1. Chapter 11
Queuing Models

2. A Basic Queueing System
Figure A basic queueing system, where each customer is indicated by C and each server by S. Although this figure shows four servers, some queueing systems (including the example in this section) have only a single server.

3. Where is There Waiting? Service Facility Disneyland Highway traffic
Fast-food restaurants
Post office
Grocery store
Bank
Disneyland
Highway traffic
Manufacturing
Equipment awaiting repair
Phone or computer network
Product orders
Slides 11.56–11.74 are based upon the lecture “Models for Queueing Theory” from the MBA elective course “Modeling with Spreadsheets” at the University of Washington (as taught by one of the authors).
With each example of where there is waiting, discuss who are the customers and who are the servers.

4. Examples of Commercial Service Systems That Are Queueing Systems
Type of System
Customers
Server(s)
Barber shop
People
Barber
Bank teller services
Teller
ATM machine service
ATM machine
Checkout at a store
Checkout clerk
Plumbing services
Clogged pipes
Plumber
Ticket window at a movie theater
Cashier
Check-in counter at an airport
Airline agent
Brokerage service
Stock broker
Gas station
Cars
Pump
Call center for ordering goods
Telephone agent
Call center for technical assistance
Technical representative
Travel agency
Travel agent
Automobile repair shop
Car owners
Mechanic
Vending services
Vending machine
Dental services
Dentist
Roofing Services
Roofs
Roofer
Table Examples of commercial service systems that are queueing systems.

5. Examples of Internal Service Systems That Are Queueing Systems
Type of System
Customers
Server(s)
Secretarial services
Employees
Secretary
Copying services
Copy machine
Computer programming services
Programmer
Mainframe computer
Computer
First-aid center
Nurse
Faxing services
Fax machine
Materials-handling system
Loads
Materials-handling unit
Maintenance system
Machines
Repair crew
Inspection station
Items
Inspector
Production system
Jobs
Machine
Semiautomatic machines
Operator
Tool crib
Machine operators
Clerk
Table Examples of internal service systems that are queueing systems.

6. Examples of Transportation Service Systems That Are Queueing Systems
Type of System
Customers
Server(s)
Highway tollbooth
Cars
Cashier
Truck loading dock
Trucks
Loading crew
Port unloading area
Ships
Unloading crew
Airplanes waiting to take off
Airplanes
Runway
Airplanes waiting to land
Airline service
People
Airplane
Taxicab service
Taxicab
Elevator service
Elevator
Fire department
Fires
Fire truck
Parking lot
Parking space
Ambulance service
Ambulance
Table Examples of transportation service systems that are queueing systems.

7. System Characteristics
Number of servers
Arrival and service pattern
rate of arrivals and service
distribution of arrivals and service
Maximum size of the queue
Queue disciplince
FCFS?
Priority system?
Population size
Infinite or finite?

8. Measures of System Performance
Average number of customers waiting
in the system
in the queue
Average time customers wait
Which measure is the most important?
When customers are internal to the organization, the first measure tends to be more important: Having such customers wait causes lost productivity.
Commercial service systems tend to place greater importance on the second measure: Outside customers are typically more concerned with how long they have to wait than with how many customers are there.

9. Population Source
Most queueing models assume an infinite population source.
If the number of potential customers is small, a finite source model can be used.
Number in system affects arrival rate (fewer potential arrivals when more in system)
Okay to assume infinite if N > 20.
Example with a finite source: Equipment awaiting repair.

10. Maximum Size of Queue
Most queueing models assume an infinite queue length is possible.
If the queue length is limited, a finite queue model can be used.
Examples with a finite queue: call center (telephone lines limited), drive through lane.

11. The Queue
The number of customers in the queue (or queue size) is the number of customers waiting for service to begin.
The number of customers in the system is the number in the queue plus the number currently being served.
The queue capacity is the maximum number of customers that can be held in the queue.
An infinite queue is one in which, for all practical purposes, an unlimited number of customers can be held there.
When the capacity is small enough that it needs to be taken into account, then the queue is called a finite queue.
The queue discipline refers to the order in which members of the queue are selected to begin service.
The most common is first-come, first-served (FCFS).
Other possibilities include random selection, some priority procedure, or even last-come, first-served.

12. Number of Servers Single Server Multiple Servers
Discuss examples of each:
Single Server -- Wendys
Multiple server (one line): bank, airport
Multiple server (line for each server): McDonalds, grocery store
These two multiple-server setups are identical from a modeling perspective. The average line lengths and waiting times will be identical. However, the left one is more fair, because it always is FCFS. In the right one, a customer can get “unlucky” and get in a slow line, and be served after someone who arrived after them but got in a quicker line. Studies have shown that waiting does not bother customers so much as perceived unfairness does.
Something else to discuss: What’s better, a single-server with many people working together to make the service go fast (the model at Wendys), vs. multiple servers, with one person at each that does the whole service more slowly (the model at McDonalds). We will return to examine this question in the following lecture.

13. Arrivals
The time between consecutive arrivals to a queueing system are called the interarrival times.
The expected number of arrivals per unit time is referred to as the mean arrival rate.
The symbol used for the mean arrival rate is
l = Mean arrival rate for customers coming to the queueing system
where l is the Greek letter lambda.
The mean of the probability distribution of interarrival times is
1 / l = Expected interarrival time
Most queueing models assume that the form of the probability distribution of interarrival times is an exponential distribution.

14. Arrival Pattern A Poisson distribution is usually assumed.
A good approximation of random arrivals.
Lack-of-memory property: Probability of an arrival in the next instant is constant, regardless of the past.

15. Service
When a customer enters service, the elapsed time from the beginning to the end of the service is referred to as the service time.
Basic queueing models assume that the service time has a particular probability distribution.
The symbol used for the mean of the service time distribution is / m = Expected service time where m is the Greek letter mu.
The interpretation of m itself is the mean service rate. m = Expected service completions per unit time for a single busy server

16. Some Service-Time Distributions
Exponential Distribution
The most popular choice.
Much easier to analyze than any other.
Although it provides a good fit for interarrival times, this is much less true for service times.
Provides a better fit when the service provided is random than if it involves a fixed set of tasks.
Standard deviation: s = Mean
Constant Service Times
A better fit for systems that involve a fixed set of tasks.
Standard deviation: s = 0.

17. Service Pattern Either an exponential distribution is assumed,
Implies that the service is usually short, but occasionally long
If service time is exponential then service rate is Poisson
Lack-of-memory property: The probability that a service ends in the next instant is constant (regardless of how long its already gone).
Decent approximation if the jobs to be done are random.
Not a good approximation if the jobs to be done are always the same.
Or any distribution
Only single-server model is easily solved.

18. Properties of the Exponential Distribution
There is a high likelihood of small interarrival times, but a small chance of a very large interarrival time. This is characteristic of interarrival times in practice.
For most queueing systems, the servers have no control over when customers will arrive. Customers generally arrive randomly.
Having random arrivals means that interarrival times are completely unpredictable, in the sense that the chance of an arrival in the next minute is always just the same.
The only probability distribution with this property of random arrivals is the exponential distribution.
The fact that the probability of an arrival in the next minute is completely uninfluenced by when the last arrival occurred is called the lack-of-memory property.

19. Notation
Parameters:
l = customer arrival rate m = service rate (1/m = average service time) s = number of servers
Performance Measures
Lq = average number of customers in the queue L = average number of customers in the system Wq = average waiting time in the queue W = average waiting time (including service) Pn = probability of having n customers in the system r = system utilization
Arrival rate and service rate must be in the form of a rate (customers per unit time) and must be in the same units.

20. Application of Queueing Models
We can use the results from queueing models to make the following types of decisions:
How many servers to employ.
How large should the waiting space be.
Whether to use a single fast server or a number of slower servers.
Whether to have a general purpose server or faster specific servers.
Slides 11.75–11.88 are based upon a lecture “Application of Queueing Models” from the MBA elective course “Modeling with Spreadsheets” at the University of Washington (as taught by one of the authors).

21. Herr Cutter’s Barber Shop
Herr Cutter is a German barber who runs a one-man barber shop.
Herr Cutter opens his shop at 8:00 A.M.
The table shows his queueing system in action over a typical morning.
Customer
Time of Arrival
Haicut Begins
Duration of Haircut
Haircut Ends 1
8:03
17 minutes
8:20 2
8:15
21 minutes
8:41 3
8:25
19 minutes
9:00 4
8:30
15 minutes
9:15 5
9:05
20 minutes
9:35 6
9:43 —
Table The data for Herr Cutter’s first five customers.

22. Evolution of the Number of Customers
Figure The evolution of the number of customers in Herr Cutter’s barber shop over the first 100 minutes (from 8:00 to 9:40), given the data in Table 11.1.

23. Labels for Queueing Models
To identify which probability distribution is being assumed for service times (and for interarrival times), a queueing model conventionally is labeled as follows: Distribution of service times — / — / — Number of Servers Distribution of interarrival times
The symbols used for the possible distributions are M = Exponential distribution (Markovian) D = Degenerate distribution (constant times) GI = General independent interarrival-time distribution (any distribution) G = General service-time distribution (any arbitrary distribution)

24. Summary of Usual Model Assumptions
Interarrival times are independent and identically distributed according to a specified probability distribution.
All arriving customers enter the queueing system and remain there until service has been completed.
The queueing system has a single infinite queue, so that the queue will hold an unlimited number of customers (for all practical purposes).
The queue discipline is first-come, first-served.
The queueing system has a specified number of servers, where each server is capable of serving any of the customers.
Each customer is served individually by any one of the servers.
Service times are independent and identically distributed according to a specified probability distribution.

25. Choosing a Measure of Performance
Managers who oversee queueing systems are mainly concerned with two measures of performance:
How many customers typically are waiting in the queueing system?
How long do these customers typically have to wait?
When customers are internal to the organization, the first measure tends to be more important.
Having such customers wait causes lost productivity.
Commercial service systems tend to place greater importance on the second measure.
Outside customers are typically more concerned with how long they have to wait than with how many customers are there.

26. Defining the Measures of Performance
L = Expected number of customers in the system, including those being served (the symbol L comes from Line Length).
Lq = Expected number of customers in the queue, which excludes customers being served.
W = Expected waiting time in the system (including service time) for an individual customer (the symbol W comes from Waiting time).
Wq = Expected waiting time in the queue (excludes service time) for an individual customer.
These definitions assume that the queueing system is in a steady-state condition.

27. Relationship between L, W, Lq, and Wq
Since 1/m is the expected service time W = Wq + 1/m
Little’s formula states that L = lW and Lq = lWq
Combining the above relationships leads to L = Lq + l/m

28. The Dupit Corp. Problem
The Dupit Corporation is a longtime leader in the office photocopier marketplace.
Dupit’s service division is responsible for providing support to the customers by promptly repairing the machines when needed. This is done by the company’s service technical representatives, or tech reps.
Current policy: Each tech rep’s territory is assigned enough machines so that the tech rep will be active repairing machines (or traveling to the site) 75% of the time.
A repair call averages 2 hours, so this corresponds to 3 repair calls per day.
Machines average 50 workdays between repairs, so assign 150 machines per rep.
Proposed New Service Standard: The average waiting time before a tech rep begins the trip to the customer site should not exceed two hours.

29. Alternative Approaches to the Problem
Approach Suggested by John Phixitt: Modify the current policy by decreasing the percentage of time that tech reps are expected to be repairing machines.
Approach Suggested by the Vice President for Engineering: Provide new equipment to tech reps that would reduce the time required for repairs.
Approach Suggested by the Chief Financial Officer: Replace the current one-person tech rep territories by larger territories served by multiple tech reps.
Approach Suggested by the Vice President for Marketing: Give owners of the new printer-copier priority for receiving repairs over the company’s other customers.

30. The Queueing System for Each Tech Rep
The customers: The machines needing repair.
Customer arrivals: The calls to the tech rep requesting repairs.
The queue: The machines waiting for repair to begin at their sites.
The server: The tech rep.
Service time: The total time the tech rep is tied up with a machine, either traveling to the machine site or repairing the machine. (Thus, a machine is viewed as leaving the queue and entering service when the tech rep begins the trip to the machine site.)

31. Notation for Single-Server Queueing Models
l = Mean arrival rate for customers = Expected number of arrivals per unit time 1/l = expected interarrival time
m = Mean service rate (for a continuously busy server) = Expected number of service completions per unit time 1/m = expected service time
r = the utilization factor = the average fraction of time that a server is busy serving customers = l / m

32. The M/M/1 Model Assumptions
Interarrival times have an exponential distribution with a mean of 1/l.
Service times have an exponential distribution with a mean of 1/m.
The queueing system has one server.
The expected number of customers in the system is
L = r / (1 – r) = l / (m – l)
The expected waiting time in the system is
W = (1 / l)L = 1 / (m – l)
The expected waiting time in the queue is
Wq = W – 1/m = l / [m(m – l)]
The expected number of customers in the queue is
Lq = lWq = l2 / [m(m – l)] = r2 / (1 – r)

33. The M/M/1 Model
The probability of having exactly n customers in the system is
Pn = (1 – r)rn Thus, P0 = 1 – r P1 = (1 – r)r P2 = (1 – r)r2 : :
The probability that the waiting time in the system exceeds t is
P(W > t) = e–m(1–r)t for t ≥ 0
The probability that the waiting time in the queue exceeds t is
P(Wq > t) = re–m(1–r)t for t ≥ 0

34. M/M/1 Queueing Model for the Dupit’s Current Policy
Figure This spreadsheet shows the results from applying the M/M/1 model with l = 3 and m = 4 to the Dupit case study under the current policy.

35. John Phixitt’s Approach (Reduce Machines/Rep)
The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq ≤ 1/4 day).
John Phixitt’s suggested approach is to lower the tech rep’s utilization factor sufficiently to meet the new service requirement. Lower r = l / m, until Wq ≤ 1/4 day, where l = (Number of machines assigned to tech rep) / 50.

36. M/M/1 Model for John Phixitt’s Suggested Approach (Reduce Machines/Rep)
Figure This application of the spreadsheet in Figure 11.4 shows that, when m = 4, the M/M/1 model gives an expected waiting time to begin service of Wq = 0.25 days (the largest value that satisfies Dupit’s proposed new service standard) when l is changed from l = 3 to l = 2.

37. The M/G/1 Model Assumptions
Interarrival times have an exponential distribution with a mean of 1/l.
Service times can have any probability distribution. You only need the mean (1/m) and standard deviation (s).
The queueing system has one server.
The probability of zero customers in the system is
P0 = 1 – r
The expected number of customers in the queue is
Lq = [l2s2 + r2] / [2(1 – r)]
The expected number of customers in the system is
L = Lq + r
The expected waiting time in the queue is
Wq = Lq / l
The expected waiting time in the system is
W = Wq + 1/m

38. The Values of s and Lq for the M/G/1 Model with Various Service-Time Distributions
Mean s
Model Lq
Exponential
1/m
M/M/1
r2 / (1 – r)
Degenerate (constant)
M/D/1
(1/2) [r2 / (1 – r)]
Erlang, with shape parameter k
(1/k) (1/m)
M/Ek/1
(k+1)/(2k) [r2 / (1 – r)]
Table The values of s and Lq for the M/G/1 model with various service-time distributions.

39. VP for Engineering Approach (New Equipment)
The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq ≤ 1/4 day).
The Vice President for Engineering has suggested providing tech reps with new state-of-the-art equipment that would reduce the time required for the longer repairs.
After gathering more information, they estimate the new equipment would have the following effect on the service-time distribution:
Decrease the mean from 1/4 day to 1/5 day.
Decrease the standard deviation from 1/4 day to 1/10 day.

40. M/G/1 Model for the VP of Engineering Approach (New Equipment)
Figure This Excel template for the M/G/1 model shows the results from applying this model to the approach suggested by the Dupit’s vice president for engineering to use state-of-the-art equipment.

41. The M/M/s Model Assumptions
Interarrival times have an exponential distribution with a mean of 1/l.
Service times have an exponential distribution with a mean of 1/m.
Any number of servers (denoted by s).
With multiple servers, the formula for the utilization factor becomes r = l / sm but still represents that average fraction of time that individual servers are busy.

42. Values of L for the M/M/s Model for Various Values of s
Figure Values of L for the M/M/s model for various values of s, the number of servers.

43. CFO Suggested Approach (Combine Into Teams)
The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq ≤ 1/4 day).
The Chief Financial Officer has suggested combining the current one-person tech rep territories into larger territories that would be served jointly by multiple tech reps.
A territory with two tech reps:
Number of machines = 300 (versus 150 before)
Mean arrival rate = l = 6 (versus l = 3 before)
Mean service rate = m = 4 (as before)
Number of servers = s = 2 (versus s = 1 before)
Utilization factor = r = l/sm = 0.75 (as before)

44. M/M/s Model for the CFO’s Suggested Approach (Combine Into Teams of Two)
Figure This Excel template for the M/M/s model shows the results from applying this model to the approach suggested by Dupit’s chief financial officer with two tech reps assigned to each territory.

45. M/M/s Model for the CFO’s Suggested Approach (Combine Into Teams of Three)
Figure This Excel template modifies the results in Figure 11.8 by assigning three tech reps to each territory.

46. Comparison of Wq with Territories of Different Sizes
Number of Tech Reps
Number of Machines l m s r Wq 1
150 3 4
0.75
0.75 workday (6 hours) 2
300 6
0.321 workday (2.57 hours)
450 9
0.189 workday (1.51 hours)
Table Comparison of Wq values with territories of different sizes for the Dupit problem.

47. Values of L for the M/D/s Model for Various Values of s
Figure Values of L for the M/D/s model for various values of s, the number of servers.

48. Priority Queueing Models
General Assumptions:
There are two or more categories of customers. Each category is assigned to a priority class. Customers in priority class 1 are given priority over customers in priority class 2. Priority class 2 has priority over priority class 3, etc.
After deferring to higher priority customers, the customers within each priority class are served on a first-come-fist-served basis.
Two types of priorities
Nonpreemptive priorities: Once a server has begun serving a customer, the service must be completed (even if a higher priority customer arrives). However, once service is completed, priorities are applied to select the next one to begin service.
Preemptive priorities: The lowest priority customer being served is preempted (ejected back into the queue) whenever a higher priority customer enters the queueing system.

49. Preemptive Priorities Queueing Model
Additional Assumptions
Preemptive priorities are used as previously described.
For priority class i (i = 1, 2, … , n), the interarrival times of the customers in that class have an exponential distribution with a mean of 1/li.
All service times have an exponential distribution with a mean of 1/m, regardless of the priority class involved.
The queueing system has a single server.
The utilization factor for the server is
r = (l1 + l2 + … + ln) / m

50. Nonpreemptive Priorities Queueing Model
Additional Assumptions
Nonpreemptive priorities are used as previously described.
For priority class i (i = 1, 2, … , n), the interarrival times of the customers in that class have an exponential distribution with a mean of 1/li.
All service times have an exponential distribution with a mean of 1/m, regardless of the priority class involved.
The queueing system can have any number of servers.
The utilization factor for the servers is
r = (l1 + l2 + … + ln) / sm

51. VP of Marketing Approach (Priority for New Copiers)
The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq ≤ 1/4 day).
The Vice President of Marketing has proposed giving the printer-copiers priority over other machines for receiving service. The rationale for this proposal is that the printer-copier performs so many vital functions that its owners cannot tolerate being without it as long as other machines.
The mean arrival rates for the two classes of copiers are
l1 = 1 customer (printer-copier) per workday (now)
l2 = 2 customers (other machines) per workday (now)
The proportion of printer-copiers is expected to increase, so in a couple years
l1 = 1.5 customers (printer-copiers) per workday (later)
l2 = 1.5 customers (other machines) per workday (later)

52. Nonpreemptive Priorities Model for VP of Marketing’s Approach (Current Arrival Rates)
Figure This Excel template applies the nonpreemptive priorities queueing model to the Dupit problem now under the approach suggested by the vice president for marketing to give priority to the printer-copiers.

53. Nonpreemptive Priorities Model for VP of Marketing’s Approach (Future Arrival Rates)
Figure The modification of Figure that applies the same model to the later version of the Dupit problem.

54. Some Insights About Designing Queueing Systems
When designing a single-server queueing system, beware that giving a relatively high utilization factor (workload) to the server provides surprisingly poor performance for the system.
Decreasing the variability of service times (without any change in the mean) improves the performance of a queueing system substantially.
Multiple-server queueing systems can perform satisfactorily with somewhat higher utilization factors than can single-server queueing systems. For example, pooling servers by combining separate single-server queueing systems into one multiple-server queueing system greatly improves the measures of performance.
Applying priorities when selecting customers to begin service can greatly improve the measures of performance for high-priority customers.

55. Total Cost
The goal is to minimize total cost = cost of servers + cost of waiting
The cost of customers waiting is often difficult to determine (except when customers are internal to the company). Sometimes a surrogate goal is used again (e.g., see Example #2).
Another approach to reduce the cost of waiting (besides reducing the actual waiting time) is to try to make the waiting more pleasant: e.g., musak in elevators, entertainment in line at Disneyland.

56. Effect of High-Utilization Factors (Insight 1)
Figure This data table demonstrates Insight 1: When designing a single-server queueing system, beware that giving a relatively high utilization factor (workload) to the server provides surprisingly poor performance for the system.

57. Economic Analysis of the Number of Servers to Provide
In many cases, the consequences of making customers wait can be expressed as a waiting cost.
The manager is interested in minimizing the total cost. TC = Expected total cost per unit time SC = Expected service cost per unit time WC = Expected waiting cost per unit time The objective is then to choose the number of servers so as to Minimize TC = SC + WC
When each server costs the same (Cs = cost of server per unit time), SC = Cs s
When the waiting cost is proportional to the amount of waiting (Cw = waiting cost per unit time for each customer), WC = Cw L

58. Acme Machine Shop
The Acme Machine Shop has a tool crib for storing tool required by shop mechanics.
Two clerks run the tool crib.
The estimates of the mean arrival rate l and the mean service rate (per server) m are l = 120 customers per hour m = 80 customers per hour
The total cost to the company of each tool crib clerk is $20/hour, so Cs = $20.
While mechanics are busy, their value to Acme is $48/hour, so Cw = $48.
Choose s so as to Minimize TC = $20s + $48L.

59. Excel Template for Choosing the Number of Servers
Figure This Excel template for using economic analysis to choose the number of servers with the M/M/s model is applied here to the Acme Machine Shop example with s = 3.

60. Model 1 (M/M/1)
Customers arrive to a small-town post office at an average rate of 10 per hour (Poisson distribution). There is only one postal employee on duty and he can serve customers in an average of 5 minutes (exponential distribution).
l = 10 customers per hour
m = (60 min/hr) / (5 min per service) = 12 per hour

61. Model 2 (M/G/1)
ABC Car Wash is an automated car wash. Each customer deposits four quarters in a coin slot, drives the car into the auto-washer, and waits while the car is automatically washed. Cars arrive at an average rate of 20 cars per hour (Poisson). The service time is exactly 2 minutes.
Note that this is actually an M/D/1 model (a special case of M/G/1)
l = 20 cars per hour
1/m = (2 min) / (60 min/hr) = 1/30 hours
s = 0

62. Model 3 (M/M/s)
A grocery store has three registers open. Customers arrive to check out at an average of 1 per minute (Poisson). The service time averages 2 minutes (exponential).
l = 1 customer per minute
m = (1/2) customers per minute
s = 3

63. One Fast Server or Many Slow Servers
A McDonalds is considering changing the way that they serve customers.
Customers arrive at an average rate of 50 per hour.
Current System: For most of the day (all but their lunch hour), they have three registers open. Each cashier takes the customer’s order, collects the money, and then gets the burgers and pours the drinks. This takes an average of 3 minutes per customer (exponential distribution).
Proposed System: They are considering having just one cash register. While one person takes the order and collects the money, another will pour the drinks, and another will get the burgers (like Wendys). The three together think they can serve a customer in an average of 1 minute.
Question: Should they switch to the proposed system?
Current System (McDonalds)
l = 50 customers per hour
m = (60 min/hour) / (3 min/customer) = 20 customers per hour
s = 3
(Note r = 50/60)
Proposed System
m = 60 customers per hour
s = 1
(Note r = 50/60).

64. 3 Slow Servers (McDonalds)
1 Fast Server (Wendys)
Lq and Wq are smaller with the current system (McDonalds).
L and W are smaller with the proposed system (Wendys).
Discuss: Which measures of performance are more important?
As for pure efficiency, L and W are more important. The proposed system is better. Why? It is more efficient because all servers are fully utilized when there are only one or two customers. The current system has idle servers in this situation.

65. LL Bean LL Bean’s mail order business
Mail order phone lines open 24 hours per day, 365 days per year
78,000 calls per week (average)
Seasonal variations as well as variability during each day
How LL Bean estimates the number of servers needed
Each of the week’s 168 hours in a week is modeled separately as a period to be staffed
Each hour modeled as an M/M/s queue
Arrival rates and service rates estimated from historical data
Service standard: no more than 15% of calls wait more than 20 seconds
Full-time, part-time, and temporary workers scheduled to meet service standard