queing sol taylor

8/11/2019 Queing Sol Taylor

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Waiting lines

12. The Peachtree Airport in Atlanta serves light aircraft. It has a single runway and oneair traffic controller to land planes. It takes an airplane 12 minutes to land and clear therunway. Planes arrive at the airport at the rate of 4 per hour.

a. Determine the average numer of planes that will stack up waiting to land.. !ind the average time a plane must wait in line efore it can land.c. "alculate the average time it takes a plane to clear the runway once it has notified

the airport that it is in the vicinity and want to land.d. The !AA has a rule that an air traffic controller can on the average land planes a

ma#imum of 4$ minutes out of every hour. There must e 1$ minutes of idle timeavailale to relieve the tension. %ill this airport have to hire an e#tra air trafficcontroller&

'olution


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1(. The !irst American )ank of *apid "ity has one outside drive+up teller. It takes theteller an average of 4 minutes to serve a ank customer. "ustomers arrive at the drive+upwindow at the rate of 12 per hour. The ank operations officer is currently analy,ing thepossiility of adding a second drive+up window at an annual cost of -2/. It isassumed that arriving cars would e e0ually divided etween oth windows. The

operations officer estimates that each minutes reduction in customer waiting time wouldincrease the anks revenue y -2/ annually. 'hould the second drive+up window einstalled&

'olution

1$. All truck traveling on Interstate 4 etween Alu0uer0ue and Amarillo are re0uired tostop at a weigh station. Trucks arrive at the weigh station at a rate of 2 per +hour day/and the station can weigh/ on the average/ 22 trucks per day.

a. Determine the average numer of trucks waiting/ the average time spend waitingand eing weighed at the weigh station y each truck/ and the average waitingtime efore eing weighed for each truck.

b. If the truck drivers find out they must remain at the weigh station longer than 1$

minutes on the average/ the will start taking a different route or traveling at night/thus depriving the state of ta#es. The state of 3ew e#ico estimates it loses-1/ in ta#es per year for each e#tra minute that trucks must remain at theweigh station. A new set of scales would have the same service capacity as thepresent set of scales/ and it is assumed that arriving trucks would line up e0uallyehind the two set of scales. It would cost -$/ per year to operate the newscales. 'hould the state install the new set of scales&


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'olution

15. The acme ache 'hop has five machines that periodically reak down and re0uireservice. The average time etween reakdowns is four days/ distriuted according to ane#ponential distriution. The average time to repair a machine is one day/ distriutedaccording to an e#ponential distriution. 6ne mechanic repairs the machines in the order

in which they reak down.

a. Determine the proaility of the mechanic eing idle.. Determine the mean numer of machines waiting to e repaired.c. Determine the mean time machines wait to e repaired.d. Determine the proaility that three machines are not operating 7are eing

repaired or waiting to e repaired8.


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Solution


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2$. The )ay "ity Police Department has eight patrol cars that are on constant call 24hours per day. A patrol car re0uires repairs every 2 days on average according to ane#ponential distriution. %hen a patrol car is in need of repair/ it is driving into the motorpool/ which has a repair person on duty at all times. The average time re0uired to repair apatrol car is 1 hours 7e#ponentially distriuted8. Determine the average time a patrol car

is not availale for use and the average numer of patrol cars out of service at any onetime/ and indicate if the repair service seems ade0uate.

25. "akes akes y the !reshfood )akery are transported from the ovens to e packagedy one of three wrappers. >ach wrapper can wrap an average of 2 cakes per hour. The


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cakes are rought to the wrappers at the rate of $ per hour. If a cake sits longer thanfive minutes efore eing wrapped/ it will not e fresh enough to meet the akerys0uality control standards. Does the akery need to hire another wrapper&

(2. The *egency


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((. The local )urger Doodle fast+food restaurant has a drive+through window. "ustomersin cars arrive at the window at the rate of 1 per hour 7Poisson distriuted8. It re0uires anaverage of 4 minutes 7e#ponentially distriuted8 to take and fill and order. The restaurantchain has a service goal of an average waiting time of ( minutes.

a. %ill the current system meet the restaurants service goal&. If the restaurant is not meeting its service goal/ it can add a second drive+through

window that will reduce the service time per customer to 2.$ minutes. %ill the

additional window enale the restaurant to meet its service goal&c. During the 2+hour lunch period/ the arrival rate of drive+in customers increases to2 per hour. %ill the two+window system e ale to achieve the restaurantsservice goal during this rush period&


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$. im "arter uilds custom furniture/ primarily cainets/ ookcases/ small tales/ andchairs.


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$1. udith Bewis is a doctoral student at 'tate Cniversity and she also works full+time asan academic tutor for scholarshiped student athletes. 'he took the :o hoping it wouldleave her free time etween tutoring to devote to her own studies. An athlete visits her fortutoring an average of every 1 hours/ and she spends an average o 1.$ hours7e#ponentially distriuted8 with him or her. 'he is ale to tutor only one athlete at a time

and athletes study while they are waiting.

a. Determine how long a player must wait to see her and the percentage of timeudith is usy. Does the :o seem to meet udiths e#pectations and does thesystem seem ade0uate to meet the athletes needs&

. If the results in 7a8 indicate the tutoring arrangement is ineffective/ suggest andad:ustment that could improve it for oth the athletes and udith.

queing sol taylor

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