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QMD: Waiting-line
analysis
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Overview
Terminology
Characteristics of Waiting-lines
Operating characteristics
M/M/1 model – single channel•Example: video store
M/M/S model – multi-channel
•Example: Multiplex theater
Psychology of waiting
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‘The other linealways moves faster.’
‘If you change lines,the one you left willstart to move faster
than the one you’rein.’
© 1995 Corel Corp.
Thank you for holding.
Hello...are you there?
You’ve Been There Before!
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Service
FacilityWaiting Line
Waiting Line System
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Situation Arrivals Servers Service Process
Waiting Line Examples
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Queue: Waiting line Arrival: 1 person, machine, part,
etc. that arrives and demands
service
Queue discipline: Rules for
determining the order that arrivals
receive service
Channel: Number of servers
Phase: Number of steps in service
Waiting Line Terminology
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Behavior of the arrivals• join the queue, and wait
until served
• No balking; refuse to jointhe line
• No reneging; leave theline
Service
FacilityWaiting Line
Pattern of arrivalsrandom
scheduled
Arrival Characteristics of aWaiting Line System
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Input
source Service
facility
Waiting
line
Service system
© 1995 Corel Corp.
Line was
too long!
Balking
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Reneging
Input
source Service
facility
Waiting
line
Service system
© 1995 Corel Corp.
I give up!
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Line Characteristics of aWaiting Line System - continued
Length of the
queue limited unlimited
Service priority
FIFO other
Service
FacilityWaiting Line
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Waiting Line Arrangements
Service facilities
(a) Single line
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Waiting Line Arrangements
Service facilities
(a) Single line Service facilities
(b) Multiple lines
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Waiting Line Arrangements
Service facilities
(a) Single line Service facilities
(b) Multiple lines
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Number of channels single multiple
Number of phases in service system single
multiple
Service Facility Characteristics of aWaiting Line System - continued
Service
FacilityWaiting Line
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Service Facility Arrangements
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Service Facility Arrangements
Servicefacility
(a) Single channel, single phase
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Arrivals
Served
unitsService
facility
Queue
Service system
DockWaiting ship line
Ships at
sea
Ship unloading system Empty
ships
Single-Channel, Single-Phase System
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Service Facility Arrangements
(b) Single channel, multiple phase
Servicefacility 1
Servicefacility 2
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Cars
& food
Single-Channel, Multi-Phase System
Arrivals
Served
unitsService
facility
Queue
Service system
Pick-upWaiting cars
Cars
in area
McDonald’s drive-through
Pay
Service
facility
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Service Facility Arrangements
(c) Multiple channel, single phase
Servicefacility 1
Servicefacility 2
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Arrivals
Served
unitsService
facilityQueue
Service system
Service
facility
Example: Bank customers wait in single line for one of
several tellers.
Multi-Channel, Single Phase System
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Decision Areas
Arrival rates
Number of service
channels Number of phases
Service time
Priority rule
Line arrangement
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Average queue length, Lq
Average number in system, Ls
Average queue time, W q
Average time in system, W s
System utilization,
Performance Measuresor
Operating Characteristics
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Simple (M/M/1) •Example: Information booth atmall
Multi-channel (M/M/S)
•Example: Airline ticket counter
Types of Queuing Models
Constant Service (M/D/1)
Example: Automated car wash
Limited Population Example: Department with only 7 drills
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Type: Single-channel, single-phase system
Queue: Unlimited; single line
Queue discipline: FIFO (FCFS)
Service rate > arrival rate
Simple (M/M/1) ModelCharacteristics
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Queuing Models
- ‘lambda’
•arrival rate of the customers
- ‘mu’
•servicerate
for a single server• If the service time is 15 minutes
per customer, what is the service
rate??
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Simple (M/M/1) Model Equations
s LAverage number of
customers in the
system
2
q LAverage numberof customers in the
queue
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Simple (M/M/1) Model Equations
1
sW
qW
Average time
customers are in
the system
Average time
customers are in
the queue
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Simple (M/M/1) Model Equations
Utilization factor –
percent of time
that the single
server is busy; also
the probability that
a customer will
wait
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Example: Video Store
The manager of a video store isinterested in providing good service.On a Friday or Saturday night, on
average 30 customers per hourarrive at the counter to check out avideo. The customers are served at
an average rate of 35 customers perhour from a single cash register.
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Example: Video Store
Determine the operatingcharacteristics for the video store.
• Average number of customers in line
• Average number of customers in thesystem
• Average wait time in line
• Average time in the system• Server utilization
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Average number of customers in line
2
q L
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Average number of customers in the system
s L
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Average wait time in line
qW
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Average time in the system
1
sW
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Server utilization
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Example: Video Store
•What service rate would berequired to have customers
average only 5 minutes in thesystem?
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What service rate would be required to havecustomers average only 5 minutes in the
system?
1
sW
GO TO EXCEL OM
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Type: Multichannel system
Queue discipline: FIFO(FCFS)
Service rates > arrivalrate
Multichannel (M/M/S) ModelCharacteristics
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Multichannel M/M/S Equations
Probability of zero
customers in the
system:
Average number of
customers in the
system:
S
S
Sn
PSn
S
n !
1
!
1
1
1
0
0
02
!1
PSS
L
S
s
Average number of
customers waiting for
service:
sq L L
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Multichannel M/M/S Equations
Average time a person or
unit spends in the queue
Average time a unit
spends in the system:
s
S
s
LP
SSW
1
!102
q
sq
LW W
1
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Example: Movie Theater
A multiplex movie theater has 3concession clerks serving customers on afirst come, first served basis. The servicetime per customer is exponentially
distributed with an average of 2 minutesper customer. Concession customersarrive at a rate of 81 customers per hour.10 minutes of previews run in the lobby.
If the average time in the concession areaexceed 10 minutes, customers becomedissatisfied.
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Example: Movie theater
What is the average number of customers in the concession area?
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What is the average number of customers in theconcession area?
GO TO EXCEL OM
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Multiplex Movie Theater
The distribution is reducing thelength of the previews to 8 minutes.How many servers will be required to
so that the average wait is no longerthan 8 minutes?
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Total Cost per Hours
Total cost per Hour= waiting cost +service cost
TC = CwL
s+ KC
s
• Cw – cost of waiting per hour per
customer
• Ls – number of customers in the system
• K – number of servers
• Cs- service cost per hour per server
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Total Cost: Video Store
Considering adding another cashierfor Friday and Saturday nights.
Part-time cashier will cost $6.00 per
hour
Customers waiting time (cost of
goodwill, etc) is $10.00 per hour.
Is it cost effective to hire the
cashier?
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Total Cost: Video Store M/M/1
TC = CwLs + KC
Cw = $10/hour
Ls = 6
K = 1
CS = $6.00/hr
TC = 10(6) +1(6)
= $66/hour
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Total Cost: Video Store M/M/2
TC = CwL + KC
Cw = $10/hour
Ls = 1.05
K = 2
Cs = $6.00/hr
TC = 10(1.05)+ 2(6)
= $22.50/hr
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= Mean number of arrivals per timeperiod
• e.g., 3 units/hour
= Mean number of people or items
served per timeperiod
• e.g., 4 units/hour
1/ = 15
minutes/unit
Remember: & Are Rates
© 1984-1994 T/Maker Co.
If average service time is 15
minutes, then μ is 4
customers/hour
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The Psychology of Waiting
Maister’s (1985) Propositions of
customer satisfaction in queues
•Unoccupied time feels longer thanoccupied time
•Pre-process wait feels longer than
in-process waits.•Anxiety makes waits feel longer
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The Psychology of Waiting
•Uncertain waits seem longer than
certain waits
•Unexplained waits seem longer
than explained waits
•Unfair waits seem longer thanequitable waits
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Psychology of Waiting
•More valuable the service, the
longer people will wait.
•Solo waiting feels longer than
group waiting.
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