waiting lines problems usc copy

8/18/2019 Waiting Lines Problems Usc Copy

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Waiting Lines

Waiting Lines Example

1. Students arrive at the Administrative Services Office at anaverage of one every 15 minutes, and their request take on

average 10 minutes to be processed. The service counter is

staffed by ony one cerk, !udy "umshoes, #ho #orks eight

hours per day. Assume $oisson arrivas and e%ponentiaservice times.

&'&'1 System

λ ( ) customers'hour

µ ( * customers'hour


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&'&'1 System



a+ hat percentage of time is !udy ide-

1λ'µ ( //.//

b+ o# much time, on average, does a student spend #aiting

in ine-

q ( λ ' µ2µλ+

( 1'/ hours or 30 minutes


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&'&'1 System



c+ o# ong is the 2#aiting ine+ on average-

4q ( λq

( )'/ customers

d+ hat is the probabiity that an arriving student 2ust

before entering the Administrative Services Office+ #i

find at east one other student #aiting in ine-

(1$0$1

( 1 6 21λ'µ+ λ'µ21λ'µ+

( )'7, or 0.))


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3. At the 8aifornia border inspection station, vehices arrive atthe rate of 10 per minute in a $oisson distribution. 9or

simpicity in this probem, assume that there is ony one ane

and one inspector, #ho can inspect vehices at the rate of 13

per minute in an e%ponentiay distributed fashion.The 8aifornia border inspection station is considering the

addition of a second inspector. The vehices #oud #ait in

one ane and then be directed to the first avaiabe inspector.

Arriva rates #oud remain the same 210 per minute+ and thene# inspector #oud process vehices at the same rate as the

first inspector 213 per minute+.


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&'&'S System

λ ( 10 vehices'minute

µ ( 13 vehices'minute

a+ hat #oud be the average ength of the #aiting ine-

0.1:5 vehices

b+ hat #oud be the average time that a vehices must #ait

to get through the system-

0.101 minutes, or *.0* seconds


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;f a second ane #as added 2one ane for each inspector+<3 &'&'1 Systems

λ ( 5 vehices'minute

µ ( 13 vehices'minute

c+ hat #oud be the average ength of the #aiting ine-

0.37= vehices per ine 0.37=>3 ( 0.57*

d+ hat #oud be the average time that a vehice must #ait

to get through the system-

0.1)/ minutes, or =.5= seconds


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W iti Li


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S ( *λ ( 30 customers'hour

µ ( ) customers'hour

ρ ( λ ' Sµ ( 30 ' 2)>*+ ( 0.=/

8a ( 1, 8s ( 5'15 ( 0.//

a+ On average ho# many customer #oud be #aiting in ine-

4q ( 1.*=

b+ On average ho# ong #oud a customer spend in the bank-

s ( q @ 21'µ+ ( 24q'λ+ @ 21'µ+

( 0.//) hours ( 30.05 minutes

31

33+123

sa

S

q

C C L

+×

−=

+

ρ

ρ

W iti Li


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). 8onsider a ca center that empoys = agents. $ast datacoected on the customer interarriva times has sho#n that

the mean time bet#een customer arrivas is 1 minute, and has

a standard deviation of 1'3 minute. The amount of time in

minutes the past 10 caers have spent taking to an agent is asfoo#s<

− ).1, *.3, 5.5, /.5, /.3, :./, =.), *./, 3.*, ).7.

a+ hat is the coefficient of variation for the interarriva

times-2standard deviation+ ' mean ( 0.5'1 ( 0.5

b+ hat is the mean time a caer spends taking to an agent-

( 5.3 minutes

Waiting Lines


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c+ hat is the standard deviation of the time a caer spendstaking to an agent-

( 1.== minutes

d+ hat is the coefficient of variation for the times a caer

spends taking to an agent-

2standard deviation+ ' mean ( 5.3'1.== ( 3.::

2standard deviation+ ' mean ( 1.=='5.3 ( 0.173

Waiting Lines


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e+ hat is the e%pected number of caers on hod, #aiting totak to an agent-

( 1 per minute

B ( 1'5.3 ( 0.173 per minute

S ( =

C ( '2SB + ( 1'20.173>=+ ( 0.*5

Dpart ( 0.)* Epart ( 0.17

4q ( Dpart>Epart ( 0.0=:

4q ( 1.=

31

33+123

sa

S

q

C C L

+

×−

=

+

ρ

ρ


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Waiting Lines


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5. es 9argo operates one AT& machine in a certain Trader!oeFs. There is on average = customers that use the AT&

every hour, and each customer spends on average * minutes

at the AT&. Assume customer arrivas foo# a $oisson

process, and the amount of time each customer spends at theAT& foo#s as e%ponentia distribution.

a+ hat is the percentage of time the AT& is in use-

( = per hour

B ( 10 per hour

C ( ='10 ( 0.=

(=0

Waiting Lines


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b+ Suppose that the percentage of time the AT& is in use is 0.=.2This may or may not be the ans#er you found in 2a+.+

b1+ hat is the probabiity there is more than one customer

#aiting in the ine to use the AT&-

G ( number of customers either using the AT& or #aiting to

use the AT&

$2 G(n + ( C n21 C+

$2 G H 3 + ( 1 6 $2G(0+ 6 $2G(1+ 6 $2G(3+

( 1 6 21 C+ 6 C21 C+ 6 C321 C+ ( 121 C @ C C 3 @ C 3 C / +

( C / ( 0.513

Waiting Lines


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b+ Suppose that the percentage of time the AT& is in use is 0.=.2This may or may not be the ans#er you found in 2a+.+

b3+ On average, ho# many customers are in ine #aiting to use

the AT&- L

3./3.0

=.0

31

333+123

==+

×−

=+

sa

S

q

C C L

ρ

ρ

Waiting Lines


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c+ Suppose that the number of customers in ine #aiting to use theAT& is /. 2This may or may not be the ans#er you found in

parts b or c+. A information remains as originay stated.

hat is the average time a customer must #ait to use the

AT&- State your ans#er in minutes.

q ( 4q'λ ( /'= I/2*0+J'= ( 33.5 minutes

Waiting Lines


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*. &atador housing office has one customer representative for #akin students. Thearriva rate 10 customers per hour and the average service time is 5 minutes. Koth

interarriva time and service time foo# e%ponentia distributions.

a+ hat is the average #aiting time in ine-

( 10 customers'hour, B ( 30 customers'hour

q ( '2B2B 6 ++ ( 0.05 hour ( / minutes

or q ( ma C3'21 6 C+>21@1+'3 ( 0.05 hour

b+ hat is the probabiity that an arriving student 2ust before entering the

housing office+ #i find at east t#o other student #aiting in ine-

1 6 $0 6 $1 ( 0.35

Waiting Lines


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:. &onterey post station has : teers from &onday to Saturday.8ustomers arrive to the station foo#ing a $oisson process

#ith rate /* customers per hour. The service time is

e%ponentiay distributed #ith mean 10 minutes.

a+ hat is the utiiLation rate of the teers- ( /* customers'hour, B ( * customers'hour

C ( 'SB ( /*')3 ( *': ( =5.:

b+ hat is the average number of customers #aiting in ine-

24q (C)'21 6 C+>21@1+'3 ( /.:=

Waiting Lines


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On Sunday, instead of teers, the post station ony opens / automaimachines to provide automatic service. Mach machine can #eight

different siLe of package, print sefadhesive abes and accept

payments. Arriva is $oisson #ith rate 30 customers per hour. The

service time is ) minutes #ith probabiity 0.:5 and 30 minutes #ith

probabiity 0.35.

c+ hat is the mean service time-

ms ( )>0.:5 @ 1* > 0.35 ( = minutes

d+ hat is the coefficient of variation of service time-Ss ( sqrt 20.:5 > /3 @ 0.35 > 73 + ( *.7/

8s ( *.7/ ' : ( 0.=**

Waiting Lines


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g


e+ hat is the utiiLation rate-

( 30 customers'hour, B ( *0'= ( :.5 customers'hour

C ( 'SB ( 30'2/>:.5+ ( ==.7

f+ hat is the average number of customers #aiting in ine-

*).531

33+123

=+

×

−

=

+

sa

S

q

C C L

ρ

ρ

Waiting Lines


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g


=. Kank of San $edro has ony one teer. On average, one customer comes every *minutes, and it takes the teer an average of / minutes to serve a customer. To

improve customer satisfaction, the bank is going to impement a unique poicy

caed, Ne $ay hie ?ou ait. Once impemented, the bank #i pay each

customer P/ per minute #hie she or he #aits in ine. 2So the cock starts #hen a

customer comes to the end of the ine, and stops #hen he or she begins to tak to

the teer.+ Kank of San $edro hired you as a consutant and you are responsibe forestimating ho# much the Ne $ay hie ?ou ait program #i cost. ?our

preiminary study indicates there are, on average, 0.5 customers #aiting in ine

Assume inear cost. ;f a customer #aits for 30 seconds in ine, Kank of San $edro

#i pay P1.

Waiting Lines


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g


a+ 8acuate the capacity of the teer. State the unit.*0'/(30 customers per hour.

b+ 8acuate proportion of the time the teer is busy.

0.5 or 50

c+ o# ong, on average, does a customer #ait in ine- State the unit.

0.5'10 ( 0.05 hours or / minutes.

Waiting Lines


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g


d+ 8acuate the e%pected houry cost of the Ne $ay hie ?ouait program.

Mach customer #aits, on average, / minutes.

So he or she receives, on average, />/(P7.

There are 10 customers arriving per hour.

So the overa cost of this program is 7>10(P70

waiting lines problems usc copy

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