1. Queuing Theory Models By
Nancy Hutchins

2. Agenda What is queuing Why is queuing important
How can this help our company
Explanation
How it works
Summary
Reading list

3. What is Queuing?
A queue is a line of waiting people, vehicles, products, etc.
Queuing theory models use a mathematical approach to study queues and make them as efficient as possible

4. Video Clip Office Space Grid Lock
Here is a link for the opening scene of “Office Space.” It portrays the main character stuck in traffic (waiting in a queue).
Review the clip and determine if it would enhance your queue training.
(The movie this clip is from is rated R, but there is nothing inappropriate in this short clip)

5. Why is this important? Inadequate queue management may lead to
Customers leaving before completing their transaction
Decrease in customer satisfaction
Reduction in number of return customers
Ask the participants if they have ever left a store or service provider because they did not have time to wait in line or wait for the service.
Ask if someone would like to tell a story about how long waits decreased their satisfaction about a business
If possible, share a short story about a
Customers may not be willing to wait in line to make a small purchase.
Customers may choose another business or service provider with shorter wait times.
Dissatisfied customers usually tell more people about bad experiences than satisfied customers tell others about their good experiences.
Dissatisfied customers are significantly less likely to return than satisfied customers.
It is about 6 times more cost effective to retain customers than it is to obtain new customers.

6. Why is this important?
Retaining customers much more cost effective than finding new customers
Many businesses depend on revenue from repeat customers
Customers are ultimately the ones who provide value to a business.
Most businesses have a small percentage of repeat customers who provide the majority of their revenue.
Increasing customer satisfaction can help a company increase their number of profitable repeat customers.

7. How can this help your company?
Decrease average customer wait time
Increase customer satisfaction
Increase number of return customers
Increase revenue
Increase positive word-of-mouth customer advertising
Several benefits can be gained by properly managing queues at businesses. Here are a few examples of some potential benefits.
Ask attendees if they can think of any other benefits.

8. Basic Ways to Manage Queues
Train employees to be friendly
Segment customers by needs
Ensure customers know what to expect
Divert the customer’s attention during wait times
Encourage customers to come during slack times
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.
Train employees to be friendly
Providing customers with special attention can help overcome a customer’s negative attitude toward waiting. Friendly employees can also help to increase customer satisfaction and encourage customers to return.
Segment customers by needs
Certain customers can be served quickly while other require a longer service time. Try to segment (separate) customers who can be served quickly so that they will not have excessive wait times. For example, most grocery stores have lines specifically for customers with 10 items or less. Another example, some banks have two lines; one where customers wait for quick transactions (deposits, withdrawals, fund transfers, etc.) and another line for customers with more specialized needs (open a new account, apply for a loan, etc.)
Ensure customers know what to expect
Customer time is valuable, ensure they know about how long it will take to be served. If the expected wait time is longer than the customer is able or willing to wait, encourage them to return at a later time or provide them with alternatives. This is especially important if the expected wait time will be longer than normal. Also, keep customers informed about what is being done to decrease their wait time.
Divert the customer’s attention during wait times
Provide music, magazines, television, toys for children, paperwork to complete, etc. to waiting customers. This will help distract the customer and can decrease their negative attitude toward waiting.
Encourage customers to come during slack times
Encouraging customers to come during slack (slow) times will help reduce the number of waiting customers during peak (busy) hours. Inform customers what your slack times are so they can decide to come during those times. Provide discounts that are only valid during your slack times. For example, movie theaters offer discounted movie tickets for afternoon movies to encourage attendance during these slow periods.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

9. Condition of Exiting Customers
The Queuing System
Source Population &
Arrival Rate
Servicing System
Condition of Exiting Customers
Source population is the number of potential customers that will use your service and will form a line and wait if necessary. Source population can be finite or infinite. Arrival rates are necessary for some models.
Arrival rate refers how customers arrive at the queue.
Servicing system topics include the length of the line, number of lines, queue discipline, service time distribution, and line structure.
Condition of exiting customers determines if the customers will return to the source population.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

10. Source Population Finite Infinite Limited size
Probabilities affected by an increase/decrease in the population
Large size
Probabilities not affected by an increase/decrease in the population
Finite population sources
Refers to the limited-size customer pool that will use the service and, at times, form a line
The reason this finite classification is important is that when a customer leaves its position as a member of the population, the size of the user group is reduced by one, which reduces the probability of the next occurrence
Conversely, when a customer is serviced and returns to the user group, the population increases and the probability of a user requiring service also increases
Infinite population sources
Is large enough in relation to the service system so that the population size caused by subtractions or additions to the population does not significantly affect the system probabilities.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

11. Distribution of Arrivals
Arrival Rate: is the number of units per period
Constant
Variable
When describing a waiting system, we need to define the manner in which customers or waiting units arrive for service.
Waiting line formulas generally require an arrival rate.
A constant arrival distribution is periodic, with exactly the same time between successive arrivals (the only systems that truly meet this criteria are machine controlled).
Variable arrival rates are much more common.
There are 2 ways we can analyze arrivals, 1) we can analyze the time between successive arrivals to see if the times follow some statistical distribution. It’s usually assumed that the time between arrivals is exponentially distributed. 2) use some length of time to try and determine how many arrivals might enter the system within T. This approach assumes that the number of arrivals per time unit is Poisson distributed.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

12. Exponential Distribution
t (minutes)
Probability that the next arrival will occur in t minutes or more
Probability that the next arrival will occur in t minutes or less
1.00
0.5
0.61
0.39
1.0
0.37
0.63
1.5
0.22
0.78
2.0
0.14
0.86
When arrivals at a service facility occur in a purely random fashion, a plot of the interval times yields an exponential distribution. The line chart depicts what the probability function may look like and the probability function is shown below the chart where λ is the mean number of arrivals per time period.
The cumulative area beneath the curve is the summation of the equation over its positive range, which is e ^ (-λt). This integral allows us to compute the probabilities of arrivals within a specified time.
Assuming λ equals 1 (an arrival rate of one customer per unit of time), then the middle column in the table represents how many customers can be expected to arrive within a specific amount of time or more. Row 1 in the middle column says that the probability of 1 customer arriving in more than 0 minutes is 1, or 100%. Row 5 says that the probability of 1 customer arriving in 2 minutes or more is .14, or 14%.
Because the area under the curve adds to 1, if 1 customer will arrive in 0 minutes or more, then 0 customers will arrive in 0 minutes or less. Likewise, there is a .86 probability that 1 customer will arrive in 2 minutes or less.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.
f(t)= λe ^ (-λt)
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

13. Customer Arrivals in Queues
Arrival Characteristics
Distribution
Pattern
Size of Arrival
Degree of Patience
Many aspects of queues can be analyzed and controlled.
We just reviewed customer arrival distributions. Distributions can be constant, exponential, Poisson, or in some other random sequence.

14. Poisson Distribution PT(n) = 𝜆𝑇 𝑛 𝑒 −𝜆𝑇 𝑛! Mean = 𝜆 = 3
Variance = 𝜆 = 3
Poisson distributions are useful to estimate the arrivals during some time period T. The distribution appears as a skewed belled curve and is obtained by finding the probability of exactly n arrivals during T. If the arrival process is random, the distribution is the Poisson and the formula is shown. The equation shows the probability of exactly n arrivals in time T.
For example, it the mean arrival rate of units into a system is three per minute (𝜆 = 3) and you want to find the probability that exactly 5 units will arrive within a 1 minute period (n = 5, T = 1), then substitute the numbers into the equation and solve. There will be a 10.1 percent chance that there will be 5 arrivals in any 1 minute interval.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

15. Pattern of Arrivals
Pattern
Controllable
Uncontrollable
The arrivals at a system are far more controllable than is generally recognized.
For example, prices may be higher during periods of high arrival rates. This can encourage customers to come at other times. This may explain why some ticket prices are higher on weekends on holidays when compared to weekday prices.
Businesses may also lower prices to entice customers to visit during periods with lower arrival rates. For example, businesses may advertise sales during their off-seasons to increase the number of customers who visit their location.
Another popular way to control arrivals is to schedule appointments for customers.
Businesses can also post their hours to inform customers when they should arrive for service.
But not all arrivals may be controlled. For example, customers seeking emergency medical services at one hospital can not be controlled. But in some areas, multiple hospitals post average wait times online and on billboards for various locations. This is one way some hospitals try to control arrival rates.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

16. Size of Arrival Units
Size of Arrival Units
Single
Batch
A single arrival may be thought of as one unit. (A unit is the smallest number handled) A single arrival on the floor of the New York Stock Exchange is 100 shares of stock.
A batch arrival is some multiple of the unit, such as a block of 1,000 shares of stock (10 units) on the New York Stock Exchange.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

17. Degree of Patience
Degree of Patience
Patient (in line and stay)
Impatient
Arrive, View, and Leave
Arrive, Wait Awhile, then Leave
A patient arrival is one who wait as long as necessary until the service facility is ready to serve them. (Even if arrivals grumble and behave impatiently, the fact that they wait is sufficient to label them as patient arrivals for purposes of waiting line theory.)
There are 2 classes of impatient arrivals. Members of the first class arrive, survey both the service facility and the length of the line, and then decide to leave. Those in the second class, view the situation, join the waiting line, and then, after some period of time, depart.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

18. Queuing System Factors
Length
Infinite potential length
Limited capacity
Number of Lines
Single
Multiple
Queue Discipline
An infinite line is simply one that is very long in terms of the capacity of the service system. Examples infinite potential length are a line of vehicles backed up for miles like there was in the “Office Space” video clip.
Certain places may have limited line capacities caused by legal restrictions or physical space characteristics. This complicates the waiting line problem not only in service system utilization and waiting line computations but also in the shape of the actual arrival distribution. The arrival denied entry into the line because of lack of space may rejoin the population for a later try or may seek service elsewhere. Either action makes an obvious difference in the finite population case.
A single line or single file is one line only. The term multiple lines refers to the single lines that form in front of two or more servers or to single lines that converge at some central redistribution point. The disadvantage of multiple lines in a busy facility is that arrivals often shift lines if several pervious services have been of short duration or if those customers currently in other lines appear to require a short service time. We also saw this idea in the video clip when the driver changes lanes multiple times only to find that he had just switched into the slowest moving lane of traffic.
Queue discipline is a priority rule or set of rules for determining the order of service to customers in a waiting line.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

19. Queue Discipline First Come, First Served (FCFS)
Shortest Processing Time
Reservations First
Emergencies First
Limited Needs
Other
The rules selected can have a dramatic effect on the system’s overall performance. The number of customers in line, the average waiting time, the range of variability in waiting time, and the efficiency of the service facility are just a few of the factors affected by the choice of priority rules.
Probably the most common priority rule is first come, first served. This rule states that customers in line are served on the basis of their chronological arrival; no other characteristics have any bearing on the selection process. This is generally accepted as the fairest rule, although in practice it discriminates against the arrival requiring a short service time.
Other methods may also include: highest-profit customer first, largest order first, best customers first, longest waiting time in line, and soonest promised date.
There are two major practical problems in using any rule: One is ensuring that customers know and follow the rule. The other is ensuring that a system exists to enable employees to manage the line.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

20. Service Time Distribution
Service rate: the capacity of the server in number of units per time period and not as service time.
Another important feature of the waiting line structure is the time the customer or unit spends with the server once the service has started. Waiting line formulas generally specify service rate as the capacity of the server in number of units per time period (such as 12 completions per hour) and not as service time, which might average five minutes each. A constant service time rule states that each service takes exactly the same time. As in constant arrivals, this characteristic is generally limited to machine-controlled operations.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

21. Line Structures Single Channel, Single Phase
Single Channel, Multiphase
Multichannel, Single phase
Multichannel, multiphase
Mixed
Single channel, single phase
This is the simplest type of waiting line structure, and straightforward formulas are available to solve the problem for standard distribution patterns of arrival and service. When the distributions are nonstandard, the problem is easily solved by computer simulation. A typical example of a single channel, single phase situation is the one-person barbershop.
Single channel, multiphase
A car wash is an illustration because a series of services (vacuuming, wetting, washing, rinsing, drying, window cleaning, and parking) is performed in a fairly uniform sequence. A critical factor in the single channel case with service in series is the amount of buildup of items allowed in front of each service, which in turn constitutes separate waiting lines.
Multichannel, single phase
Checkout counters in high-volume department stores exemplify this type of structure. The difficulty with this format is that the uneven service time given each customer results in unequal speed of flow among the lines. This results in some customers being served before others who arrived earlier, as well as in some degree of line shifting. Varying this structure to ensure the servicing of arrivals in chronological order would require forming a single line, from which, as a server becomes available, the next customer in the queue is assigned.
Multichannel, multiphase
This case is similar to multichannel, single phase except that two or more services are performed in sequence.
Mixed
Under this general heading are two subcategories: (1) multiple-to-single channel structures and (2) alternative path structures. Under (1), we find either lines that merge into one for single-phase service or lines that merge into one for multiphase service. Under (2), we encounter two structures that differ in directional flow requirements. The first is similar to the multichannel, multiphase case, except that (a) there may be switching from one channel to the next after the first service has been rendered and (b) the number of channels and phases may vary—again—after the performance of the first service.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

22. Exiting the Queuing System
Low Probability of Re-service
Return to Source Population
Once a customer is served, two exit fates are possible: (1) The customer may return to the source population and immediately become a competing candidate for service again or (2) there may be a low probability of re-service.
It should be apparent that when the population source is infinite, any change in the service performed on customers who return to the population modifies the arrival rate at the service facility. This, of course alters the characteristics of the waiting line under study and necessitates reanalysis of the problem.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

23. Properties of Some Specific Line Models
Layout
Service
Phase
Population
Source
Arrival Pattern
Discipline
Queue
Service Pattern
Permissible Queue Length
Example 1
Single
Channel
Infinite
Poisson
FCFS
Exponential
Unlimited
One-lane toll bridge 2
Constant
Roller coaster rides in amusement park 3
Multi-channel
Parts counter in auto agency
This slide and the following slides would be good handouts to provide for future reference.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

24. Infinite Queuing Notation: Models 1-3
λ = arrival rate
µ = service rate
1/µ = average service time
1/λ = average time between arrivals
ρ = ratio of total arrival rate to service rate for a single server (λ/µ)
Lq = average number waiting in line
Ls = average number in system (including and being served)
This slide and the following slides would be good handouts to provide for future reference.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

25. Infinite Queuing Notation: Models 1-3
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
This slide would be good handouts to provide for future reference.

26. Equations for Model 1 Lq = 𝜆 2 𝜇(𝜇−𝜆) Ls = 𝜆 𝜇−𝜆 Wq = Lq 𝜆 Ws = Ls 𝜆
Pn = (1− 𝜆 𝜇 ) ( 𝜆 𝜇 ) 𝑛
ρ = 𝜆 𝜇
Po = (1− 𝜆 𝜇 )

27. Equations for Model 2 and 3
Lq = 𝜆 2 2𝜇(𝜇−𝜆)
Ls = Lq + 𝜆 𝜇
Wq = Lq 𝜆
Ws = Ls 𝜆
Ls = Lq + 𝜆 𝜇
Wq = Lq 𝜆
Ws = Ls 𝜆
Pw = Lq ( 𝑆𝜇 𝜆 −1)

28. Brainstorming Exercise
What are inexpensive ways
our company can reduce
customer wait times?
Spend 5 to 10 minutes brainstorming. To effectively brainstorm consider
-having each participant write down their thoughts individually
-break the participants into small groups to brainstorm ideas
-conduct a group brainstorm session
Consider the number of participants and the amount of time available

29. Summary
Effective queue management may lead to improved customer satisfaction and increased revenue
Many queue management methods require little money to implement
Software is available to help with queue analysis
Waiting line analysis is relevant to mane service situations. The basic objective is to balance the cost of waiting with the cost of adding more resources. For a service system this means that the utilization of a server may be quite low to provide a short waiting time to the customer. Many queuing problems appear quite simple until an attempt is made to solve them. This presentation provided a brief overview of simpler concepts. When situations become more complex, when there are multiple phases, or when services are performed only in a particular sequence, computer simulation may be necessary.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.

30. Reading List
An Introduction to Queuing Theory: Modeling and Analysis in Applications (Statistics for Industry and Technology) by U. Narayan Bhat
Introduction to Queuing Networks by Erol Gelenbe and Guy Pujolle
Optimal Design of Queuing Systems by Shaler Stidham
Fundamentals of Queuing Theory by Donald Gross and Carl M. Harris
Operations and Supply Management The Core by Jacobs, F. Robert, and Richard B. Chase

31. Reference
** Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, Print.