Title: Spreadsheet Modeling
1Spreadsheet Modeling Decision Analysis
- A Practical Introduction to Management Science
- 4th edition
- Cliff T. Ragsdale
2Queuing Theory
Chapter 13
3Introduction to Queuing Theory
- It is estimated that Americans spend a total of
37 billion hours a year waiting in lines. - Places we wait in line...
- ? stores ? hotels ? post offices
- ? banks ? traffic lights ? restaurants
- ? airports ? theme parks ? on the phone
- Waiting lines do not always contain people...
- ? returned videos
- ? subassemblies in a manufacturing plant
- ? electronic message on the Internet
- Queuing theory deals with the analysis and
management of waiting lines.
4The Purpose of Queuing Models
- Queuing models are used to
- describe the behavior of queuing systems
- determine the level of service to provide
- evaluate alternate configurations for providing
service
5Queuing Costs
Total Cost
Cost of providing service
Cost of customer dissatisfaction
Service Level
6Common Queuing System Configurations
7Characteristics of Queuing SystemsThe Arrival
Process
- Arrival rate - the manner in which customers
arrive at the system for service.
- where l is the arrival rate (e.g., calls arrive
at a rate of l5 per hour) - See file Fig13-3.xls
8Characteristics of Queuing SystemsThe Service
Process
- Service time - the amount of time a customer
spends receiving service (not including time in
the queue).
- where m is the service rate (e.g., calls can be
serviced at a rate of m7 per hour) - The average service time is 1/m.
- See file Fig13-4.xls
9Comments
- If arrivals follow a Poisson distribution with
mean l, interarrival times follow an Exponential
distribution with mean 1/l. - Example
- Assume calls arrive according to a Poisson
distribution with mean l5 per hour. - Interarrivals follow an exponential distribution
with mean 1/5 0.2 per hour. - On average, calls arrive every 0.2 hours or every
12 minutes. - The exponential distribution exhibits the
Markovian (memoryless) property.
10Kendall Notation
- Queuing systems are described by 3 parameters
- 1/2/3
- Parameter 1
- M Markovian interarrival times
- D Deterministic interarrival times
- Parameter 2
- M Markovian service times
- G General service times
- D Deterministic service times
- Parameter 3
- A number Indicating the number of servers.
- Examples,
- M/M/3 D/G/4 M/G/2
11Operating Characteristics
- Typical operating characteristics of interest
include - U - Utilization factor, of time that all
servers are busy. - P0 - Prob. that there are no zero units in the
system. - Lq - Avg number of units in line waiting for
service. - L - Avg number of units in the system (in line
being served). - Wq - Avg time a unit spends in line waiting for
service. - W - Avg time a unit spends in the system (in
line being served). - Pw - Prob. that an arriving unit has to wait
for service. - Pn - Prob. of n units in the system.
12Key Operating Characteristics of the M/M/1 Model
13The Q.xls Queuing Template
- Formulas for the operating characteristics of a
number of queuing models have been derived
analytically. - An Excel template called Q.xls implements the
formulas for several common types of models. - Q.xls was created by Professor David Ashley of
the Univ. of Missouri at Kansas City.
14The M/M/s Model
- Assumptions
- There are s servers.
- Arrivals follow a Poisson distribution and occur
at an average rate of l per time period. - Each server provides service at an average rate
of m per time period, and actual service times
follow an exponential distribution. - Arrivals wait in a single FIFO queue and are
serviced by the first available server. - llt sm.
15An M/M/s Example Bitway Computers
- The customer support hotline for Bitway Computers
is currently staffed by a single technician. - Calls arrive randomly at a rate of 5 per hour and
follow a Poisson distribution. - The technician services calls at an average rate
of 7 per hour, but the actual time required to
handle a call follows an exponential
distribution. - Bitways president, Rod Taylor, has received
numerous complaints from customers about the
length of time they must wait on hold for
service when calling the hotline. - Continued
16Bitway Computers (continued)
- Rod wants to determine the average length of time
customers currently wait before the technician
answers their calls. - If the average waiting time is more than 5
minutes, he wants to determine how many
technicians would be required to reduce the
average waiting time to 2 minutes or less.
17Implementing the Model
18Summary of Results Bitway Computers
- Arrival rate 5 5
- Service rate 7 7
- Number of servers 1 2
-
- Utilization 71.43 35.71
- P(0), probability that the system is empty 0.2857
0.4737 - Lq, expected queue length 1.7857 0.1044
- L, expected number in system 2.5000 0.8187
- Wq, expected time in queue 0.3571 0.0209
- W, expected total time in system 0.5000 0.1637
- Probability that a customer waits 0.7143 0.1880
19The M/M/s Model With Finite Queue Length
- In some problems, the amount of waiting area is
limited. - Example,
- Suppose Bitways telephone system can keep a
maximum of 5 calls on hold at any point in time. - If a new call is made to the hotline when five
calls are already in the queue, the new call
receives a busy signal. - One way to reduce the number of calls
encountering busy signals is to increase the
number of calls that can be put on hold. - If a call is answered only to be put on hold for
a long time, the caller might find this more
annoying than receiving a busy signal. - Rod wants to investigate what effect adding a
second technician to answer hotline calls has on - the number of calls receiving busy signals
- the average time callers must wait before
receiving service.
20Implementing the Model
21Summary of ResultsBitway Computers With Finite
Queue
- Arrival rate 5 5
- Service rate 7 7
- Number of servers 1 2
- Maximum queue length 5 5
- Utilization 68.43 35.69
- P(0), probability that the system is
empty 0.3157 0.4739 - Lq, expected queue length 1.0820 0.1019
- L, expected number in system 1.7664 0.8157
- Wq, expected time in queue 0.2259 0.0204
- W, expected total time in system 0.3687 0.1633
- Probability that a customer waits 0.6843 0.1877
- Probability that a customer balks 0.0419 0.0007
22The M/M/s Model With Finite Population
- Assumptions
- There are s servers.
- There are N potential customers in the arrival
population. - The arrival pattern of each customer follows a
Poisson distribution with a mean arrival rate of
l per time period. - Each server provides service at an average rate
of m per time period, and actual service times
follow an exponential distribution. - Arrivals wait in a single FIFO queue and are
serviced by the first available server.
23M/M/s With Finite Population Example The Miller
Manufacturing Company
- Miller Manufacturing owns 10 identical machines
that produce colored nylon thread for the
textile industry. - Machine breakdowns follow a Poisson distribution
with an average of 0.01 breakdowns per operating
hour per machine. - The company loses 100 each hour a machine is
down. - The company employs one technician to fix these
machines. - Service times to repair the machines are
exponentially distributed with an avg of 8 hours
per repair. (So service is performed at a rate of
1/8 machines per hour.) - Management wants to analyze the impact of adding
another service technician on the average time
to fix a machine. - Service technicians are paid 20 per hour.
24Implementing the Model
25Summary of Results Miller Manufacturing
- Arrival rate 0.01 0.01 0.01
- Service rate 0.125 0.125 0.125
- Number of servers 1 2 3
- Population size 10 10 10
- Utilization 67.80 36.76 24.67
- P(0), probability that the system is
empty 0.3220 0.4517 0.4623 - Lq, expected queue length 0.8463 0.0761 0.0074
- L, expected number in system 1.5244 0.8112
0.7476 - Wq, expected time in queue 9.9856 0.8282 0.0799
- W, expected total time in system 17.986 8.8282
8.0799 - Probability that a customer waits 0.6780 0.1869
0.0347 - Hourly cost of service technicians 20.00 40.00
60.00 - Hourly cost of inoperable machines 152.44
81.12 74.76 - Total hourly costs 172.44 121.12 134.76
26The M/G/1 Model
- Not all service times can be modeled accurately
using the Exponential distribution. - Examples
- Changing oil in a car
- Getting an eye exam
- Getting a hair cut
- M/G/1 Model Assumptions
- Arrivals follow a Poisson distribution with mean
l. - Service times follow any distribution with mean m
and standard deviation s. - There is a single server.
27An M/G/1 Example Zippy Lube
- Zippy-Lube is a drive-through automotive oil
change business that operates 10 hours a day, 6
days a week. - The profit margin on an oil change at Zippy-Lube
is 15. - Cars arrive at the Zippy-Lube oil change center
following a Poisson distribution at an average
rate of 3.5 cars per hour. - The average service time per car is 15 minutes
(or 0.25 hours) with a standard deviation of 2
minutes (or 0.0333 hours). - Continued
28Zippy Lube (continued)
- A new automated oil dispensing device costs
5,000. - The manufacturer's representative claims this
device will reduce the average service time by 3
minutes per car. (Currently, employees manually
open and pour individual cans of oil.) - The owner wants to analyze the impact the new
automated device would have on his business and
determine the pay back period for this device.
29Implementing the Model
30Summary of Results Zippy Lube
- Arrival rate 3.5 3.5 4.371
- Average service TIME 0.25 0.2 0.2
- Standard dev. of service time 0.0333 0.0333 0.333
- Utilization 87.5 70.0 87.41
- P(0), probability that the system is
empty 0.1250 0.3000 0.1259 - Lq, expected queue length 3.1168 0.8393 3.1198
- L, expected number in system 3.9918 1.5393 3.9939
- Wq, expected time in queue 0.8905 0.2398 0.7138
- W, expected total time in system 1.1405 0.4398 0.9
138
31Payback Period Calculation
- Increase in
- Arrivals per hour 0.871
- Profit per hour 13.06
- Profit per day 130.61
- Profit per week 783.63
- Cost of Machine 5,000
- Payback Period 6.381 weeks
32The M/D/1 Model
- Service times may not be random in some queuing
systems. - Examples
- In manufacturing, the time to machine an item
might be exactly 10 seconds per piece. - An automatic car wash might spend exactly the
same amount of time on each car it services. - The M/D/1 model can be used in these types of
situations where the service times are
deterministic (not random). - The results for an M/D/1 model can be obtained
using the M/G/1 model by setting the standard
deviation of the service time to 0 ( s 0).
33Simulating Queues
- The queuing formulas used in Q.xls describe the
steady-state operations of the various queuing
systems. - Simulation is often used to analyze more complex
queuing systems. - See file Fig13-21.xls
34End of Chapter 13