Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
The black curve shows the arrival rate in customers per hour.
At 9AM the box office opens and customers are served at a
rate of 200 per hour.
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
The length of the line at 9AM is closest to
A) 75 B) 125 C) 175 D) 225 E) 275
2
3
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
The length of the line at 9AM is closest to
A) 75 B) 125 C) 175 D) 225 E) 275
Answer: C. Each square represents 100 people so the area under the
curve from 8AM to 9AM is closest to 175 people.
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
The length of the line at 10AM is closest to
A) 50 B) 100 C) 200 D) 300 E) 400
2
3
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
The length of the line at 10AM is closest to
A) 50 B) 100 C) 200 D) 300 E) 400
Answer: C. The area under the curve minus the area under the line
between 8AM and 10AM is closest to 200 people.
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
The length of the line at 11AM is closest to
A) 50 B) 100 C) 200 D) 300 E) 400
2
3
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
The length of the line at 11AM is closest to
A) 50 B) 100 C) 200 D) 300 E) 400 Answer: C.
2
3
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
The rate (in customers per hour) the line is changing at 10AM is
A) -50 B) 50 C) 100 D) 200 E) 250
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
The rate (in customers per hour) the line is changing at 10AM is
A) -50 B) 50 C) 100 D) 200 E) 250 Answer: B. arrival rate minus
service rate at 10AM.
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
The line is longest at
A) 9AM B) 9:30AM C) 10AM D) 10:30AM E) 11AM
3
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
The line is longest at
A) 9AM B) 9:30AM C) 10AM D) 10:30AM E) 11AM Answer: D.
When arrival rate = service rate … critical point for length of line.
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
The wait time if you get in line at 9AM is closest to
A) 30 min B) 1 hr C) 1.5 hrs D) 2 hrs E) 2.5 hrs
3
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
The wait time if you get in line at 9AM is closest to
A) 30 min B) 1 hr C) 1.5 hrs D) 2 hrs E) 2.5 hrs Answer: B.
175 in line divided by 200 per hour service rate
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
The earliest time when there is no wait is closest to
A) 11AM B) 1PM C) 3PM D) never
3
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
The earliest time when there is no wait is closest to
A) 11AM B) 1PM C) 3PM D) never Answer: B. When area under
curve is close to area under line.
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
Given that the arrival rate is r(t) write an expression for the length of
line at 10AM.
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
Given that the arrival rate is r(t) write an expression for the length of
10
line at 10AM. Answer:
∫
8
r ( t ) dt
−
200
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
Given that the arrival rate is r(t) write an expression for the rate line is
increasing at 10AM.
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
Given that the arrival rate is r(t) write an expression for the rate line is
increasing at 10AM. Answer: r(10) − 200
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
Given that the arrival rate is r(t) write an expression for the time when
line is longest.
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
Given that the arrival rate is r(t) write an expression for the time when
line is longest. Answer: T such that r(T) = 200
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
Given that the arrival rate is r(t) write an expression for the wait time
at 9AM.
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
Given that the arrival rate is r(t) write an expression for the wait time
9
at 9AM. Answer:
∫
8
r ( t ) dt / 200
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
Given that the arrival rate is r(t) write an expression for the time when
there is no line.
Standing in Line
300
200
100
8AM
9
10
11
12
1PM
2
3
Given that the arrival rate is r(t) write an expression for the time when
there is no line. Answer: military time T such that
T
∫
8
r(t )dt = 200(T − 9)