1. Spreadsheet Modeling & Decision Analysis
A Practical Introduction to Management Science
5th edition
Cliff T. Ragsdale

2. Chapter 13
Queuing Theory

3. Introduction 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.

4. The 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

5. Queuing Costs $ Service Level Total Cost Cost of providing service
Cost of customer dissatisfaction
Service Level

6. Common Queuing System Configurations
CustomerArrives
...
Waiting Line
Server
CustomerLeaves
CustomerLeaves
...
Waiting Line
Server 1
Server 2
Server 3
CustomerArrives
Waiting Line
Server 1
Server 2
Server 3
CustomerLeaves
...
CustomerArrives

7. Characteristics of Queuing Systems: The Arrival Process
Arrival rate - the manner in which customers arrive at the system for service.
Arrivals are often described by a Poisson random variable:
where l is the arrival rate (e.g., calls arrive at a rate of l=5 per hour)
See file Fig13-3.xls

8. Characteristics of Queuing Systems: The Service Process
Service time - the amount of time a customer spends receiving service (not including time in the queue).
Service times are often described by an Exponential random variable:
where m is the service rate (e.g., calls can be serviced at a rate of m=7 per hour)
The average service time is 1/m.
See file Fig13-4.xls

9. Comments
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 l=5 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.

10. Kendall Notation Queuing systems are described by 3 parameters: 1/2/3
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

11. Operating 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.

12. Key Operating Characteristics of the M/M/1 Model

13. The 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.

14. The 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.
l< sm.

15. An 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.
Bitway’s 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.
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.

16. Implementing the Model
See file Q.xls

17. Summary 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
Lq, expected queue length
L, expected number in system
Wq, expected time in queue
W, expected total time in system
Probability that a customer waits

18. The M/M/s Model With Finite Queue Length
In some problems, the amount of waiting area is limited.
Example,
Suppose Bitway’s 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.

19. Implementing the Model
See file Q.xls

20. Summary of Results: Bitway 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
Lq, expected queue length
L, expected number in system
Wq, expected time in queue
W, expected total time in system
Probability that a customer waits
Probability that a customer balks

21. The 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.

22. M/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.

23. Implementing the Model
See file Q.xls

24. Summary of Results: Miller Manufacturing
Arrival rate
Service rate
Number of servers
Population size
Utilization 67.80% 36.76% 24.67%
P(0), probability that the system is empty
Lq, expected queue length
L, expected number in system
Wq, expected time in queue
W, expected total time in system
Probability that a customer waits
Hourly cost of service technicians $ $ $60.00
Hourly cost of inoperable machines $ $ $74.76
Total hourly costs $ $ $134.76

25. The 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.

26. An 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 hours).
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.

27. Implementing the Model
See file Q.xls

28. Summary of Results: Zippy Lube
Arrival rate
Average service TIME
Standard dev. of service time
Utilization 87.5% 70.0% 87.41%
P(0), probability that the system is empty
Lq, expected queue length
L, expected number in system
Wq, expected time in queue
W, expected total time in system

29. Payback 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 weeks

30. The 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).

31. Simulating 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

32. End of Chapter 13