3. Components of the Queuing
System
Servicing System
Servers
Queue or
Customer
Arrivals
Waiting Line
Exit
3
4. Customer Service Population
Sources
Population Source
Finite
Infinite
Example: Number of
Example: Number of
machines needing
machines needing
repair when a
repair when a
company only has
company only has
three machines.
three machines.
Example: The
Example: The
number of people
number of people
who could wait in
who could wait in
a line for
a line for
gasoline.
gasoline.
4
8. Suggestions for Managing
Queues
1. Determine an acceptable waiting
time for your customers
2. Try to divert your customer’s
attention when waiting
3. Inform your customers of what to
expect
4. Keep employees not serving the
customers out of sight
5. Segment customers
8
9. 9
Suggestions for Managing Queues
(Continued)
6. Train your servers to be friendly
7. Encourage customers to come during
the slack periods
8. Take a long-term perspective toward
getting rid of the queues
10. 10
Waiting Line Models
Model Layout
1
Single channel
Source
Population
Infinite
Service Pattern
Exponential
2
Single channel
Infinite
Constant
3
Multichannel
Infinite
Exponential
4
Single or Multi
Finite
Exponential
These four models share the following characteristics:
• Single phase
• Poisson arrival
• FCFS
• Unlimited queue length
11. 11
Notation: Infinite Queuing: Model
1
λ = Arrival rate
µ = Service rate
1
= Average service time
µ
1
= Average time between arrivals
λ
λ
ρ = = Ratio of total arrival rate to sevice rate
µ
for a single server
Lq = Average number waiting in line
12. Infinite Queuing Model 1
(Continued)
Ls = Average number in system
(including those being served)
Wq = Average time waiting in line
Ws = Average total time in system
(including time to be served)
n = Number of units in the system
S = Number of identical service channels
Pn = Probability of exactly n units in system
Pw = Probability of waiting in line
12
13. Example: Model 1
Assume a drive-up window at a fast food restaurant.
Customers arrive at the rate of 25 per hour.
The employee can serve one customer every two
minutes.
Assume Poisson arrival and exponential service
rates.
Determine:
Determine:
A) What is the average utilization of the employee?
A) What is the average utilization of the employee?
B) What is the average number of customers in line?
B) What is the average number of customers in line?
C) What is the average number of customers in the
C) What is the average number of customers in the
system?
system?
D) What is the average waiting time in line?
D) What is the average waiting time in line?
E) What is the average waiting time in the system?
E) What is the average waiting time in the system?
F) What is the probability that exactly two cars will be
F) What is the probability that exactly two cars will be
in the system?
in the system?
13
14. 14
Example: Model 1
A) What is the average utilization of the
employee?
λ = 25 cust / hr
1 customer
µ =
= 30 cust / hr
2 mins (1hr / 60 mins)
λ
25 cust / hr
ρ =
=
= .8333
µ
30 cust / hr
15. 15
Example: Model 1
B) What is the average number of customers in
line?
λ
(25)
Lq =
=
= 4.167
µ ( µ - λ ) 30(30 - 25)
2
2
C) What is the average number of customers in the
system?
λ
25
Ls =
=
=5
µ - λ (30 - 25)
16. 16
Example: Model 1
D) What is the average waiting time in line?
Lq
Wq =
= .1667 hrs = 10 mins
λ
E) What is the average waiting time in the system?
Ls
Ws =
= .2 hrs = 12 mins
λ
17. Example: Model 1
F) What is the probability that exactly two cars
will be in the system (one being served and the
other waiting in line)?
pn
λ λ
= (1- )( )
µ µ
n
25 25 2
p 2 = (1- )( ) = .1157
30 30
17
18. 18
Question Bowl
The central problem for virtually all queuing
problems is which of the following?
a.
Balancing labor costs and equipment costs
b.
Balancing costs of providing service with the
costs of waiting
c.
Minimizing all service costs in the use of
equipment
d.
All of the above
e.
None of the above
Answer: b. Balancing
costs of providing
service with the costs
of waiting
19. 19
Question Bowl
Customer Arrival “populations” in a queuing
system can be characterized by which of the
following?
a.
Poisson
b.
Finite
c.
Patient
d.
FCFS
e.
None of the above
Answer: b. Finite
20. 20
Question Bowl
Customer Arrival “rates” in a queuing system
can be characterized by which of the
following?
a.
Constant
b.
Infinite
c.
Finite
d.
All of the above
e.
None of the above
Answer: a. Constant
21. 21
Question Bowl
An example of a “queue discipline” in a queuing
system is which of the following?
a.
Single channel, multiphase
b.
Single channel, single phase
c.
Multichannel, single phase
d.
Multichannel, multiphase
None of the above
Answer: e. None of the above (These are the rules for
determining the order of service to customers, which
include FCFS, reservation first, highest-profit customer
first, etc.)
e.
22. 22
Question Bowl
Withdrawing funds from an automated teller machine
is an example in a queuing system of which of
the following “line structures”?
a.
Single channel, multiphase
b.
Single channel, single phase
c.
Multichannel, single phase
d.
Multichannel, multiphase
e.
None of the above
Answer: b. Single channel, single phase
23. 23
Question Bowl
Refer to Model 1 in the textbook. If the service
rate is 15 per hour, what is the “average
service time” for this queuing situation?
a.
16.00 minutes
b.
0.6667 hours
c.
0.0667 hours
d.
16% of an hour
e.
Can not be computed from data above
Answer: c. 0.0667 hours (1/15=0.0667)
