MGTSC 352
Lecture 23:
Congestion Management
Introduction: Asgard Bank example
Simulating a queue
Types of congested systems, queueing template
Ride’n’Collide example
MEC example
Manufacturing example
Analyzing a Congested System
(pg. 174)
Inputs
System Description
Model of the System
Measures of Quality of Service
Outputs
Measures important to Servers
Time between Arrivals (min.)
> 4.00
3.75 - 4.00
3.50 - 3.75
3.25 - 3.50
3.00 - 3.25
40
2.75 - 3.00
50
2.50 - 2.75
60
2.25 - 2.50
2.00 - 2.25
1.75 - 2.00
70
1.50 - 1.75
1.25 - 1.50
1.00 - 1.25
0.75 - 1.00
0.50 - 0.75
0.25 - 0.50
0.00 - 0.25
Frequency
pg. 168
Asgard Bank: Times Between
Arrivals
(pg. 173)
Average = 1.00 min.
St. dev. = 0.92 min.
Arrival rate =  = ?
30
20
10
0
Duration of Service (min.)
3.80 - 3.90
3.60 - 3.70
3.40 - 3.50
3.20 - 3.30
3.00 - 3.10
2.80 - 2.90
2.60 - 2.70
2.40 - 2.50
2.20 - 2.30
40
2.00 - 2.10
50
1.80 - 1.90
1.60 - 1.70
1.40 - 1.50
1.20 - 1.30
1.00 - 1.10
0.80 - 0.90
0.60 - 0.70
0.40 - 0.50
0.20 - 0.30
0.00 - 0.10
Frequency
Asgard Bank: Service Times
80
70
60
Average = 0.95 min. (57 sec.)
St. dev. = 0.17 min. (10 sec.)
service rate =  = ?
30
20
10
0
Including Randomness: Simulation
• Service times:
Normal distribution, mean = 57/3600 hrs, stdev
= 10/3600 hrs.
MAX(NORMINV(RAND(),57/3600,10/3600),0)
• Inter-arrival times:
Exponential distribution, mean = 1/ 60 hrs.
– (1/60)*LN(RAND())
To Excel …
Simulated Lunch Hour 1:
Customer number
61
51
Waiting time
41
Service time
71 arrivals
31
21
Press F9 to recalculate
11
1
0.00
0.10
0.20
0.30
0.40
0.50
Time (hours)
0.60
0.70
0.80
0.90
1.00
Simulated Lunch Hour 2:
Customer number
61
51
Waiting time
41
Service time
31
50 arrivals
21
Press F9 to recalculate
11
1
0.00
0.10
0.20
0.30
0.40
0.50
Time (hours)
0.60
0.70
0.80
0.90
1.00
Simulated Lunch Hour 3:
Unused capacity
Customer number
61
51
Waiting time
41
Service time
31
21
Press F9 to recalculate
11
1
0.00
0.10
0.20
0.30
0.40
0.50
Time (hours)
0.60
0.70
0.80
0.90
1.00
Causes of Congestion
• Higher than average number of arrivals
• Lower than average service capacity
• Lost capacity due to timing
Lesson: For a service where customers arrive
randomly, it is not a good idea to operate the
system close to its average capacity
Template.xls
• Does calculations for
–
–
–
–
•
M/M/s
M/M/s/s+C
M/M/s//M
M/G/1
Want to know more? Go to http://www.bus.ualberta.ca/aingolfsson/qtp/
• Asgard Bank Data
–
–
–
–
Model: M/G/1
Arrival rate: 1 per minute
Average service time: 57/60 min.
St. dev of service time: 10/60 min.
Asgard Conclusions
• The ATM is busy 95% of the time.
• Average queue length = 9.3 people
• Average no. in the system = 10.25
(waiting, or using the ATM)
• Average wait = 9.3 minutes
• What if the service rate changes to …
– 1.05 / min.?
– 1.06 / min.?
Ride’n’Collide (pg. 178)
•
•
•
•
•
Repair personnel cost: $10 per hour
Average repair duration: 30 minutes
Lost income: $50 per hour per car
Number of cars: 20 cars
A car will function for 10 hours on average
from the time it has been fixed until the
next time it needs to be repaired.
• How many repair-people should be hired?
Ride’n’Collide
• Customers =
• Servers =
• Average number in system =
•
•
•
•
Lost revenue per hour =
Arrival rate =
Service rate =
Model to use:
Waiting Line Analysis Template:
Which Model to Use?
• Who are the customers?
• Who are the servers?
• Where is the queue?
… not always obvious
• If you are told how many customers there
are
… then you should consider using the “finite
population” template
waiting room =
queue
Number is small
enough to worry
potential customers about
parallel servers
• If you are told the maximum number of
customers that can wait (the size of the
waiting room)
… then you should consider using the “finite
Q” template
waiting room =
queue
Capacity is small
enough to worry
potential customers about
parallel servers
• If you are told the standard deviation of the
service times, and there is 1 server
… then you should consider using the
“MG1” template
waiting room =
queue
one server, nonexponential service
time distribution
potential customers
• If you are told nothing about the size of the
pool of potential customers, or the
maximum number that will wait, or the
standard deviation of the service times,
… then you should probably use the “MMs”
template
MEC (p. 181)
• One operator, two lines to take orders
– Average call duration: 4 minutes exp
– Average call rate: 10 calls per hour exp
– Average profit from call: $24.76
• Third call gets busy signal
• How many lines/agents?
– Line cost: $4.00/ hr
– Agent cost: $12.00/hr
– Avg. time on hold < 1 min.
Modeling Approaches
• Simulation
• Waiting line analysis template
• We’ll use both for this example
To Excel …
Manufacturing Example (p. 184)
Machine
(1.2 or 1.8/minute)
1/minute
Poisson
arrivals
Exponential
service times
Manufacturing Example
• Arrival rate for jobs: 1 per minute
• Machine 1:
– Processing rate: 1.20/minute
– Cost: $1.20/minute
• Machine 2:
– Processing rate: 1.80/minute
– Cost: $2.00/minute
• Cost of idle jobs: $2.50/minute
• Which machine should be chosen?
To Excel …
Manufacturing Example
• Cost of machine 1 =
$1.20 / min. + ($2.50 / min. / job)  (5.00 jobs)
= $13.70 / min.
• Cost of machine 2 =
$2.00 / min. + ($2.50 / min. / job)  (1.25 jobs)
= $5.13 / min.
 Switching to machine 2 saves money –
reduction in lost revenue outweighs higher
operating cost.
Cost of waiting (Mach. 1)
• Method 1:
– Unit cost × L = ($2.50 / min. job)  (5.00 jobs)
= $13.70 / min
• Method 2:
– Unit cost ×  × W =
= ($2.50 / min. job)  (5.00 min)  (1 job/min)
= $13.70 / min
• Little’s Law
L=×W