武汉理工大学研究生生产计划与控制(2)作业2 及对应答案
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Homeworks
1. A food store has an average of 220 customers arriving per hour during peak shopping
hours . During these peak periods all eight checkout counters will be open and operating with a capacity of serving an average of 35 customers per hour per counter . All checkout counters are identical .
a. What proportion of the time would all of the checkout counters and waiting lines be
empty of customers?
b. How long would customers wait in line on the average?
c. How many customers would be waiting in each line on the average? Solution:
From the question, we know: λ=220,μ=35, N=8,
a. compute 0P , probability that the system would be empty: n=0,
()()01
n 01
P //!!1n N
N n N N λμλμλμ-==
⎡⎤+⎢⎥⎛⎫⎢⎥⎣⎦- ⎪
⎝⎭∑=
()()
881n 01
220/35220/35220!8!135*8n n -=⎡⎤+⎢⎥⎛⎫⎢⎥⎣⎦- ⎪
⎝⎭
∑ =
()()
()
87n 01
44/744/7!8!3/14n n =⎡⎤+⎢⎥⎢⎥⎣⎦∑= 10.0015377.831685208282.048=+ b. Compute 1t :
()()80122
2200.0015352203588!1!1358N P t N N N λμλμμ⎡⎤⎡⎤
⎢⎥⎢⎥
⎛⎫⎛⎫⎢⎥⎢⎥== ⎪ ⎪⎢⎥⎢⎥⎝⎭⎝⎭⎛⎫⎛⎫⎢⎥⨯--⎢⎥ ⎪ ⎪⨯⎝⎭⎢⎥⎣⎦⎝⎭⎣
⎦ 8
2200.0015
0.007135518469⎛⎫=⨯
= ⎪⎝⎭ c. compute 1n :
()()()()()81022
/22035220/350.0015 1.55131!7!835220N n P N N λμλμμλ⎡⎤⎡⎤⨯⨯==⨯=⎢⎥⎢⎥--⨯-⎢⎥⎢⎥⎣⎦⎣⎦
There are about 1.5513 customers would be waiting in each line on the average.
Use POM software to solve this problem. The solution is as follows: MODEL: Multiple Channels Arrival Rate (lambda) = 220
Service Rate (mu) = 35 Number of Channels = 8
Average Number of Units in Waiting Line = 1.5672 Average Number of Units in System = 7.8529 Average Waiting Time in Line = 0.0071 Average Time in System = 0.0357
Probability of Idle System = 0.0015
2. The Financial Aid Center at Rhode Island State University would like to improve its
responsiveness to students who come to the center for assistance. One idea is to hire additional financial aid counselors . A recent study of students using the Financial Aid Center found that an average of 17 students come to the center each hour . The study also found that each of the center's three financial aid counselors saw 4 students per hour on average . Try to determine how many total counselors it would take so that the average waiting time for students before they see a counselor is less than 10 minutes .
Solution: Compute 1t : λ= 17, μ=4,
From the question, we need to hire N counselors to server the students:
1μ=N μ, ρ=
μλN =N
417<1, N>4.25 If N=5, then
()(
)()
(
)05
1
4
n 0n 01
1
P //17/417/417!
!
5!1!154n
N
n
N n n N N λμλμλμ-===
=
⎡⎤⎡⎤+
+
⎢⎥⎢⎥⎛⎫⎛⎫⎢⎥⎢⎥⎣⎦⎣⎦-- ⎪ ⎪⨯⎝⎭
⎝⎭
∑∑
=
1
0.008577.032240.66986276
=+
()()50122
170.0085417455!1!145N P t N N N λμλμμ⎡⎤⎡⎤
⎢⎥⎢⎥
⎛⎫⎛⎫⎢⎥⎢⎥== ⎪ ⎪⎢⎥⎢⎥⎝⎭⎝⎭⎛⎫⎛⎫⎢⎥⨯--⎢⎥ ⎪ ⎪⨯⎝⎭⎢⎥⎣⎦⎝⎭⎣
⎦ 0.0085
1386.57
910.218
13.1m i n
10m i n
54
hour =⨯==> if N=6, then