Maths Quest Maths A Year 12 for Queensland
WorkSHEET 8.2
Chapter 8 Directed graphs and networks WorkSHEET 8.2
1
Critical path analysis and queuing
Name: ___________________________
1
2
In one sentence describe each of the following
terms relating to people’s behaviour in queues:
(a) Baulking
(b) Reneging
(c) Jockeying
What is the meaning of the following terms?
(a) Inter-arrival time
(b) Service Time.
(a)
Baulking occurs when a customer decides
to leave the store because the queue
appears too long.
(b)
Reneging occurs when a customer leaves
a queue that he or she has joined.
(c)
Jockeying describes the behaviour of
those who move from queue to queue
trying to find the shortest.
(a)
The inter-arrival time is the time between
the arrival of one customer and the next
at a service point.
(b)
The service time is the time it takes to
serve a customer. It does not include time
spent waiting in the queue.
/50
6
4
Questions 3 to 7 refer to the following table
which provides information about customers in
a queue.
Time
11.00
11.02
11.04
11.06
11.08
11.10
11.12
11.14
11.16
11.18
11.20
11.22
11.24
11.26
11.28
11.30
3
Customer
served
A
A
A
B
B
B
C
C
C
D
D
D
E
E
E
F
Arrivals
B
C
D
E
F
G
H
I
-
People in
queue
B, C
C
C, D
C, D
Calculate the time taken to serve each
customer.
Length of
queue
0
1
2
1
2
2
The average time is 6 minutes.
5
Maths Quest Maths A Year 12 for Queensland
Chapter 8 Directed graphs and networks WorkSHEET 8.2
2
4
On average what is the inter-arrival time?
Nine customers arrive in 30 minutes.
The average inter-arrival time
is 30  9  3.33 minutes.
5
5
For how long did customer D wait until he or
she was served?
Customer D arrived at 11:08 and was served at
11:18. Customer D waited 10 minutes in the
queue.
5
6
Complete the table showing the people in the
queue and the length of the queue from 11:12
until 11:30.
Time
11.12
11.14
11.16
11.18
11.20
11.22
11.24
11.26
11.28
11.30
People in
queue
D, E
D, E
D, E, F
E, F
E, F, G
E, F, G
F, G, H
F, G, H
F, G, H, I
G, H, I
Length of
queue
2
2
3
2
3
3
3
3
4
3
6
7
Describe the long-term behaviour of this queue. As the inter-arrival time (3.33 min) is less than
the service time (6 min), the queue will grow.
4
8
Customers arrive at Crazy Larry’s checkouts at (a)
the rate of 2.5 customers each minute. On
average it takes 45 seconds to serve each
(b)
customer.
(a) What is the inter-arrival time?
Inter-arrival time  60  2.5  24 seconds
4
(b)
How many service points should be
operating to prevent a queue from
growing?
Two queues. Because the service time is
45 seconds and 45  2  22.5 which is
less than 24.
Maths Quest Maths A Year 12 for Queensland
Chapter 8 Directed graphs and networks WorkSHEET 8.2
3
Questions 9 and 10 refer to the following:
People arriving at the Balthorne Cinema
Complex usually take only a short time to be
served but occasionally a customer holds the
queue up as illustrated in the following table
showing the distribution of arrival times:
Service time (to the
nearest 15 seconds)
15
30
45
Probability
0.9
0.03
0.07
Suppose customers arrive at a rate of 15 each
minute.
9
10
On average, how long does it take to serve a
customer?
If the cinema complex can staff a maximum of
four service points, describe the long-term
behaviour of the queues.
5
Average service time
 0.9  15  0.03  30  0.07  45
 17.6 seconds.
6
If there are 4 service points then:
Average service time  number of service points
 17.6  4
 4.4 seconds.
This is larger than the inter-arrival time so the
queues will grow in the long term.