waiting line mgmt
TRANSCRIPT
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Waiting Line Management
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Waiting Line Characteristics Suggestions for Managing Queues Examples (Models 1, 2, 3, and 4)
OBJECTIVES
3Components of the Queuing System
CustomerArrivals
Servers
Waiting Line
Servicing System
Exit
Queue or
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Customer Service Population Sources
Population Source
Finite Infinite
Example: Number of machines needing repair when a company only has three machines.
Example: Number of machines needing repair when a company only has three machines.
Example: The number of people who could wait in a line for gasoline.
Example: The number of people who could wait in a line for gasoline.
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Service Pattern/Arrival Rate
ServicePattern
Constant Variable
Example: Items coming down an automated assembly line.
Example: Items coming down an automated assembly line.
Example: People spending time shopping.
Example: People spending time shopping.
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Arrival Rate Exponential Distribution- When arrivals at a service
facility occurs in a purely random fashion, a plot of interarrival times yields an E.D. f(t)=λe^-λt
Poisson Distribution- When one is interested in the no. of arrivals during some time period T, it is obtained by finding the probability of exactly n arrivals during T. If the arrival process is random , the distribution is Poisson.
TIME BETWEEN ARRIVALS IS E.D. & THE NO. OF ARRIVALA PER UNIT TIME IS POISSON DIST.
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Arrivals Single Arrival – A unit is the smallest no.
handled. Batch arrival - Some multiple of the unit
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The Queuing System
Queue Discipline
FCFS, SPT,bLarge Orders First, Emerg. First,Reservations
Length•Infinite•finite
Number of Lines &Line Structures
Service Time Distribution/service rate
Queuing System
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Examples of Line Structures
Single Channel
Multichannel
SinglePhase Multiphase
One-personbarber shop
Car wash
Hospitaladmissions
Bank tellers’windows
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Degree of Patience
No Way!
BALK
No Way!
RENEG
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Suggestions for Managing Queues
1. Determine an acceptable waiting time for your customers
2. Try to divert your customer’s attention when waiting
3. Inform your customers of what to expect
4. Keep employees not serving the customers out of sight
5. Segment customers
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Suggestions for Managing Queues (Continued)
6. Train your servers to be friendly
7. Encourage customers to come during the slack periods
8. Take a long-term perspective toward getting rid of the queues
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Waiting Line Models
Model LayoutSourcePopulation Service Pattern
1 Single channel Infinite Exponential
2 Single channel Infinite Constant
3 Multichannel Infinite Exponential
4 Single or Multi Finite Exponential
These four models share the following characteristics: Single phase Poisson arrival FCFS Unlimited queue length
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Notation: Infinite Queuing: Models 1-3
linein tingnumber wai Average
server single afor
rate sevice torate arrival totalof Ratio = =
arrivalsbetween timeAverage
timeservice Average
rate Service =
rate Arrival =
1
1
Lq
linein tingnumber wai Average
server single afor
rate sevice torate arrival totalof Ratio = =
arrivalsbetween timeAverage
timeservice Average
rate Service =
rate Arrival =
1
1
Lq
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Infinite Queuing Models 1-3 (Continued)
linein waitingofy Probabilit
systemin units exactly ofy Probabilit
channels service identical ofNumber =
system in the units ofNumber
served) be to time(including
systemin time totalAverage
linein waiting timeAverage =
served) being those(including
systemin number Average = s
Pw
nPn
S
n
Ws
Wq
L
linein waitingofy Probabilit
systemin units exactly ofy Probabilit
channels service identical ofNumber =
system in the units ofNumber
served) be to time(including
systemin time totalAverage
linein waiting timeAverage =
served) being those(including
systemin number Average = s
Pw
nPn
S
n
Ws
Wq
L
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Assume a drive-up window at a fast food restaurant.Customers arrive at the rate of 25 per hour.The employee can serve one customer every two minutes.Assume Poisson arrival and exponential service rates.Determine:A) What is the average utilization of the employee?B) What is the average number of customers in line?C) What is the average number of customers in the system?D) What is the average waiting time in line?E) What is the average waiting time in the system?F) What is the probability that exactly two cars will be in the system?
Determine:A) What is the average utilization of the employee?B) What is the average number of customers in line?C) What is the average number of customers in the system?D) What is the average waiting time in line?E) What is the average waiting time in the system?F) What is the probability that exactly two cars will be in the system?
Example: Model 1
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= 25 cust / hr
= 1 customer
2 mins (1hr / 60 mins) = 30 cust / hr
= = 25 cust / hr
30 cust / hr = .8333
= 25 cust / hr
= 1 customer
2 mins (1hr / 60 mins) = 30 cust / hr
= = 25 cust / hr
30 cust / hr = .8333
Example: Model 1
A) What is the average utilization of the employee?
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Example: Model 1
B) What is the average number of customers in line?
4.167 = 25)-30(30
(25) =
) - ( =
22
Lq 4.167 = 25)-30(30
(25) =
) - ( =
22
Lq
C) What is the average number of customers in the system?
5 = 25)-(30
25 =
- =
Ls 5 = 25)-(30
25 =
- =
Ls
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Example: Model 1
D) What is the average waiting time in line?
mins 10 = hrs .1667 =
=
LqWq mins 10 = hrs .1667 =
=
LqWq
E) What is the average waiting time in the system?
mins 12 = hrs .2 = =Ls
Ws mins 12 = hrs .2 = =Ls
Ws
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Example: Model 1
F) What is the probability that exactly two cars will be in the system (one being served and the other waiting in line)?
p = (1-n
n
)( )p = (1-n
n
)( )
p = (1- = 2
225
30
25
30)( ) .1157p = (1- =
2
225
30
25
30)( ) .1157
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Example: Model 2
An automated pizza vending machine heats and dispenses a slice of pizza in 4 minutes.
Customers arrive at a rate of one every 6 minutes with the arrival rate exhibiting a Poisson distribution.
Determine:
A) The average number of customers in line.B) The average total waiting time in the system.
Determine:
A) The average number of customers in line.B) The average total waiting time in the system.
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Example: Model 2
A) The average number of customers in line.
.6667 = 10)-(2)(15)(15
(10) =
) - (2 =
22
Lq .6667 = 10)-(2)(15)(15
(10) =
) - (2 =
22
Lq
B) The average total waiting time in the system.
mins 4 = hrs .06667 = 10
6667. =
=
LqWq mins 4 = hrs .06667 =
10
6667. =
=
LqWq
mins 8 = hrs .1333 = 15/hr
1 + hrs .06667 =
1 + =
WqWs mins 8 = hrs .1333 = 15/hr
1 + hrs .06667 =
1 + =
WqWs
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Example: Model 3Recall the Model 1 example:
Drive-up window at a fast food restaurant.
Customers arrive at the rate of 25 per hour.
The employee can serve one customer every two
minutes.
Assume Poisson arrival and exponential service rates.
If an identical window (and an identically trained server) were added, what would the effects be on the average number of cars in the system and the total time customers wait before being served?
If an identical window (and an identically trained server) were added, what would the effects be on the average number of cars in the system and the total time customers wait before being served?
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Example: Model 3Average number of cars in the system
ion)interpolatlinear -using-TN7.11(Exhibit
1760= .Lqion)interpolatlinear -using-TN7.11(Exhibit
1760= .Lq
1.009 = 30
25 + .176 = + =
LqLs 1.009 = 30
25 + .176 = + =
LqLs
Total time customers wait before being served
)( = mincustomers/ 25
customers .176 = = Wait! No
LqWq mins .007
)( =
mincustomers/ 25
customers .176 = = Wait! No
LqWq mins .007
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Notation: Finite Queuing: Model 4
channels service ofNumber
linein units ofnumber Average
)( system
queuingin thoseless source Population =
served being units ofnumber Average
linein wait tohaving
ofeffect theof measure a factor, Efficiency
linein must wait arrivalan y that Probabilit =
S
L
n-N
J
H
F
D
channels service ofNumber
linein units ofnumber Average
)( system
queuingin thoseless source Population =
served being units ofnumber Average
linein wait tohaving
ofeffect theof measure a factor, Efficiency
linein must wait arrivalan y that Probabilit =
S
L
n-N
J
H
F
D
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Finite Queuing: Model 4 (Continued)
required timeservice of proportionor factor, Service
linein time waitingAverage
tsrequiremen servicecustomer between timeAverage
service theperform to timeAverage =
system queuingin units exactly ofy Probabilit
source populationin units ofNumber
served) being one the(including
system queuingin units ofnumber Average =
X
W
U
T
nPn
N
n
required timeservice of proportionor factor, Service
linein time waitingAverage
tsrequiremen servicecustomer between timeAverage
service theperform to timeAverage =
system queuingin units exactly ofy Probabilit
source populationin units ofNumber
served) being one the(including
system queuingin units ofnumber Average =
X
W
U
T
nPn
N
n
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Example: Model 4
The copy center of an electronics firm has four copymachines that are all serviced by a single technician.
Every two hours, on average, the machines require adjustment. The technician spends an average of 10minutes per machine when adjustment is required.
Assuming Poisson arrivals and exponential service, how many machines are “down” (on average)?
The copy center of an electronics firm has four copymachines that are all serviced by a single technician.
Every two hours, on average, the machines require adjustment. The technician spends an average of 10minutes per machine when adjustment is required.
Assuming Poisson arrivals and exponential service, how many machines are “down” (on average)?
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Example: Model 4N, the number of machines in the population = 4M, the number of repair people = 1T, the time required to service a machine = 10 minutesU, the average time between service = 2 hours
X =T
T + U
10 min
10 min + 120 min= .077X =
T
T + U
10 min
10 min + 120 min= .077
From Table TN7.11, F = .980 (Interpolation)From Table TN7.11, F = .980 (Interpolation)
L, the number of machines waiting to be serviced = N(1-F) = 4(1-.980) = .08 machines
L, the number of machines waiting to be serviced = N(1-F) = 4(1-.980) = .08 machines
H, the number of machines being serviced = FNX = .980(4)(.077) = .302 machines
H, the number of machines being serviced = FNX = .980(4)(.077) = .302 machines
Number of machines down = L + H = .382 machinesNumber of machines down = L + H = .382 machines
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Question Bowl
The central problem for virtually all queuing problems is which of the following?
a. Balancing labor costs and equipment costsb. Balancing costs of providing service with the
costs of waitingc. Minimizing all service costs in the use of
equipmentd. All of the abovee. None of the above Answer: b. Balancing
costs of providing service with the costs of waiting
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Question Bowl
Customer Arrival “populations” in a queuing
system can be characterized by which of the
following?
a. Poisson
b. Finite
c. Patient
d. FCFS
e. None of the above
Answer: b. Finite
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Question Bowl
Customer Arrival “rates” in a queuing system
can be characterized by which of the
following?
a. Constant
b. Infinite
c. Finite
d. All of the above
e. None of the above
Answer: a. Constant
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Question Bowl
An example of a “queue discipline” in a queuing
system is which of the following?
a. Single channel, multiphase
b. Single channel, single phase
c. Multichannel, single phase
d. Multichannel, multiphase
e. None of the above
Answer: e. None of the above (These are the rules for determining the order of service to customers, which include FCFS, reservation first, highest-profit customer first, etc.)
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Question Bowl
Withdrawing funds from an automated teller machine
is an example in a queuing system of which of
the following “line structures”?
a. Single channel, multiphase
b. Single channel, single phase
c. Multichannel, single phase
d. Multichannel, multiphase
e. None of the above
Answer: b. Single channel, single phase
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Question Bowl
Refer to Model 1 in the textbook. If the service
rate is 15 per hour, what is the “average
service time” for this queuing situation?
a. 16.00 minutes
b. 0.6667 hours
c. 0.0667 hours
d. 16% of an hour
e. Can not be computed from data above
Answer: c. 0.0667 hours (1/15=0.0667)
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Question BowlRefer to Model 1 in the textbook. If the arrival rate
is 15 per hour, what is the “average time
between arrivals” for this queuing situation?
a. 16.00 minutes
b. 0.6667 hours
c. 0.0667 hours
d. 16% of an hour
e. Can not be computed from data above
Answer: c. 0.0667 hours (1/15=0.0667)
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Question Bowl
Refer to Model 4 in the textbook. If the “average time to
perform a service” is 10 minutes and the “average
time between customer service requirements” is 2
minutes, which of the following is the “service factor”
for this queuing situation?
a. 0.833
b. 0.800
c. 0.750
d. 0.500
e. None of the above
Answer: a. 0.833 (10/(10+2)=0.833)
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