1. Management of Waiting Lines
Stevenson 18
Management of Waiting Lines

2. Learning Objectives Explain why waiting lines form in systems.
Implications of waiting line
Goal of waiting line management
Characteristics of waiting line
Measures of waiting line performance
Queuing models – Infinite Source and Finite Source
Constraint management
Psychology of waiting
Operations strategy

3. Why do waiting lines form?
Waiting lines occur when there is a temporary imbalance between supply (capacity) and demand.
Waiting lines add to the cost of operation and they reflect negatively on customer service,
It is important to balance the cost of having customers wait with the cost of providing service capacity.
From a managerial perspective, the key is to determine the balance that will provide an adequate level of service at a reasonable cost.

4. Waiting lines are non-value added occurrences
Disney World
Waiting in lines does not add enjoyment
Waiting in lines does not generate revenue
Waiting lines are non-value added occurrences

5. Waiting lines
Waiting Time: Operators and machines waiting for parts or work to arrive from suppliers or other operations. Customers waiting in line.
One of the “seven wastes”

6. Waiting Lines
Queuing theory: Mathematical approach to the analysis of waiting lines.
Goal of queuing analysis is to minimize the sum of two costs
Customer waiting costs
Service capacity costs

7. Implications of Waiting Lines
Cost to provide waiting space
Loss of business
Customers leaving
Customers refusing to wait
Loss of goodwill
Reduction in customer satisfaction
Congestion may disrupt other business operations

8. Goal of waiting line management
Minimize the sum of two costs: customer waiting costs and service capacity costs
Total
cost
Customer
waiting cost
Capacity
cost = +
Total cost
Cost
Cost of service capacity
Cost of customers
waiting
Service capacity
Optimum

9. System Characteristics
Population Source
Infinite source: customer arrivals are unrestricted
Finite source: number of potential customers is limited
Number of Servers (channels)
Arrival and service patterns
Queue discipline (order of service)

10. System Characteristics: Number of servers/phases
Multiple channel
Multiple phase
Channel: A server in a service system

11. System Characteristics: Arrival and Service Patterns
The Poisson distribution often provides a reasonably good description of customer arrivals per unit of time (e.g., per hour).
A discrete probability distribution
probability of a given number of events occurring in a fixed interval of time
events occur with a known average rate and independently.
The expected value and variance of (random) variable is equal to λ
P(r) = (e-λ λr ) / r! where
r = arrivals/time unit and
λ = mean arrivals/time unit
f(t) = μe – μt
where t = service time and
μ = mean service time
m = the average number of customers who can be served per time period.
Mean service time = 1/m

12. System Characteristics: Queue Discipline
First-come, first-served
Priority
Preferred (loyalty programs/fee-based)
Reservation (appointment)

13. Waiting line Models Patient Reneging Jockeying Balking
Customers enter the waiting line and remain until served
Reneging
Waiting customers grow impatient and leave the line
Jockeying
Customers may switch to another line
Balking
Upon arriving, decide the line is too long and decide not to enter the line

14. Waiting Line Performance
Waiting line performance relates to potential customer dissatisfaction and cost:
The average number of customers, either in line or in the system
The average time the customer waits, either in line or in the system
System utilization
The implied cost of a given level of capacity and its related waiting line
The probability that an arrival would have to wait

15. Waiting Time vs. Utilization
System utilization reflects the extent to which servers are busy
100% utilization is not a realistic goal
System Utilization
Average number on time waiting in line
100%

16. Queuing Models: Infinite-Source
Single channel, exponential service time
Single channel, constant service time
Multiple channel, exponential service time
Multiple priority service, exponential service time
All of the above assume Poisson arrival rate, and average arrival and service rates are steady (i.e., steady state)

17. Basic Relationships of Infinite Source Queuing Model

18. Infinite Source

19. Infinite source
Lq is a key value, and generally one of the first values we should calculate
Equation will be different for M/M/1, M/D/1 models, M/M/S models
Little’s Law applied in Waiting Line:
Number of People Waiting in Line = (Average Customer Arrival Rate) x (Average Time in the System)

20. LEGENDS
M/M/1 – Poisson Arrival Rate (the first M), Poisson (or exponential) service rate (the second M), and 1 server (the 1)
M/D/1 – Poisson Arrival Rate (the M), deterministic (constant) service rate (the D), and 1 server (the 1)
M/M/S - Poisson Arrival Rate (the first M), Poisson (or exponential) service rate (the second M), and multiple servers (the S)

21. Example: Infinite Source
Customers arrive at a bakery at an average rate of 18 per hour on weekday mornings. The arrival distribution can be described by a Poisson distribution with a mean of 18. Each clerk can serve a customer in an average of three minutes; this time can be described by an exponential distribution with a mean of 3.0 minutes.
What are the arrival and service rates?
Compute the average number of customers being served at any time.
Suppose it has been determined that the average number of customers waiting in line is 8.1. Compute the average number of customers in the system (i.e., waiting in line or being served), the average time customers wait in line, and the average time in the system.
Determine the system utilization for M = 1, 2, and 3 servers.

22. Example: Infinite Source
λ=18 customers per hour
We need to change the service time to comparable hourly rate. 60 minutes per hour, and 3 minutes per customer means 20 customers per hour. μ = 20 customers/per hour

23. Different models within Infinite Source
Single Server, Exponential Service Time, M/M/1
The queue discipline is first-come, first-served, and it is assumed that the customer arrival rate can be approximated by a Poisson distribution and service time by a negative exponential distribution. There is no limit on length of queue

24. M/M/1 Example
An airline is planning to open a satellite ticket desk in a new shopping plaza, staffed by one ticket agent. It is estimated that requests for tickets and information will average 15 per hour, and requests will have a Poisson distribution. Service time is assumed to be exponentially distributed. Previous experience with similar satellite operations suggests that mean service time should average about three minutes per request. Determine each of the following:
a. System utilization.
b. Percentage of time the server (agent) will be idle.
c. The expected number of customers waiting to be served.
d. The average time customers will spend in the system.
e. The probability of zero customers in the system and the probability of four customers in the system.

25. M/M/1 Example

26. Different models within Infinite Source
Single Server, Constant Service Time, M/D/1
Waiting lines are a consequence of random, highly variable arrival and service rates. If a system can reduce or eliminate the variability of either or both, it can shorten waiting lines noticeably. A case in point is a system with constant service time. The effect of a constant service time is to cut in half the average number of customers waiting in line
The average time customers spend waiting in line is also cut in half. Similar improvements can be realized by smoothing arrival times (e.g., by use of appointments).

27. M/D/1 Example
Wanda's Car Wash & Dry is an automatic, five-minute operation with a single bay. On a typical Saturday morning, cars arrive at a mean rate of eight per hour, with arrivals tending to follow a Poisson distribution. Find
a. The average number of cars in line.
b. The average time cars spend in line and service

28. M/D/1 Example

29. Different models within Infinite Source
Multiple Servers, M/M/S
Two or more servers are working independently. Use of the model involves the following assumptions:
1. A Poisson arrival rate and exponential service time.
2. Servers all work at the same average rate.
3. Customers form a single waiting line (in order to maintain first-come, first-served processing).
The multiple-server formulas are more complex than the single-server formulas, especially the formulas for Lq and P0..

30. M/M/S Equations

31. Queuing Model: Finite Source
Calling population is limited to a relatively small
One person may be responsible for handling breakdowns on 15 machines
There may be more than one server or channel
Arrival rates are required to be Poisson and service times exponential.
arrival rate of customers here is affected by the length of the waiting line
the arrival rate decreases as the length of the line increases

32. Finite Source Queuing Formulas and Notations

33. Constraint Management
Managers may be able to reduce waiting times by actively managing one or more system constraints.
Use temporary workers
Shift demand
Standardize the service
Look for a bottleneck

34. Psychology of waiting Occupied time feels shorter than unoccupied time
People want to get started
Anxiety makes waits seem longer
Uncertain waits are longer than known, finite waits
Unexplained waits are longer than explained waits
Unfair waits are longer than equitable waits
The more valuable the service, the longer the customer will wait
Solo waits feel longer than group waits
Source: David H. Maister, “The Psychology of Waiting Lines,” davidmaister.com, blog, September 8, 2008

35. Two key items to remember
Waiting lines forms because arrival rates and services rates are different. In other words, there is mismatch between supply and demand, or waiting lines happen due to the variability in arrival and service rates.
Increase in system utilization results in an exponential increase in customer waiting time.

36. Disney’s approach to managing waiting lines
Provide distractions
Provide alternatives for those willing to pay a premium
Keep customers informed
Exceed expectations
Comfortable waiting environment and other distractions
A form of reservations (Fast Pass)

37. Operations Strategy
Carefully assess the costs and benefits of various alternatives for capacity of service systems.
Increase the processing rate vs. number of servers.
New processing equipment/methods
Standardization (reduce variability in processing)
Shift some arrivals to “off-times” by using reservations systems
“early-bird” specials
senior discounts