D-1 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Operations Management...
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Transcript of D-1 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Operations Management...
D-1 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Operations Operations ManagementManagement
Waiting-Line ModelsWaiting-Line ModelsModule DModule D
D-2 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
You manage a call center which can answer an average of 20 calls per hour. Your call center gets 17.5 calls in an average hour. On average, what is the time a customer spends on hold waiting for service?
a) on average, customers should not have to wait on hold since capacity is greater than demand.
b) average customer wait will be less than 10 minutes. c) average customer wait will be between 10 and 20 minutes. d) average customer wait will be greater than 20 minutes. e) who knows? There’s no way to tell.
Queues / LinesQueues / Lines
© 1995 Corel Corp.
D-3 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Two ways to address waiting linesTwo ways to address waiting lines
Queuing theory Certain types of lines can be described mathematically Requires that assumptions are valid for your situation Systems with multiple lines that feed each other are
too complex for queuing theory Simulation
Building mathematical models that attempt to act like real operating systems
Real-world situations can be studied without imposing on the actual system
D-4 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Why do we have to wait?Why do we have to wait?
© 1995 Corel Corp.
Why do services (and most non-MTS manufacturers) have queues? Processing time and/or arrival time variance Costs of capacity – can we afford to always have more
people/servers than customers? Efficiency – e.g. scheduling at the Doctor’s office
D-5 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Waiting Costs and Service CostsWaiting Costs and Service Costs
Total expected cost
Cost of waiting time
Cost
Low level of service
Optimal service level
High levelof service
Minimum total cost
Cost of providing service
D-6 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Costs of QueuesCosts of Queues
Too Little Queue Too Much Queue
Cost of capacity
Wasted capacity
Annoyed customers
Lost customers
Space
Possible opportunity:e.g. wait in the bar for a restaurant table
D-7 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Bank Customers Teller Deposit, etc.
Doctor’s Patient Doctor Treatmentoffice
Traffic Cars Light Controlledintersection passage
Assembly line Parts Workers Assembly
1–800 software User call-ins Tech support Technical supportsupport personnel
Situation Arrivals Servers Service Process
Waiting Line ExamplesWaiting Line Examples
D-8 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Service FacilityPopulation
Pattern of arrivals Scheduled Random – estimated by Poisson distribution
Arrival Characteristics
Characteristics of a Waiting Line SystemCharacteristics of a Waiting Line System
Size of the source population Limited Unlimited
Behavior of arrivals Join the queue, wait until served Balk – refuse to join the line Renege – leave the line
Waiting Line
= average arrival rate
D-9 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Poisson Distributions for Arrival RatesPoisson Distributions for Arrival Rates
=2 =4
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 1 2 3 4 5 6 7 8 9 10 11 12x
Prob
abilit
y
Prob
abilit
y
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 1 2 3 4 5 6 7 8 9 10 11 12x
Prob
abilit
y
= average arrival rate
D-10 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Service Facility
Waiting LinePopulation
Waiting Line CharacteristicsLength of the queue
Limited Unlimited
Queue discipline FIFO Other
Characteristics of a Waiting Line SystemCharacteristics of a Waiting Line System
D-11 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Service FacilityWaiting LinePopulation
Service Characteristics Number of channels
Single Multiple
Number of phases in service system
Single Multiple
Service time distribution Constant Random – estimated by negative exponential distribution
Characteristics of a Waiting Line SystemCharacteristics of a Waiting Line System
= average service rate
D-12 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Negative Exponential DistributionNegative Exponential Distribution
Average Service Rate () = 3 customers per hourAverage Service Time = 20 minutes per customer
Average Service Rate () = 1 customer per hour
Probability that Service Time is greater than t=e-t, for t > 0
Time t in Hours
Prob
abili
ty th
at S
ervi
ce T
ime
t
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
Average Service Rate () = 3 customers per hourAverage Service Time = 20 minutes per customer
Average Service Rate () = 1 customer per hour
Probability that Service Time is greater than t = e for t > 0–t
D-13 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Basic Queuing System DesignsBasic Queuing System Designs
ArrivalsServed unitsService
facility
Queue
Single-Channel, Single-PhaseSingle-Channel, Single-Phase
Service facility
Arrivals
Served units
Service facilityQueue
Service facility
Service facility
Multi-Channel, Multi-PhaseMulti-Channel, Multi-Phase
Arrivals
Served unitsService
facilityQueue
Service facility
Multi-Channel, Single-PhaseMulti-Channel, Single-Phase
e.g. U.S. Post Office
e.g. drive-through bank
e.g. Suds & Suds Laundromat
ArrivalsServed units
Service facility
QueueService facility
Single-Channel, Multi-PhaseSingle-Channel, Multi-Phase
e.g. McDonald’s drive-through
D-14 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
You manage a call center which can answer an average of 20 calls per hour. Your call center gets 17.5 calls in an average hour. On average, what is the time a customer spends on hold waiting for service?
Call Center – Solution (1)Call Center – Solution (1)
= average arrival rate = 17.5 calls/hr
= average service rate = 20 calls/hr
ρ = average utilization of system = / = 87.5%
D-15 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
You manage a call center which can answer an average of 20 calls per hour. Your call center gets 17.5 calls in an average hour. On average, what is the time a customer spends on hold waiting for service?
Call Center – Solution (2)Call Center – Solution (2)
= average arrival rate = 17.5 calls/hr = average service rate = 20 calls/hrρ = average utilization of system = / = 87.5%
L = average number of customers in service
system (line and being served) = / ( - ) = 7 calls Lq = average number waiting in line = ρL = 6.125 calls
D-16 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
You manage a call center which can answer an average of 20 calls per hour. Your call center gets 17.5 calls in an average hour. On average,what is the time a customer spends on hold waiting for service?
Call Center – Solution (3)Call Center – Solution (3)
= average arrival rate = 17.5 calls/hr
= average service rate = 20 calls/hr
ρ = average utilization of system = / = 87.5%
W = average time in system
(wait and service) = 1 / ( - ) = .4 hr = 24 min
Wq = average time waiting = ρW = 21 min
D-17 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Single server equationsSingle server equations = average arrival rate
= average service rate
ρ = average utilization of system = / Pn = probability that n customers are in the system = (1- ρ) ρn
L = average number of customers in service system (line and being served) = / ( - )
Lq = average number waiting in line = ρL
W = average time in system (wait and service) = 1/ ( - )
Wq = average time waiting = ρW
D-18 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
You are opening an ice cream stand that has a single employee (you). You expect to see about 25 customers an hour. It takes you an average of 2 minutes to serve each customer. Customers are served in a FCFS manner.
Your research suggests that if there is a line of more than 4 people that some customers will leave without buying anything. In addition, if customers have to wait more than 6 minutes to get their order filled they are not likely to come back.
How well will this system do at satisfying customers? What assumptions are you making to answer this question?
Izzy’s Ice Cream StandIzzy’s Ice Cream Stand
D-19 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Izzy’s Ice Cream Stand (2)Izzy’s Ice Cream Stand (2) = 25 customers/hr = 30 customers/hr ρ = / = .833
Pn = probability that n customers are in the system = (1- ρ) ρn
P0 = (1 – .833) x .833 = .167 .167 P1 = (1 – .833) x .833 = .139 .306 P2 = (1 – .833) x .833 = .116 .422 P3 = (1 – .833) x .833 = .097 .519 P4 = (1 – .833) x .833 = .080 .599
Pmore than 4 = 1 – .599 = .401
1
2
3
4
0cumulative
L = / ( - ) = 5 customers
Customers in the system
D-20 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Izzy’s Ice Cream Stand (3)Izzy’s Ice Cream Stand (3) = 25 customers/hr = 30 customers/hr ρ = / = .833
L = / ( - ) = 25 / (30 – 25) = 5 customersLq = ρL = .833 x 5 = 4.17 customers
W = 1 / ( - ) = 1 / (30 – 25) = .2 hr = 12 minWq = ρW = .833 x 12 min = 10 min
Why not L – 1 ?
D-21 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
Why do we have to wait?Why do we have to wait?
© 1995 Corel Corp.
Why do services (and most non-MTS manufacturers) have queues? Processing time and/or arrival time variance Costs of capacity – can we afford to always have more
people/servers than customers? Efficiency – e.g. scheduling at the Doctor’s office
D-22 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458
x
Time t
Prob
abili
ty th
at
Serv
ice
Tim
e
t
Probability
ARRIVALS(Poisson)
SERVICE(Exponential)
CUSTOMERS IN THE
SYSTEM
Prob
abili
ty
n