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Math Help - Queuing Problem

  1. #1
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    Unhappy Queuing Problem

    Kirkland Backpackers
    Will and Helen Kirkland operate a successful outdoor-wear store in Colorado called Kirkland Backpackers. They stock mostly cold weather outdoor items such as hiking and backpacking clothes, gear, and accessories. They established an excellent reputation throughout Colorado for quality products and service. Eventually, Will and Helen noticed that more and more of their sales were to customers who did not live in the immediate vicinity but were calling in orders on the telephone. As a result, the Kirklands decided to distribute a catalog and establish a phone-order service. The order department consisted of five operators working eight hours per day from 10:00am to 6:00pm, Monday through Friday. For a few years the mail-order service was only moderately successful; the Packers just about broke even on their investment. However, during the holiday season of the third year of the catalog-order service, they were over-whelmed with phone orders. Although they made a substantial profit, they were concerned about the large number of lost sales they estimated they incurred. Based on information provided by the telephone company regarding call volume and complaints from customers, the Kirklands estimated they lost sales of approximately $100,000. Also, they felt they had lost a substantial number of old and potentially new customers because of the poor service of the catalog order department.
    Prior to the next holiday season, the Kirklands explored several alternatives for improving the catalog-order service. The current system includes the five original operators with computer terminals who work eight-hour days, five days per week. The Kirklands have hired a consultant to study this system, and she reported that the time for an operator to take a customer order is exponentially distributed with a mean of 3.6 minutes. Calls are expected to arrive at the telephone center during the six-week holiday season according to a Poisson distribution with a mean rate of 175 calls per hour. When all operators are busy, callers are put on hold, listening to music until an operator can answer. Waiting calls are answered on a FIFO basis. Based on her experience with other catalog telephone-order operations and data from Kirkland Backpackers, the consultant has determined that if Kirkland Backpackers can reduce customer call-waiting time to approximately one-half minute or less the company will save $135,000 in lost sales during the coming holiday season.
    Therefore, the Kirklands have adopted this level of call service as their goal. However, in addition to simply avoiding lost sales, the Kirklands believe it is important to reduce waiting time in order to maintain their reputation for good customer service. Thus, they would like for about 70% of their callers to receive immediate service.
    The Kirklands can maintain the same number of workstations and computer terminals they currently have and increase their service to 16 hours per day with two operator shifts running from 8:00am to midnight. The Kirklands believe when customers become aware of their extended hours the calls will spread out uniformly, resulting in a new call average arrival rate of 87.5 calls per hour (still Poisson distributed). This schedule change would cost Kirkland Backpackers approximately $11,500 for the six-week holiday season.
    Another alternative for reducing customer-waiting times is to offer weekend service. However, the Kirklands believe that if they do offer weekend service it must coincide with whatever service they offer during the week. In other works, if they have phone order service 8 hours per day during the week. They must have the same service during the weekend; the same is true with 16-hours-per-day service. They feel that if weekend hours differ from weekday hours it will confuse customers. If 8-hour service is offered seven days per week, the new call arrival rate will be reduced to 125 calls per hour at a cost of $3,600. If they offer 16-hour service, the mean call arrival rate will be reduced to 62.5 hours, at a cost of $7,200.
    Still another possibility is to add more operator stations. Each station includes a desk, an operator, a phone, and a computer terminal. An additional station that is in operation five days per week, 8 hours per day, will cost $2,900 for the holiday season. For a 16-hour day the cost per new station is $4,700. For 7-day service the cost of an additional station for 8-hour-per-day service is $3,800; for 16-hour-per-day service the cost is $6,300.
    The facility Kirkland Backpackers uses to hour its operators can accommodate a maximum of 10 stations. Additional operators in excess of 10 would require the Kirklands to lease, remodel, and wire a new facility, which is a capital expenditure they do not want to undertake this holiday season. Alternatively, the Kirklands do not want to reduce their current number of operator stations.
    Determine what order service configuration the Kirklands should use to achieve their goal, and explain your recommendation.
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  2. #2
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    Quote Originally Posted by ouazdi View Post
    Kirkland Backpackers
    Will and Helen Kirkland operate a successful outdoor-wear store in Colorado called Kirkland Backpackers. They stock mostly cold weather outdoor items such as hiking and backpacking clothes, gear, and accessories. They established an excellent reputation throughout Colorado for quality products and service. Eventually, Will and Helen noticed that more and more of their sales were to customers who did not live in the immediate vicinity but were calling in orders on the telephone. As a result, the Kirklands decided to distribute a catalog and establish a phone-order service. The order department consisted of five operators working eight hours per day from 10:00am to 6:00pm, Monday through Friday. For a few years the mail-order service was only moderately successful; the Packers just about broke even on their investment. However, during the holiday season of the third year of the catalog-order service, they were over-whelmed with phone orders. Although they made a substantial profit, they were concerned about the large number of lost sales they estimated they incurred. Based on information provided by the telephone company regarding call volume and complaints from customers, the Kirklands estimated they lost sales of approximately $100,000. Also, they felt they had lost a substantial number of old and potentially new customers because of the poor service of the catalog order department.
    Prior to the next holiday season, the Kirklands explored several alternatives for improving the catalog-order service. The current system includes the five original operators with computer terminals who work eight-hour days, five days per week. The Kirklands have hired a consultant to study this system, and she reported that the time for an operator to take a customer order is exponentially distributed with a mean of 3.6 minutes. Calls are expected to arrive at the telephone center during the six-week holiday season according to a Poisson distribution with a mean rate of 175 calls per hour. When all operators are busy, callers are put on hold, listening to music until an operator can answer. Waiting calls are answered on a FIFO basis. Based on her experience with other catalog telephone-order operations and data from Kirkland Backpackers, the consultant has determined that if Kirkland Backpackers can reduce customer call-waiting time to approximately one-half minute or less the company will save $135,000 in lost sales during the coming holiday season.
    Therefore, the Kirklands have adopted this level of call service as their goal. However, in addition to simply avoiding lost sales, the Kirklands believe it is important to reduce waiting time in order to maintain their reputation for good customer service. Thus, they would like for about 70% of their callers to receive immediate service.
    The Kirklands can maintain the same number of workstations and computer terminals they currently have and increase their service to 16 hours per day with two operator shifts running from 8:00am to midnight. The Kirklands believe when customers become aware of their extended hours the calls will spread out uniformly, resulting in a new call average arrival rate of 87.5 calls per hour (still Poisson distributed). This schedule change would cost Kirkland Backpackers approximately $11,500 for the six-week holiday season.
    Another alternative for reducing customer-waiting times is to offer weekend service. However, the Kirklands believe that if they do offer weekend service it must coincide with whatever service they offer during the week. In other works, if they have phone order service 8 hours per day during the week. They must have the same service during the weekend; the same is true with 16-hours-per-day service. They feel that if weekend hours differ from weekday hours it will confuse customers. If 8-hour service is offered seven days per week, the new call arrival rate will be reduced to 125 calls per hour at a cost of $3,600. If they offer 16-hour service, the mean call arrival rate will be reduced to 62.5 hours, at a cost of $7,200.
    Still another possibility is to add more operator stations. Each station includes a desk, an operator, a phone, and a computer terminal. An additional station that is in operation five days per week, 8 hours per day, will cost $2,900 for the holiday season. For a 16-hour day the cost per new station is $4,700. For 7-day service the cost of an additional station for 8-hour-per-day service is $3,800; for 16-hour-per-day service the cost is $6,300.
    The facility Kirkland Backpackers uses to hour its operators can accommodate a maximum of 10 stations. Additional operators in excess of 10 would require the Kirklands to lease, remodel, and wire a new facility, which is a capital expenditure they do not want to undertake this holiday season. Alternatively, the Kirklands do not want to reduce their current number of operator stations.
    Determine what order service configuration the Kirklands should use to achieve their goal, and explain your recommendation.
    You have a better chance of gettting a reply if you break your problem down into more easily read chunks and posted clear questions about what you can and can't do.

    Furthermore, this looks like it might be some sort of project that will contribute towards your final grade. If that is the case, then you should not be asking for help with it.
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  3. #3
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    No this is not a project, I am going back to college next fall, and I am trying to refresh my memory doing some problems, and I did have some problem with this one, and I want some help.
    I did break the problem as followed so tell me what you think:
    Information on current system:
    Five operators working eight hours a day, five days a week.
    Estimated lost sales of approximately $100,000.
    lamda = 175 customer / hr = 2.92 customer / min
    = 83.33 customer / hr = 1.40 customer / min
    Waiting calls are answered on a FIFO basis.
    Goals:
    Reduce customer call-waiting time to approximately one-half minute or less the company will save $135,000 in lost sales during the coming holiday season.
    Would like about 70% of their callers to receive immediate service.
    Alternatives for improving the catalog-order service
    1.Alternative 1: Increase their service to 16 hours per day with two operators. Result:

    lamda = 87.5 customer / hr = 1.46 customer / min
    = 83.33 customer / hr = 1.40 customer / min
    Cost $11,500 for the six-week holiday season.

    2.Alternative 2: Offer weekend service

    If 8-hour service is offered seven days per week, Result:
    lamda = 125 customer / hr = 2.08 customer / min
    = 83.33 customer / hr = 1.40 customer / min
    Cost: $3,600

    If they offer 16-hour service, result:
    lamda = 62.5 customer / hr = 1.04 customer / min
    = 83.33 customer / hr = 1.40 customer / min
    Cost: $7,200

    3.Alternative 3: add more operator stations to a maximum of 10 stations.
    An additional station for:
    5 days / 8 hours: cost $2,900
    5 days / 16 hour: cost $4,700.
    7 days / 8 hours: cost $3,800
    7 days / 16 hour: cost $6,300
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  4. #4
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    the Kirklands explored several alternatives for improving the catalog-order service. The current system includes the five original operators with computer terminals who work eight-hour days, five days per week. The Kirklands have hired a consultant to study this system, and she reported that the time for an operator to take a customer order is exponentially distributed with a mean of 3.6 minutes. Calls are expected to arrive at the telephone center during the six-week holiday season according to a Poisson distribution with a mean rate of 175 calls per hour. When all operators are busy, callers are put on hold, listening to music until an operator can answer. Waiting calls are answered on a FIFO basis.

    Please help I just want to know what is lamda ƛ, and u for this problem tha't' all i want
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