Managing Waiting Lines NKFUST The Economies of Waiting Features - - PDF document

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Managing Waiting Lines

• The Economies of Waiting
• Features of Queuing Systems
• Waiting Time Formula
• Waiting Line Management

Shin‐Ming Guo NKFUST

Where the Time Goes

In a life time, the average person will spend‐‐ SIX MONTHS Waiting at stoplights EIGHT MONTHS Opening junk mail ONE YEAR Looking for misplaced 0bjects TWO YEARS Unsuccessfully returning phone calls FOUR YEARS Doing housework FIVE YEARS Waiting in line SIX YEARS Eating

Japanese know waiting

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Waiting Realities

• Inevitability of Waiting: Waiting results from

variations in arrival times and service times

• Economics of Waiting: High server utilization at the

expense of customer waiting. Make waiting productive (salad bar) or profitable (drinking bar).

Total Cost minimum

The Economies of Waiting

4

Cost Cost of Waiting Cost of Capacity Service Capacity Large Small Total cost per hour = Cost of capacity per hour + Cost of Waiting Time

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• I. The Operation of a Typical Call Center

5

Blocked calls (busy signal) Abandoned calls (tired of waiting) Calls

• n Hold

Sales reps processing calls Answered Calls Incoming calls

Call center

Lost throughput Holding cost Lost goodwill Lost throughput (abandoned) $$$ Revenue $$$ Cost of capacity Cost per customer

• At peak, 80% of calls

dialed received a busy signal.

• Customers getting

through had to wait on average 10 min. .

A “Perfect or Somewhat Odd” Call Center

6

7:00 7:10 7:20 7:30 7:40 7:50 8:00

Caller Arrival Time Service Time

1 2 3 4 5 6 7 8 9 10 11 12 5 10 15 20 25 30 35 40 45 50 55 4 4 4 4 4 4 4 4 4 4 4 4

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A More Realistic Service Process

7 Caller Arrival Time Service Time

1 2 3 4 5 6 7 8 9 10 11 12 7 9 12 18 22 25 30 36 45 51 55 5 6 7 6 5 2 4 3 4 2 2 3 Time 7:10 7:20 7:30 7:40 7:50 8:00 7:00

caller 1 caller 3 caller 5 caller 7 caller 9 caller 11 caller 2 caller 4 caller 6 caller 8 caller 10 caller 12 1 2 3 2 min. 3 min. 4 min. 5 min. 6 min. 7 min.

Service times Number of cases

Variability Leads to Waiting Time

8

7:00 7:10 7:20 7:30 7:40 7:50

Inventory (callers on line)

5 4 3 2 1

8:00

7:00 7:10 7:20 7:30 7:40 7:50 8:00

Wait time Service time

Caller Arrival Time Service Time

1 2 3 4 5 6 7 8 9 10 11 12 7 9 12 18 22 25 30 36 45 51 55 5 6 7 6 5 2 4 3 4 2 2 3

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Ignoring Variability Leads to Over-Confidence

9

A: You cannot inventory services — Capacity can never run ahead of demand. Random arrivals and varying demands are common in services. In the presence of variability,

• ne cannot estimate the process

performance based on averages. Q: Why does variability not average out over time?

Variability: Where does it come from?

10

Input

• Unpredicted

Volume swings

• Random arrivals

(randomness is the rule, not the exception)

• Incoming quality
• Product Mix

Resources

• Breakdowns / Maintenance
• Operator absence
• Set‐up times

Tasks

• Inherent variation
• Lack of SOPs
• Quality (scrap / rework)

Buffer Processing

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• II. Essential Features of Queuing Systems

11

Departure Queue discipline Arrival process Queue configuration Service process Renege Balk Calling population No future need for service

Calling Population

12

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Customer Arrival Process

Static Dynamic Appointments Price Accept/Reject Balking Reneging Random arrivals with constant rate

Random arrival rate varying with time Random arrival rate varying with time

Facility‐ controlled Customer‐ exercised control

Arrival process Arrival process

Analyzing an Arrival Process

14

Call Arrival Time, ATi

1 2 3 4 5 6 7 6:00:29 6:00:52 6:02:16 6:02:50 6:05:14 6:05:50 6:06:28

Interarrival Time IAi=ATi+1 -ATi

00:23 01:24 00:34 02:24 00:36 00:38

Time 6:01 6:02 6:03 6:04 6:05 6:06 6:00

Call 1 Call 2 Call 3 Call 4 Call 5 Call 6 Call 7 IA1 IA2 IA3 IA4 IA5 IA6

mean deviation standard Variation

• f

t Coefficien

2

 

a

CV

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Distribution of Caller Inter-arrival Times

Random (Poisson) arrivals

 Customers arriving independently from each

• ther follow exponential inter‐arrival times.

t

e t f







 ) (

1/ = average inter‐arrival time = arrival rate

1

2  a

CV

Temporal Variation in Arrival Rates

An arrival process is not stationary if the average number of arrivals in any given time interval is not fixed over the entire time period.

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Queue Configuration Multiple Queues vs. Single Queue

18

Multiple Queues Take a Number

3 4 8 2 6 10 12 11 5 7 9 Enter

Single queue

?

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Queue Discipline

Queue discipline Static (FCFS rule)

Dynamic selection based on status

• f queue

selection based on status

• f queue

Selection based

• n individual

customer attributes Selection based

• n individual

customer attributes

Number of customers waiting Round robin Priority Preemptive Processing time

• f customers

(SPT rule)

Priority Rules in Waiting Time Systems

20

A: 9 minutes B: 10 minutes C: 4 minutes D: 8 minutes

D A C B 9 min. 19 min. 23 min.

Total wait time: 9+19+23=51min

B D A C 4 min. 13 min. 21 min.

Total wait time: 4+13+21=38 min

 First‐Come‐First‐Serve: easy to implement + perceived fairness  Shortest Processing Time Rule: Minimizes average waiting time  Sequence based on importance: emergency or profitable customers

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Service Process and Customer Involvement

21

Histograms of Service Times

22

1/μ = mean service time μ = service rate   1

2  s

CV

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• III. Single Server Model: M/G/1
• 1. Mean arrival rate:
• 2. Mean service rate:
• 3. Mean number in service (utilization):
• 4. Mean number of customers in queue:
• 5. Mean number of customers in the system:
• 6. Mean time in system:
• 7. Mean time in queue:





   

s s

L W  1    1 1   

s q q

W L W ) 1 ( 2

2 2 2

      

q

L   

q s

L L

long time averages

Reducing Variability to Reduce Waiting

Reduce Arrival Variability

• appointment/reservation: how to handle late arrivals or

no‐shows

• encourage customers to avoid peak hours.

Reduce Service Time Variability

• training and technology
• limit service selection
• reduce customer involvement

24

Service time factor Utilization factor Variability factor

2 1 1

2 2 s a q

CV CV W        

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Multiple, Parallel Resources with One Queue

25

arrival departure number waiting Lq Entry to system Departure Begin Service Time in queue Wq Service Time 1/μ Time in system Ws=Wq + 1/μ number in service

Waiting Time for Multiple, Parallel Resources

26

                          

 

2 1 time service mean queue in Time

2 2 1 ) 1 ( 2 s a m

CV CV n utilizatio n utilizatio m

Under the assumption that we approximate the average waiting time as

1       m rate service rate arrival n Utilizatio

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The Power of Pooling

27

Independent Resources 2x(m=1) Pooled Resources (m=2)

0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 60% 65%

m=1 m=2 m=5 m=10

70% 75% 80% 85% 90% 95%

Waiting Time Wq Utilization

Pooling benefits are lower if queues are not truly independent

Staffing over the Course of a Day

28 20 40 60 80 100 120 140 160 0:15 2:00 3:45 5:30 7:15 9:00 10:45 12:30 14:15 16:00 17:45 19:30 21:15 23:00 20 40 60 80 100 120 140 160 0:15 2:00 3:45 5:30 7:15 9:00 10:45 12:30 14:15 16:00 17:45 19:30 21:15 23:00

Time Number of customers Per 15 minutes Number of CSRs

17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

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More on Reducing Wait Times

29

 Appointment (no show?)  Segment customers (pay more)  Encourage customers to come during

the slack periods

 Provide limited service or self‐service  Provide 24x7 service or internet access  Flexible servers and flexible work hours  Technology: bar‐code, OCR, RFID (ETC),

Fast Lane, customer database

• IV. Waiting Times & Customer Satisfaction

30

Reducing average waiting time does not guarantee customer satisfaction. A small percentage of customers may experience long waits and complain bitterly. Solution: service guarantee and/or service recovery

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Service Levels in Waiting Systems

31

Target Wait Time (TWT) depends on your market position and the importance of incoming calls for your business Service Level = Probability{Waiting TimeTWT}

0.2 0.4 0.6 0.8 1 50 100 150 200 Waiting time [seconds] Fraction of customers who have to wait x seconds or less

Waiting times for those customers who do not get served immediately Fraction of customers who get served without waiting at all 90% of calls had to wait 25 seconds or less

TWT

Psychology of Waiting

• That Old Empty Feeling: Unoccupied time goes slowly
• A Foot in the Door: Pre‐service waits seem longer that

in‐service waits

• The Light at the End of the Tunnel: Reduce anxiety with

attention

• Excuse Me, But I Was First: Social justice with FCFS

queue discipline

• They Also Serve, Who Sit and Wait: Avoids idle service

capacity

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Perception of Waiting

33

Perceived Wait Time

 Amount of time customers

believe they have waited prior to receiving service.

 Has a greater effect on

customer satisfaction than actual waiting time

t*

0.2 0.4 0.6 0.8 1

perceived wait

threshold sensitivity Satisfaction

Factors Affecting Perceived Wait Times

34

Customer‐Related Factors

 Solo versus group waits  Waits for more valuable

versus less valuable services

 Customer’s own tolerance

Server‐Related Factors

 Passive vs. active waits  Unfair vs. fair waits  Uncomfortable vs.

comfortable waits

 Unexplained vs. explained

waits

 Unproductive vs.

Productive waits

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Suggestions for Managing Queues

35

• 1. Determine an acceptable waiting time for your customers
• 2. Try to divert your customer’s attention
• 3. Inform your customers of what to expect
• 4. Keep idle employees out of sight
• 5. Segment customers (Din Tai Fung)
• 6. Train your servers to be friendly
• 7. Encourage customers to come during the slack periods
• 8. Take a long‐term perspective (redesign the system)

Summary

36  Variability is the norm, not the exception

‐ understand where it comes from and eliminate what you can

 Variability leads to waiting times although utilization<100%

Operations benefit from flexibility in capacity

 Demand can exhibit seasonality → Time varying capacity  Pooling resources can reduce waiting times

Managing customers’ perceived wait times