Barnacle Cleaners Case Study
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AD605LectureWeek084.pdf
AD 605: Operations Management
Week 8: Waiting Line Management
Boston University Metropolitan College
Spring 2019
Topics
● Motivation & Background – Capacity Buffering
● Waiting Lines – Poisson Arrivals, Exponential Service Times
● Queuing Models – M/M/s (Mathematical) Model
● The 85% Threshold – Implications for Service Operations Managers
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Capacity Buffering
● At times, determining capacity buffers is done using “waiting line” (or “queuing”) models or simulations. – especially true when customer arrivals and activity times are
subject to high levels of uncertainties, and when no inventory buffering exists.
● The approaches are typically used for the analysis of services or when the operating system consists of a complicated set of interconnected processes.
● Waiting line models tend to be used in single-process cases with high levels of uncertainties, and simulation models tend to be used in the case of interconnected processes.
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Modeling Queues
Arrivals Exit
Queue
Servers
Modeling is done using mathematics (simple queues) or simulation (complex queues).
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Describe Various Waiting Lines
Retail store (e.g., Walmart)
Call center (e.g., credit card, banking)
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Describe Various Waiting Lines
R&D organization (e.g., Gates Foundation)
IT support (e.g., in-company troubleshooting)
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Arrival Uncertainty
Examples: Repair, Emergencies,
Troubleshooting.
RANDOM Examples: Dentist, Maintenance, and
Reporting.
APPOINTMENTS
Random arrivals pose a special challenge.
Poisson Arrivals
● Customers arrive independently of one another.
● TBA follows an “exponential distribution.”
● Customers seem to arrive in bunches, but are independent.
● Average time until next arrival is constant.
● TBA for two-thirds of arrivals are less than the average.
Easily simulated with a six-sided die, where “6” represents an arrival. Time Between Arrivals (TBA)
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The Exponential Distribution
● Typically applies in waiting line modeling – only parameter is mean (average) – common in real life systems, especially for arrivals – easy to work with mathematically
● Has interesting intuitive interpretation – called memoryless property – e.g., chance of arrival in next instant independent of time
of previous arrival
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Memoryless property is easy to appreciate when applying to the roll of a 6-sided die.
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Exponential Distribution (Average = 6)
0 5 10 15 20 25 30 35 40
TBA
For example, the most likely next “6” when rolling a 6- sided die is the next roll.
Data are collected for the time between arrivals (in seconds) for cars at a turnpike toll booth:
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0 2 4 6 8
10 12 14 16
5 15 25 35 45 55 65 75
TBA (seconds)
Frequency The histogram shows that the TBA follows an exponential distribution.
In this case the average TBA is estimated to be 20 seconds. 12
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Service Time Uncertainty
Most service activities.
RANDOM Most manufacturing
activities.
STANDARD
Random service times pose a special challenge.
Service times often follow “skewed” distributions.
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Mathematical Queuing Models
● Models available for a variety of scenarios: – based on “assumptions”
● “Kendall notation” exists to describe nature of model- A/B/s/D/N/K, where:
A is the arrival time distribution B is the service time distribution s is the number of parallel servers D is the queue discipline N is the system capacity K is the size of the population
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M/M/s* Queue (Waiting Line) Model
Assumes Poisson arrivals & service to a single queue – allows for multiple servers – any queue discipline – assumes infinite customer population and size queues
Key input parameters: – is arrival rate (e.g., customers/hour or customers/min) – is the service rate (e.g., customers/hour or customers/min) – s is the number of servers
* “M” refers to “Markovian” (or memoryless) because knowledge of historical events not necessary to predict future events. In queuing theory, “M/M” refers to Poisson arrivals & exponential service times.
Note on Rates – Very Important
● Lambda () and mu () are expressed in number of arrivals or service completions per unit time.
● For example, based on the turnpike time between arrival data set, may be expressed as: – 0.05 arrivals per second, or – 3 arrivals per minute, or – 180 arrivals per hour.
● For example, if =4 there are an average of 4 customers served per unit time (e.g., seconds, minutes, hours).
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Consider the operation of a new walk-in financial center. The anticipated arrival rate is 14 customers per hour (Poisson arrivals) and the anticipated average service time is 3.6 minutes (exponentially distributed). Use a M/M/1 model to analyze the queuing system. We have:
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Important note: and are always expressed in customers per time unit (e.g., minutes, hours, days). In this case:
Key M/M/1 Formulae
fraction of time server is idle (this is equal to the capacity buffer)
fraction of time n customers are in the system
average waiting time in the queue
average number of customers in the queue
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𝑃 = 1 − 𝜆
𝜇 = 1 −
14
16.7 = 0.162
server is idle 16.2% of the time (i.e., 16% capacity buffer)
𝑃 = 1 − 𝜆
𝜇
𝜆
𝜇 = 1 −
14
16.7
14
16.7 = 0.136
one customer is in the system 13.6% of the time
𝑊 = 𝜆
𝜇 𝜇 − 𝜆 =
14
16.7 16.7 − 14 = 0.3105
average wait time in queue is 0.3105 hours
𝐿 = 𝜆
𝜇 𝜇 − 𝜆 =
14
16.7 16.7 − 14 = 4.4
average queue length is 4.4 customers
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Important note: and could also have been expressed as:
Of the key model results, only Wq is affected by this change of time units.
𝑊 = .
. . . =18.6
average wait time in queue is 18.6 minutes
Notes
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Analysis of Queuing Systems
● Analysis tradeoff: – cost of system (e.g., maximum queue size, number of servers,
complexity of queue), and – cost of waiting (opportunity costs, lost customers, etc.).
● Main analysis criteria: – average & variation of queue size, – idle time (capacity buffer) or utilization of servers, and/or – average & worst case waiting times.
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Queuing system can be analyzed with a mathematical model (simpler queues) or with a computer simulation (complex queues).
Consider the operation of a new walk-in financial center. The anticipated arrival rate is 14 customers per hour (Poisson arrivals) and the anticipated average service rate is 3.6 minutes (exponentially distributed). Use a M/M/1 model to analyze the queuing system: compare a one service option with a two server option.
Use Excel template for performing the analyses.
Note that, when using the template, the option exists to enter arrival and service rates as customers per hour or customers per minute (the next slides show inputs/outputs based on an hourly scale).
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25
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Important note: When and could also have been expressed in customers/min (0.233 & 0.278 respectively), the following values are obtained:
From the output, the most important values are:
The capacity buffer is 16.2% and the server utilization is 83.8%.
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When determining capacity (in this case, driven by the number of servers), a decision maker chooses between the following alternatives:
One server resulting in a 83.8% utilization, an average queue size of 4.35 and an average waiting time of 18.6 minutes, or
Two servers resulting in a 41.9% utilization, an average queue size of 0.18 and an average waiting time of 0.8 minutes.
This decision is difficult because server “cost” is well-known but the “cost” of customer waiting time is impossible to quantity precisely.
Exercise: Coffee Shop Considering a Drive-up Window
● Arrival rates vary according to time of day, and follow a Poisson process because customer arrive independently of one another.
● Demand patterns will change by drive-up window : – surveys show 30% current customers will use window, and – anticipated 20% new customers (window only).
● Current service time data are available.
● Service times will change with drive-up window: – increased in store by 25%; decreased at window by 50%.
● Currently 4 servers are on duty during the morning timeframe, although current waiting times are not known.
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Analysis Procedure
1. Analyze data on current arrival and service times and determine their distribution: – show that the M/M/s model is valid.
2. Use M/M/s model to determine the queue characteristics for the current morning period (w/4 servers). – follow-up with similar analysis to determine best number of
servers in current system during other time periods. 3. Use the M/M/s model to analyze system with window:
– queue characteristics for the window (s=1), and – queue characteristics & number of servers in store.
4. Make recommendation to either keep the current system or install the drive-up window.
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Arrival Times
● Current arrival rates vary according to time of day (we can assume that an analysis confirmed exponential arrivals because customers arrive independently of one another): – morning (7-10): 105 arrivals per hour, – mid-day (10-4): 90 arrivals per hour, and – late afternoon (4-10): 75 arrivals per hour.
● Recall that demand patterns expected to change from current: – surveys show 30% current customers will use window, and – anticipated 20% new customers (window only).
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Average arrival rates (per hour) are:
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Current Proposed
Timeframe In Store At Window In Store
Morning 105
Mid-Day 90 45* 63+
Late Afternoon 75
* (0.3×90)+(0.2× 90) + 0.7× 90
52.5
37.5
73.5
52.5
Current Service Times (60 Customers)
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Analysis of Service Times
0
5
10
15
20
25
30
35
0.7 2.1 3.5 4.9 6.3 7.7 9.1 10.5
Service Time (Minutes)
Histogram
Average service time was 2.0 minutes, distribution is exponential.
● Current service times changed by drive-up window: – increased in store by 25% (from 2 min/customer to 2.5
min/customer), and – decreases at drive-up window by 50% (from 2
min/customer to 1 min/customer).
● Average Service Rates (per hour):
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Current Proposed
Timeframe In Store At Window In Store
Morning 30
Mid-Day 30
Late Afternoon 30
Notes
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60
60
60
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24 24
Current System Analysis
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Determine server utilization and average wait time for peak (morning) period, with 4 servers. The average wait time will be the benchmark for all evaluations. Then determine the best number of servers for mid-day and late afternoon.
Timeframe Servers Utilization Avg Wait
Morning 4
Mid-Day
Mid-Day
Late Afternoon
Late Afternoon
Notes
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87.5 3
4 75 1
5 60 0.2 3 83.3 2.8
4 63 0.4
Future System Analysis (Window)
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Determine how well a one-server window will operate relative to average wait time.
Timeframe Servers Utilization Avg Wait
Morning 1
Mid-Day 1
Late Afternoon 1
Future System Analysis (In Store)
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Determine the number of servers required in the store for each timeframe that approximates average wait time in the current system.
Timeframe Servers Utilization Avg Wait
Morning
Morning
Mid-Day
Mid-Day
Late Afternoon
Late Afternoon
88% 7 75%
63%
3
1.7
4 77 1.4
3 83 5.2 4 0.766
3 73 1.6
Recommendation
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Summarize results and recommend whether or not to change the store’s configuration, including how many servers would be needed.
Utilization vs. Waiting Relationship
● When arrivals or service times are random (i.e., uncertain) a non-linear relation will exist between resource utilization and customer waiting time. – in these cases, once a certain threshold utilization is reached,
waiting times will increase dramatically.
● Therefore, when customers arrive according to an exponential distribution and service times are skewed, a certain amount of idleness should be planned: – 85% utilization is often used as an effective threshold.
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Example
● Consider a call center where during peak periods, customer calls arrive at a rate of 40 per hour: – studies have shown that the arrival distribution is
exponential.
● A manager wishes to use an M/M/s model to determine the number of servers necessary to maintain satisfactory customer service: – studies have shown that service time is exponential,
averaging 15 minutes per customer.
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Analysis Results
S=11
S=12
S=13 S=14S=15S=16S=18S=20
0
2
4
6
8
10
12
40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95%
Server Utilization (Capacity Buffer)
W ai
ti ng
T im
e (M
in ut
es )
S = # Servers
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(60%) (55%) (50%) (45%) (40%) (35%) (30%) (25%) (20%) (15%) (10%) (5%)
85% Rule: Management Implications
15% idle time goes against management inclination to avoid
idle resources.
Exceeding 85% resource
utilization leads to long queues.
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BUT
Notes
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Case Study
Motivated by an increase in Emergency Department (ED) patient wait times at a hospital serving 33,000 patients annually.
47Overall Average TAT (Turnaround Time) 6 Hours
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ED Flow & Arrival/Service Information
Arrival DischargeTriage Register Treatments
Physician/BedNurse Clerk
6.5 9.0
4/hr
24.0^
135.0# Service Times*
Bed
# Known to include non-value-added activities
* Average service time are shown – all service times followed a gamma distribution with standard deviation equal to 50% of its mean value.
Due to the system’s complexity, a Monte Carlo simulation was developed to model the queues; it was written in Excel.
^ Physician Service Time
Queue Analysis: Current State
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Hospital personnel considered the service time for discharge to be especially inefficient (i.e., too long).
Current State Result: Unstable System w/Long TAT
0
100
200
300
400
500
600
700
800
TA T
(M in
)
Patient Number (1-1000)
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Key Results: Average ED TAT: 373 Minutes (6 hour 13 min) Average Bed Queue Time: 174 Minutes (2 hour 54 min)
Queue Analysis: Target State
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Simulation used to see if reduction in discharge service time would improve system (and how much reduction was needed).
Target State Result: Stable System w/Reduced TAT
0
100
200
300
400
500
600
Patient Number (1-1000)
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Key Results: Average ED TAT: 225 Minutes (3 hour 45 min) Average Bed Queue Time: 42 Minutes
● When utilizations exceeded a threshold (about 85%), two problems occurred: – queues get long on average, and – queues become erratic and unstable.
● Action plan would include studying discharge service time to remove wasteful activities, lowering bed utilization to about 80%: – use of Lean approaches.
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A service time decrease of 15 minutes resulted in approximately 2.5 hour decrease in turnaround time (373 min to 225 min).
Scenario Discharge Svc Time Bed Utilization Average TAT Average Bed Queue
Current State 135 Min 88% 373 Min 174 Min
Target State 120 Min 80% 225 Min 42 Min
Notes
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The Waiting Time Analysis Challenge......
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It is difficult to quantify the financial impact of customer wait times (e.g., differences between customers’ perception of the wait could cause some customers to be unhappy with a certain wait time and others satisfied).
Waiting in line can also seem longer for some customers when certain factors are present, such as uncertainty (of the wait duration), unfairness (if others appears to be given higher priority or the queue is disorganized), and boredom (because waiting feels longer for customers without reading or listening material).
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In conjunction with configuring queues to have reasonable wait times: 1. Keep customers informed regarding the anticipated wait time. 2. Divert customer attention by creating diversions that entertain or
educate customers. 3. Keep non-serving employees out of sight so that it does not
appear that the firm is indifferent to line of waiting customers. 4. Encourage demand during non-peak hours, which is common in
restaurants and hotels. 5. Segment customers so that customers with “common” or short
duration service needs will not wait with other customers (such as the express line in a grocery store).
The Psychology of Queues
