queuingproblems-110227225126-phpapp01

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OPERES3

Problems in Queuing Theory

1. (Case 1) A computing system has a single printer attached to print out the output of the users. Theoperating system software sends an average of 20 requests per hour to the printer. The printer is

capable of printing out 35 jobs per hour on the average. Assume that the job arrival rate and theprinting rate are Poisson distributed.a. On the average, how many jobs are in the output queue waiting to be printed out?b. How many jobs are in the printer output system on the average?c. How long can a user expect to wait to have his or her job printed out once it has been sent to the

printer?d. Once a job enters the output queue, how long does it take before printing begins on the job?e. What percentage of time will the printer be busy doing print jobs?f. (Case 2) Assume that a second printer is added to the system. Answer questions 1.1 to 1.5,

assuming that the two printers are identical.

2. (Case 1) Telephone calls arrive at a particular small-scale paging system at the Poisson rate of 20 per

hour. The calls are transferred to an operator that can handle an average of 25 calls per hour,following an exponential distribution. The operator can handle only one caller at a time; if the operatoris busy, arriving calls are put on hold and the callers are assumed to be patient.a. Assume that no limit is placed on the number of calls that can be put on hold:

1) What percentage of the time is the operator busy?2) On the average, how many callers are on hold?3) On the average, how long can a caller expect to spend on the telephone when calling this

office?b. (Dont do this) Suppose that with the single operator, only a maximum of two calls can be put on

hold. How does this affect your answers in 2.1?c. (Case 2) Suppose that an additional operator is hired, with no limit on the number of calls on hold:

1) What percentage of the time are both operators busy?2) On the average, how many callers are on hold?

3) On the average, how long can a caller expect to spend on the telephone when calling thisoffice?

d. (Case 2) Suppose that the office management feels that, in order to provide good customer service,a caller should not have to wait more than 10 seconds on hold. How many telephone operatorsshould be hired to answer calls?

3. Orders for a firms products arrive at the rate of 200 a week. A single clerk is assigned to process theorders at an average of 250 orders per week. The cost of delaying an average shipment for 1 week isP500. If a clerk costs P300 per week, how many clerks should be assigned to process the paperwork?

4. Moonlight Travel Reservation Service has one agent and 3 telephone lines. When the agent is busy, thenext caller is placed on hold on one of the remaining lines. When all three lines are busy, callers

receive a busy signal and call elsewhere for reservations. Service appears to be exponentiallydistributed, with a mean of three minutes. Calls arrive randomly at a Poisson rate of 15/hour.a. What is the probability that a caller receives a busy signal?b. What is the average number of customers put on hold at any given time?c. What is the probability that a customer is placed on hold?d. How long, on the average, will a customer be on hold before the agent can attend to him?e. If a caller makes three attempts before finally calling elsewhere for reservations, what fraction of

the customers will be able to do business with Moonlight Travel Reservation Service?

5. (Case 1) Customers arrive at a one-person barber shop with an average interarrival time of 20 minutes.


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The average time for a haircut is 12 minutes, exponentially distributed.a. The owner wishes to have enough seats in the waiting area so that on the average no more than 5%

of the arriving customers will have to stand. How many seats should be provided?b. Suppose that there is only sufficient space in the waiting area for five seats. What is the

probability that an arriving customer will not find a seat?

6. A plant distributes its products by truck. The exponential average loading time per truck at the loadingfacility is 20 minutes. Trucks arrive at a rate of 2/hour. Management feels that the existing loadingfacility is more than adequate. However, the drivers complain that they have to wait more than half ofthe time. Analyze the situation and find how much money the company can save if the waiting time ofa truck is figured at P10/hr and the plant is in operation 8 hours per day. An automatic device that canload 10 trucks per hour is available at a cost of P90/day over the cost of the existing facility.

7. Campers arrive at the County Park at a Poisson rate of 100 per month. The time spent in the park by acamper varies according to an exponential distribution with a mean of 0.10 month. Since the park isvery large, campers have no problem finding a campsite. However, the state recreation officials have apolicy of maintaining a certain level of forest-ranger workforce for the convenience and safety of thecampers. The current policy is to have one ranger per 10 campers at any given time.a. How many campers are using the park on the average?

b. What percentage of the time is the park empty?c. How many forest rangers should the state employ at the County Park?

8. A gasoline station is served by 1 employee who is capable of serving 30 customers per hour, varyingaccording to an exponential distribution. There is a maximum space for five cars in the station,including those served and in line. Cars arrive at the station at an average rate of every 3 minutes.Cars that do not have parking spaces leave and do not return. Find:a. The average number of cars waiting for service.b. The probability of finding no car in the station.c. The average waiting time in line for each car.d. The probability of finding 3 cars in the station at any given time.e. It was proposed to increase the space so that 10 cars would be accommodated. The investment

would entail P5 per car space per hour. A car that leaves the station means a loss of P10. Shouldthe space be enlarged or not?

9. A shop utilizes 10 identical machines. The profit per machine is P40 per hour of operation. Eachmachine breaks down on the average once every 7 hours. One person can repair a machine in 4 hourson the average, but the actual repair time varies according to a Poisson distribution. The repairmanssalary is P60/hour. Determine:a. The number of repairmen that will minimize cost.b. The number of repairmen needed so that the expected number of broken machines at any given

time is less than 4.c. The number of repairmen needed so that the expected delay time until a machine is repaired is less

than 4 hours.

10. (Case 2) Students arriving at the computing center can wait either at a small table in the center (whichwill seat four students) or if the table is full they can go to a nearby student lounge or they can wait inthe hall. What percentage of time will the waiting area inside the computing center be sufficient toaccommodate the waiting students?

11. The city of Manila operates a large fleet of police squad cars. When a car breaks down due to a minormalfunction it is sent to a single-garage repair facility where it is repaired by a crew of mechanics. Thecrew foreman, Jim, must put in a request to hire her team mechanics. If n mechanics work on a singlecar together, repair time for minor repairs is approximately constant at 20/n minutes. Disabled squadcars arrive at the repair facility at the Poisson rate of five cars per hour. The police chief tells Joanne


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that on the average he does not want a squad car to be out of commission for longer than 15 minutes.a. What is the minimum number of mechanics Joanne should hire?b. On the average, how many squad cars are out of commission?

12. Solve Joannes facility design problem (exercise 11 above) under the assumption that the reapir time of20/n minutes is a mean exponential time. Answer both parts (a) and (b) of the problem.

13. The manufacturing Supervisor of Hardpress Industries must decide between three makers of forklifttrucks to use on the plant floor. The truck will be used to move raw materials from storage bins towork centers in a section of the plant. Request from these work centers occur according to a poissonprocess at a mean rate of eight per hour. The time to move raw materials is exponentially distributed.The service rates per hour for each of the truck under consideration is given in the table along withestimated costs per hour (operating costs plus depreciation). For each hour a work center is idlewaiting for raw materials, an average cost of P100 is incurred (wages plus lost productivity).

Table

Truck Service Rate (jobs/hr) Cost (P/hr)

1 10 30

2 15 50

3 18 40

Which truck would you purchase based on queueing analysis? (Be sure to specify clearly thequeueing model you are using.)

14. In exercise 13, would your purchase choice change if the Manufacturing Supervisor felt that theservice times of the forklift trucks were (more or less) constant times? That is, Truck 1 can servicework centers at the constant rate of 10 jobs per hour, Truck 2 at 15 per hour, and Truck 3 at 18 perhour. (What would be the total cost per hour of using each of the trucks?)

15. (Case 1) The state operates a weigh station for trucks traveling on the highway. Every truck must pulloff the highway, enter the weigh station, and undergo state inspection procedures before counting on.

Trucks arrive at this station at a poisson rate of seven per hour. The time to inspect/weigh a truckvaries, having an exponential probability density with a mean of 7 minutes.a. How many trucks are detained at the station on the average?b. How long can a truck expect to be detained?c. How many trucks are lined up in front of the station on the average?d. How long on the average does a truck driver have to wait in line for the truck to be inspected?

16. Refer to exercise 15, the truck driver traveling the route passing the weigh station have complained oflong waiting time at the inspection point. The major phase of the inspection procedure consists ofweighing-in the truck. The state highway commissioner has decided to look into ways of reducing theaverage time a trucker must spend at the weigh station to 25 minutes or less. The commissioner isconsidering the purchase of new digital solid waste weighing scale that should reduce the weighingtime. Three manufacturer products are available: (1) model 1 costs P10000 and could check 10

trucks per hour, (2) model 2 costs P12000 and would have an average service time of 5.5 minutes, and(3) model 3 costs P8500 and would have an average service time of 6.5 minutes.a. Determine which model to purchase.b. For the decision made in part (a), determine system operating characteristics P0, L, Lq, W and Wq.

17. A recent increase in consumer complaints at Jersey Electric Authority has led to a decision to assignone person to receive the complaints. Complaints are phoned in at a rate of 20 per hour and arepoisson distributed. Call can handled at a rate of 40 per hour. Service times vary considerably andhave been found to be exponentially distributed. Jersey Electric would like to make sure that no morethan 5% of the calls receive busy signals. If the operator is busy, calls can be placed on a hold line.


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d. A students arrives at 4:58 pm. The closing time is strictly set at 5:00 pm. Can we expect thestudent to finish the whole registration process?

24. Machinist at a firm periodically need a wide range of special purpose tools, which are stored at 3locations on the plant floor. Each tool room is staffed by one clerk, who checks the tools in and out.Arrivals at each tool room are poisson distributed with a mean of 15 employees per hour. Servicetimes are exponential and average 3 minutes. It has been proposed that the 3 tool rooms beconsolidated into a single centrally located facility staffed by 3 clerks. Travel time it takes for amachinist to walk to and from the work area to a tool room would be increased by an average of 2minutes. Each hour that a machinist spends away from the work area has been estimated to cost P10per hour. Service costs will not be reduced by the move to a single tool room. Evaluate the proposal.

25. Superchopper Supermarket operate two types of checkout lanes, an express lane (for purchases lessthan or equal to 8) and a regular lane. The person at the express lane cash register can check out 15customers per hour and a person working at the regular check out lane can service 11 customers perhour. The average arrival rate of customers at the checkout area is 15 per hour, 60% of these haveover 8 grocery items. Assume poisson arrival and exponential service rate. Analyze the steady statesystem behavior.

26. A service system has an average inter arrival time of two minutes, poisson distributed. The system hasone server. Determine and compare the steady state queueing statistics under the following cases:I. Service time is constant at 1 minute.II. Service time has a continuous density function

f(t) = t/2 0 < t < 2III. Service time is normal with a mean of 1 minute. It takes more than 1.5 minutes to service in

about 1% of the time.

27. Suppershopper Inc. has 20 customers entering per hour. The customer spends an exponentiallydistributed amount of time shopping. The average time spent shopping is 15 minutes.a. How many customers will you expect to find shopping in the aisles?b. What is the probability of 8 customers entering during the period 1:00-1:15 pm?c. What is the probability of finding at least 3 customers in the aisles in at least one of two observed

times?

28. A barber shop has 3 barbers and 2 additional chairs for waiting customers. Customers arrive at anaverage rate of 8/hour and haircuts take an average of 15 minutes with an exponential distribution.Arrivals who find the barber shop busy and 2 customers waiting leave at once.a. What fraction of the time are the barbers idle?b. Determine the mean time a customer spends in the barber shop.

29. A small store has a parking lot of 6 spaces. Potential customers arrive at a rate of 10/hour and leaveimmediately if no parking space is available. Occupancy of parking space is 30 min. on the averageand tends to be exponential.a. What is the probability that a customer leaves at once?

b. What is the average fraction of empty spaces?

30. Mildreds Tools and Die Shop has a central tool cage manned by a single clerk, who takes an averageof 5 minutes to check and carry parts to each machinist who requests them. The machinists arrive onceevery 8 min. on the average. Times between arrivals and for service are assumed to both haveexponential distributions. A machinists time is valued at P15 per hour; a clerks time is valued at P9per hour. What are the average hourly queueing system costs associated with the tool cage operation?

31. C.A. Gopher and Sons is to excavate a site from which 100,000 cubic yards of dirt must be removed.Gopher has the choice of using a scoop loader or a shovel crane. A scoop loader costs P40 per hour,


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and a shovel crane costs P60 per hour. Once work has been started, the trucks will arrive according toa poisson process at a mean rte of 7 trucks per hour. Truck-filling times are approximatelyexponentially distributed. A scoop loader can fill an average of 10 trucks per hour. The shovel craneis faster and is capable of filling 15 trucks per hour on the average, (For simplicity, we will assumethat the truck arrival rate is the same, regardless of the equipment used.)

Since the number of truck arrivals required to excavate the site is fixed by the amount of dirt to beremoved, the optimal choice of filling equipment will be the one that minimizes the combined averagehourly costs of unproductive truck time plus the cost of the filling equipment. Determine the optimalchoice.

32. The manager of a Way Safe market with 10 checkout counters wishes to determine how many countersto operate on Saturday morning. His decision will be determined in part by the costs assigned to eachadditional minute that a customer spends checking out of the store. In a special experiment in obsoletestores about to be closed, customers were forced to wait in line for an abnormally long time. The studyconcluded that an average of P0.05 in future profits is lost for every minute that a customer spendswaiting.

Assume that Way Safe customer arrivals at the checkout area are approximated by a poisson process

with a mean rate of 2 per minute, that each attendant can check out customers at a mean rate of 0.5 perminute, and that the service time is exponentially distributed.a. What is the minimum number of checker required for the service capacity to exceed the demand

for service? Determine the mean customer waiting time when that many checkers are providingservice.

b. What is the mean customer waiting time is one more checker is added?c. The salary expense for each checker is P10 per hour. What number of checkers will minimize the

total hourly queueing system cost the number in (a) or in (b)? (Ignore service time, since nopenalty applies for actual checkout time spent.)

33. A toll gate at the Highway is staffed by one attendant. The time it takes him to collect the toll from acar is exactly 30 seconds. Cars arrive at the toll booth at a poisson rate of 75 cars per hour.a. What percentage of the time is the attendant free to read magazines?b. On the average, how many cars are lined up on the highway waiting to approach the toll booth?c. What is the expected number of cars tied up at the bridge due to the toll gate bottleneck?d. What is the average length of time a traveler can expect to wait before paying the toll/e. How long on the average is a car detained on the highway due to the toll gate operation?

34. A manufacturing facility operates a central tool crib for its machinists. The crib operates like an(M/M/s) system. Arrivals occur at an average rate of 30 machinists requiring tools per hour. Eachworker behind the tool counter can service the requests of 15 machinists per hour on the average and ispaid an hourly wage of P8.50. The machinists wages run P15.75 per hour on the average.a. How many tool crib attendants should be hired?b. For the system resulting from your analysis of part (a), determine the following, how many

machinists are idle (at the tool crib) on the average? How long can a machinist expect to spend at

the tool crib?

35. Patients arrive at a medical clinic at a poisson rate of 10 per hour. The small clinic is staffed by onedoctor, assisted by several nurses. Patients wait in the waiting room until the doctor can see them. Thetime the doctor spends with a patient has an exponential probability density with a mean of 5 minutes.a. How many patients are in the clinic on the average?b. How long must a patient wait before seeing the doctor?c. How long can a patient expect to spend in the clinic?d. On the average how many patients are in the waiting room?


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36. Students arrive at the universitys computer center at a poisson rate of 20 per hour. The center has sixon-line terminals for student use. The time a student spends at a terminal appears to follow anexponential probability density with a mean of 0.20 hours (12 minutes).a. On the average, how many students are at the computing system?b. How many students are waiting to use a terminal?c. How long can a student expect to spend at the center?d. What is the average wait time for a student before having access to a terminal?e. What percentage of time is the center busy?

37. A single crew is provided for unloading and/or loading each truck that arrives at the loading dock of awarehouse. These trucks arrive according to a poisson input process at a mean rate of one per hour.The time required by a crew to unload and/or load a truck has an exponential distribution (regardless ofthe crew size). The expected time required by a one-man crew would be 1 hour.

The cost of providing each additional member of the crew is P10 per hour. The cost is attributable tohaving a truck not in use (i.e., truck standing at the loading dock) is estimated to be P15/hour.a. Assume that the mean service rate of the crew is proportional to its size. What should the size be

to minimize the expected total cost per hour?b. Assume that the mean service rate of the crew is proportional to the square root of its size. What

should the size be to minimize expected total cost per hour?

38. A machine shop contains a grinder for sharpening the machine cutting tools. A decision must now bemade on the speed at which to set the grinder.The grinding time required by a machine operator to sharpen his cutting tool has an exponential

distribution, where the mean 1/ can be set at anything from to 2 minutes, depending upon the speed

of the grinder. The running and maintenance costs goes up rapidly with the speed of the grinder, so the

estimated cost per minute for providing a mean of 1/ is P(0.102).

The machine operators arrive to sharpen their tools according to a poisson process at a mean rate ofone every 2 minutes. The estimated cost of an operator being away from his machine to the grinder isP0.20/min.

Plot the total expected cost per minute E(TC) vs. over the feasible range for to solve graphically

for the minimizing value of.

39. Two repairmen are attending five machines in a workshop. Each machine breaks down according to apoisson distribution with a mean of 3 per hour. The repair time per machine is exponential with amean of 15 minutes.a. Find the probability that the two repairmen are idle. That one repairman is idle.b. What is expected number of idle machines not being repaired?


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Sample Problems for Queueing

1. Joe's Service Station operates a single gas pump. Cars arrive according to a Poisson distribution at anaverage rate of 15 cars per hour. Joe can service cars at the rate of 20 cars per hour with service timefollowing an exponential distribution.a. What fraction of the time is Joe busy servicing cars?b. How many cars can Joe expect to find at his station?c. What is the probability that there are at least 2 cars in the station?d. How long can a driver expect to wait before his car enters the service facility?e. If you need to put in gas at Joe's service station, how long will it take you, on the average?

2. In the previous example, Joe is planning to set an area of land near the service facility to park carswaiting to be serviced. If each car requires 200 square feet on the average, how much space would berequired?

3. Students arriving at a student registration center of a university must have their registration materialsprocessed by an operator seated at a computer terminal. The systems design calls for 4 operators to beon duty, each operator performing an identical service. Students arrive according to a Poisson processat an average rate of 100 per hour. Each operator can process 40 students per hour with service time

being exponentially distributed.a. What fraction of the time that there are no students in the registration center?b. If the waiting area inside the center building will comfortably accommodate five students, what

percentage of time will there be students lined up outside the building?c. How long does an average student spend in the center building?d. On the average, can we expect a student arriving 3 minutes before closing time to make it just

before closing? (He is able to register)e. How long is the line of students waiting to register?

4. Arrivals at the AAA Transmission Repair are observed to be random at the Poisson rate of 1.5 per day.The time required to repair transmission varies and the probability distribution is given below:

Service Time (days) Probability

0.25 0.274

0.50 0.548

0.75 0.110

1.00 0.068

Determine the steady state statistics of this system.

5. Supershopper Supermarket has 15 customers entering per hour. The customer spends an exponentiallydistributed amount of time shopping. The average time spent shopping is 20 minutes.a. On the average, how many customers can be found shopping in the aisles?b. What is the average length of time a customer spends shopping in the aisles?c. Find the probability of there being 6 or more customers in the aisles.

6. The owner of a small full-service gas station operates it by himself. On Friday afternoon, customers

arrive at the rate of 12 per hour. On the average, service takes 3 minutes and tends to be exponential.The space allows the accommodation of only 3 cars at one time (including the one being serviced).a. What is the mean number of customers in the station?b. What fraction of the time will the owner be idle?c. What fraction of customers will be lost?d. What is the mean customer waiting time?e. What is the probability that a customer waits?

7. Consider a barbershop with 3 barbers and 2 additional chairs for waiting customers. Customers arriverandomly at an average rate of 8 per hour, and haircuts take an average of 15 minutes, with anexponential distribution. Arriving customers who find the barbershop busy and 2 customers waiting


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leave immediately.a. What is the mean number of customers waiting at any time?b. What is the mean time a customer spends in the barbershop?c. Determine the mean number of customers in the barbershop?d. What fraction of the time are the barbers idle?

8. A firm operates a single-server maintenance storeroom where electrical repair persons check outneeded spare parts and equipment. Repair persons arrive at a rate of 8 per hour. The service rate is 10per hour. Arrivals are Poisson distributed and service completion times follow the exponential model.The cost of waiting is Php 9 per hour. The company is considering giving the stock clerk a helperwhich would increase the service rate to 12 per hour. The cost of the helper is Php 6 per hour. Whatdo you recommend?

9. The FNB has decided to operate its lobby service according to an (M/M/S) model. Customer arrive atthe bank at an average of 40 per hour. Each bank teller can serve 10 people per hour. How manybank teller should be hired so that the average time a customer spends waiting in the queue does notexceed 0.75 minutes?

10. A machine in a plant is designed to perform various operations. Process time of a job is exponential

with a mean of 5 minutes. Job arrivals are poisson with a mean of 7 per hour. Jobs waiting are placedin a storage area in front of machine. If this area fills up, the arriving jobs must be placed in an aislenext to a wall. Each waiting machine occupies 9 square feet. How much floor space should be plannedfor the storage area if it is to be adequate 80% of the time? (It is to include job being processed)

11. Workfor Metals purchases metal to be recycled from several sources. These metals consist of steelbars and sheets aluminum shapes, and iron plates. Truckload shipments of scrap metals arrive atWorkfor in more or less random pattern at an average rate of one truckload per day. These trucks areowned and operated by Workfor Metals. Each shipment must be unloaded by a crew of workers. Acombination of unloading operations is used: simple dumping, use of crane or forklift, and directmanual labor. A crew of n workers can unload 0.8n per day. For each day that a truck is detained inunloading, a cost of Php 300 is incurred (due to lost productive use of the truck and hourly wages oftrack driver and his assistant). Each worker in the unloading crew is paid Php 105 per day. Determinethe optimal number of crew members to hire, assuming the arrival and service rates are poisson.

12. An engineering design firm is replacing its scientific computer, which is sued to solve problemsencountered in design projects. The list of alternatives has been narrowed to 3 computers.

Computer Type Lease Cost/day No. of problems solved per day

A 600 20

B 750 25

C 1050 35

Computation times are exponential and arrivals of problems at the computer center follow a poissondistribution at mean rate if 15 per day. Progress on a project is delayed until a problem is solved. Thedaily cost of delaying a project is Php 250. Which computer should be selected?

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