.85) ≈ 32 minutes
Using an M/M/3 model
l q = 0.3 patients waiting for a doctor and a bed
w q = 6.3 minutes wait for a doctor and a bed
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• Additional bed.
In terms of time to wait to see a doctor, the beds are not a constraint.
M/M/2 model
l q = 2.8 patients waiting to see a doctor
w q = 57 minutes to see a doctor
The expected number of patients in the system is
Using an
l = λw = λ(w q +1/µ) ≈ 4.45
Thus, with 4 beds the expected number waiting for a bed is 4.45 − 4=0.45, which is
larger than the 0.3 patients when adding another doctor.
It appears that adding a doctor will be better in terms of time to see a doctor and
time to reach a bed.
10. To obtain a rough-cut model we will replace all random variables by their expected
values and treat it as deterministic.
Suppose the company is open 8 hours per day. Then the period of interest is 90(8) =
720 hours.
The number of lost sales during 1 day is (8 hours) (1 customer/hour) (6 bags/customer)
= 48 bags.
If the company orders s bags, they will run out in
s bags
= s/6 hours
6 bags/hour