Waiting Lines and Queuing

8/10/2019 Waiting Lines and Queuing

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Waiting Lines and Queuing Theory Models CHAPTER 14

TRUE/FALSE

14.1 The three parts of a queuing system are the arrivals, the queue, and the service facility.

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14.# T$o characteristics of arrivals are the line length and queue discipline.

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14.% &ueuing theory models can also apply to customers placing telephone calls and 'eing placed onhold.

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14.4 The only o'(ective of queuing theory is to minimi)e customer dissatisfaction.

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14. Should a customer leave a queue 'efore 'eing served, it is said that the customer has reneged.

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14.- al/ing refers to customers $ho enter the queue 'ut may 'ecome impatient and leave $ithoutcompleting their transactions.

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14.0 ost systems use the queue discipline /no$n as the first2in, first2out rule.

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14.3 n a very comple5 queuing model, if all of the assumptions of the traditional models are not met,then the pro'lem cannot 'e handled.

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14.6 efore using e5ponential distri'utions to 'uild queuing models, the quantitative analyst shoulddetermine if the service time data fit the distri'ution.

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14.17 *or practical purposes, queue length is almost al$ays modeled $ith a finite queue length.

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14.11 The 8ree/ letter is used to represent the average service rate at each channel.

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14.1# *or a single channel model that has 9oisson arrivals and e5ponential service rates, the 8ree/

letter is the utili)ation factor.

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14.1% n a multi2channel, single2phase queuing system, the arrival $ill pass through at least t$odifferent service facilities.

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14.14 n a multi2channel model : ;< =.

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14.1 A goal of many $aiting line pro'lems is to help a firm find the ideal level of services to 'e

offered.

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14.1- Any $aiting line pro'lem can 'e investigated using an analytical queuing model.

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14.10 >ne of the difficulties in $aiting line analyses is that it is sometimes difficult to place a value oncustomer $aiting time.

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14.13 The goal of most $aiting line pro'lems is to identify the service level that minimi)es service cost.

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14.16 >ne of the limitations of analytical $aiting line models is that they do not give information one5treme cases


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14.#% >n a practical note@ if $e are using $aiting line analysis to study cars passing through a singletoll'ooth, reneging is pro'a'ly not an issue.

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14.#4 >n a practical note @ if $e are studying patrons moving through chec/out lines at a grocery store,

and $e note that these patrons sometimes move from one line to another, $e should consider'al/ing as an issue.

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14.# >n a practical note @ if $e $ere to study the $aiting lines in a hair salon $hich had only fivechairs for patrons $aiting, $e $ould have to use a finite queue $aiting line model.

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14.#- All practical $aiting line pro'lems can 'e vie$ed as having a **> queue discipline.

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14.#0 A hospital emergency room $ill usually employ a **> queue discipline.

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14.#3 f $e $ish to study a 'an/, in $hich patrons entered the 'uilding and then, depending upon theservice desired, chose one of several tellers in front of $hich to form a line, $e $ould employ aset of single2channel queuing models.

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14.#6 >n a practical note @ $e should pro'a'ly vie$ the chec/out counters in a grocery store as a set

of single channel systems.

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14.%7 A cafeteria, in $hich cold dishes are separated from hot dishes, is pro'a'ly 'est vie$ed as asingle2channel, single2phase system.

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14.%1 An emergency room might 'e vie$ed as a multi2channel, multi2phase system.

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14.%# A single high$ay $ith multiple toll'ooths should 'e vie$ed as a single2channel system.

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14.%% n a doctors office, $e $ould e5pect the arrival rate distri'ution to 'e 9oisson distri'uted, andthe service time distri'ution to 'e negative e5ponential.

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14.%4 The ;;1 queuing model assumes that the arrival rate does not change over time.

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14.% The analytical queuing models typically provide operating characteristics that are averages


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14.4% Whether or not $e use the finite population queuing model depends upon the amount of space $ehave in $hich to form the queue.

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14.44 f a $aiting line pro'lem is particularly comple5, $e may have to turn to a simulation model.

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14.4 f $e are using a simulation queuing model, $e still have to a'ide 'y the assumption of a 9oissonarrival rate, and negative e5ponential service rate.

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14.4- "sing a simulation model allo$s one to ignore the common assumptions required to useanalytical models.

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B14.40 f $e are studying the arrival of automo'iles at a high$ay toll station, $e can assume an infinitecalling population.

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B14.43 f $e are studying the need for repair of electric motors on a small assem'ly line, $e can assumean infinite calling population.

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B14.46 The difference 'et$een 'al/ing and reneging is that 'al/ing implies that the arrival never (oinedthe queue, $hile reneging implies that the arrival (oined the queue, 'ut 'ecame impatient and left.

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B14.7 When loo/ing at the arrivals at the tic/et counter of a movie theater, $e can assume an unlimitedqueue.

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B14.1 When loo/ing at the arrivals at a 'ar'ershop, $e must assume a finite queue.

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B14.# A 'an/, in $hich a single queue is used to move customers to several tellers, is an e5ample of asingle2channel system.

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B14.% A fast food drive2through system is an e5ample of a multi2channel queuing system.

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14.-7 Which of the follo$ing is not a /ey operating characteristic for a queuing systemC


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14.-6 &ueue discipline may 'e


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14.0% Dustomers enter the $aiting line at a cafeteria on a first come, first served 'asis. The arrival ratefollo$s a 9oisson distri'ution, $hile service times follo$ an e5ponential distri'ution. f theaverage num'er of arrivals is si5 per minute and the average service rate of a single server iseight per minute, $hat is the average num'er of customers in the systemC


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14.0- Dustomers enter the $aiting line to pay for food as they leave a cafeteria on a first come, firstserved 'asis. The arrival rate follo$s a 9oisson distri'ution, $hile service times follo$ ane5ponential distri'ution. f the average num'er of arrivals is si5 per minute and the averageservice rate of a single server is eight per minute, on average, ho$ much time $ill elapse from thetime a customer enters the line until he;she leaves the cafeteriaC


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14.06 A post office has a single line for customers to use $hile $aiting for the ne5t availa'le postalcler/. There are t$o postal cler/s $ho $or/ at the same rate. The arrival rate of customersfollo$s a 9oisson distri'ution, $hile the service time follo$s an e5ponential distri'ution. Theaverage arrival rate is three per minute and the average service rate is t$o per minute for each ofthe t$o cler/s. What proportion of the time are 'oth cler/s idleC


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Ta'le 1421

;;%!

ean Arrival ate! 4 occurrences per minute

ean Service ate! # occurrences per minute

Num'er of Servers! %

Solution!ean Num'er of "nits in the System! #.336

ean Num'er of "nits in the &ueue! 7.336

ean Time in the System! 7.0## minutes

ean Time in the &ueue! 7.### minutes

Service *acility "tili)ation *actor! 7.--0

9ro'a'ility of No "nits in System! 7.111

14.3% According to Ta'le 1421, $hich provides a queuing pro'lem solution, $hat proportion of the timeis the system totally emptyC


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Ta'le 142#

;;#

ean Arrival ate! occurrences per minute

ean Service ate! % occurrences per minute

Num'er of Servers! #

Solution!ean Num'er of "nits in the System! .4

ean Num'er of "nits in the &ueue! %.033

ean Time in the System! 1.761 minutes

ean Time in the &ueue! 7.03 minutes

Service *acility "tili)ation *actor! 7.3%%

9ro'a'ility of No "nits in System! 7.761

14.3- According to Ta'le 142#, $hich provides a queuing pro'lem solution, on average, ho$ manyunits are in the lineC


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Donstant Service ate! 4 occurrences per minute

Solution!

ean Num'er of "nits in the System! 1.30

ean Num'er of "nits in the &ueue! 1.1#

ean Time in the System! 7.-# minutes

ean Time in the &ueue! 7.%0 minutes

Service *acility "tili)ation *actor! 7.079ro'a'ility of No "nits in System! 7.#7

14.36 According to Ta'le 142%, $hich presents a queuing pro'lem solution for a queuing pro'lem $itha constant service rate, on average, ho$ much time is spent $aiting in lineC


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14.6# According to Ta'le 142%, $hich presents a queuing pro'lem $ith a constant service rate, onaverage, ho$ many minutes does a customer spend in the service facilityC


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14.171 The utili)ation factor for a system tells one the


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14.17 The 'ehavior of (umping from one queue to another trying to get through as quic/ly as possi'le iscalled!


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14.176 f everything else remains constant, including the mean arrival rate and service rate, e5cept thatthe service time 'ecomes constant instead of e5ponential, the


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14.11% Dustomers enter the $aiting line at a cafeteria on a first come, first served 'asis. The arrival ratefollo$s a 9oisson distri'ution, $hile service times follo$ an e5ponential distri'ution. f theaverage num'er of arrivals is four per minute and the average service rate of a single server isseven per minute, $hat is the average num'er of customers in the systemC


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14.11- Dustomers enter the $aiting line to pay for food as they leave a cafeteria on a first come, firstserved 'asis. The arrival rate follo$s a 9oisson distri'ution, $hile service times follo$ ane5ponential distri'ution. f the average num'er of arrivals is four per minute and the averageservice rate of a single server is seven per minute, on average, ho$ much time $ill elapse fromthe time a customer enters the line until he;she leaves the cafeteriaC


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14.116 A post office has a single line for customers to use $hile $aiting for the ne5t availa'le postalcler/. There are t$o postal cler/s $ho each $or/ at the same rate. The arrival rate of customersfollo$s a 9oisson distri'ution, $hile the service time follo$s an e5ponential distri'ution. Theaverage arrival rate is seven per minute and the average service rate is four per minute for each ofthe t$o cler/s. What proportion of the time are 'oth cler/s idleC


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14.1%7 According to the information provided in Ta'le 1424, $hat percentage of the total availa'leservice time is 'eing usedC


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14.1%% According to Ta'le 142, $hich presents a queuing pro'lem solution for a queuing pro'lem $itha constant service rate, on average, ho$ many customers arrive per time periodC


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B14.1%0 At a local fast food (oint, cars arrive randomly at a rate of 1# every %7 minutes. The fast food(oint ta/es an average of # minutes to serve each arrival. The utili)ation factor for this system is


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B14.141 Dars arrive at a local K+"E franchise at the rate of 1 every 1# minutes. Service times aree5ponentially distri'uted $ith an average of 1 minutes. The average customer $aits in line


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high$ay is five lanes $ide. "nder light traffic conditions, a car does not have to commit to aspecific toll lane until actually approaching the toll 'ooths. >n average, ho$ long is the line infront of a specific toll 'oothC


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PROBLEMS

14.143 A ne$ shopping mall is considering setting up an information des/ manned 'y one employee.ased upon information o'tained from similar information des/s, it is 'elieved that people $illarrive at the des/ at the rate of 1 per hour. t ta/es an average of t$o minutes to ans$er aquestion. t is assumed that arrivals are 9oisson and ans$er times are e5ponentially distri'uted.


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14.10 A company has si5 computers that are used to run an automated manufacturing facility. Each ofthese runs an average of 67 minutes $ithout requiring any attention from the technician. Eachtime the technician is required to ad(ust a computer, an average of 1 minutes


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14.1-7 The ne$ 9rovidence shopping mall has 'een considering setting up an information des/ manned'y one employee. The layout for this mall is quite comple5, leading the mall manager to e5pect ahigher than normal arrival rate for persons see/ing assistance. t appears that a reasona'lee5pectation is an arrival rate of appro5imately # patrons per hour. "nder the original plan, themanager e5pected that it $ould ta/e appro5imately 4 minutes for the nformation Fes/ employeeto help the average person. y utili)ing a ne$ map and special guide signs, he 'elieves that therequired service time can 'e reduced to an average of t$o minutes per patron. The manager has

also noticed that the people see/ing help at the information des/ may come from one of t$ogroups!


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ANSWE!


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14.1-0 At the start of 'allet season, the tic/et office gets very 'usy the day 'efore the first performance.Dustomers arrive at the rate of four every fifteen minutes, and the average time to transact'usiness is - minutes. There are t$o servers in the tic/et office, 'oth of $hom operate at thesame speed.


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SHORT ANSWER/ESSAY

14.107 With regard to queue theory, define $hat is meant 'y 'al/ing.

ANSWE! the situation $hen arriving customers choose to not enter the queue

14.101 With regard to queue theory, define $hat is meant 'y reneging.

ANSWE! the situation $hen customers in a queue choose to leave the queue unserviced

14.10# o$ is **> used in descri'ing a queuing theory pro'lemC

ANSWE! **> descri'es the queue discipline in that the first customers to arrive are the firstcustomers to 'e serviced and, hence, the first out.

14.10% +ist three /ey operating characteristics of a queuing system.

ANSWE! the average num'er of customers in line, the average $aiting times, and the percentof system idle time

14.104 What is meant 'y a single2channel queuing systemC

ANSWE! one service facility system fed 'y one queue

14.10 What is meant 'y a multi2channel queuing systemC

ANSWE! more than one service facility system all fed 'y the same queue

14.10- What is meant 'y a single2phase systemC

ANSWE! arrivals leave the system after receiving service at only one station

14.100 What is meant 'y a multi2phase systemC

ANSWE! service is provided at more than one station, one after the other

14.103 What is represented 'y C

ANSWE! the proportion of the time service facilities are in use

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Waiting Lines and Queuing