Waiting Line Mgmt

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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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