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Averages and Probability

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The Wizard Tax Service is analyzing its operations during the month prior to the April 15 tax filing date. There is currently only one tax preparer at Wizard. On the basis of past data, it has been estimated that customers arrive according to a Poisson process with and average time between arrivals of 12 minutes. When the tax return is being completed, the customer is at the tax preparer's desk working with the preparer and this is considered the service time. The time to complete a return for a customer is exponentially distributed with a mean time of 10 minutes. Customers are processed in the order of arrival. Based on this information:

A. On average and measured from when a customer arrives at Wizard, how much time does the customer spend at the tax service having his tax return completed?

B. What is the average number of customers at the tax service, whether waiting for the tax preparer or having the return prepared?

C. An arriving customer will not wait if there are at least three others waiting for the tax preparer. What is the probability that the arriving customer will not wait?

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Arrival rate per hour: λ=(60/12)/hour=5/hour
Service rate: µ=1/(10/60)=6/hour, ...

Solution Summary

The averages and probability are examined. The expert determines how much time does the customer spend at tax services to have his taxes completed.

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Loading Dock Queuing

See the attachments.

Deliveries clogging the loading dock area
The specific details regarding the delivery process at the receiving dock.

1-Receiving dock open from 7am to 3pm, 8 hours, or 480 minutes.
2-The average number of arrivals on any given day is 28, which is 3.5 arrivals per hour, average.
3-The data we have collected indicate that we can unload 4.2 trucks per hour.
This should help you get started to think about how to approach this problem.

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References/ material to use/help
Queuing systems are "stochastic", which means based on random variables. The arrival rate of the customers is random but is theorized to follow a specific probability function. The key to analyzing queues is using the theory and equations that allow you to determine the probabilities
This website provides a good general overview of Queuing and waiting lines in business.
Queuing Theory. (n.d.). Encyclopedia of Business, 2nd Ed.; Reference for Business, retrieved from: http://www.referenceforbusiness.com/encyclopedia/Pro-Res/Queuing-Theory.html
Attached in the PowerPoint file named Queuing theory provides lecture notes on queuing and queuing equations. It also has Exercises for you if you need them
Also attached is the Excel file QueueCalc. There are two Tabs - the first is for Single Server models, and the second is for Multi-Server models. You enter the relevant information of a queuing problem and it will calculate the pertinent results. The values shown in this worksheet when you open it are the Phlebotomy Examples in the PowerPoint.
You can use the QueueCalc spreadsheet to try all of the examples and exercises in the PowerPoint if need be!
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Once you have mastered the examples and exercises; need your help in writing a report tackling the company's problem. And please use the QueueCalc for the problem when writing the report.
Use the above information and the Analyze this situation and determine what we need to know and give me a report. At this point in time, I am looking only for the problem to be quantified and the unload rate determined for the current situation (single server) and possible two servers.
The current situation is a Single Server situation. Enter the arrival rate and service rate to calculate the pertinent queuing system state data. Find out the probabilities of 5 or more trucks in the system, then 6, then 7. Then use trial and error to find the greatest number of trucks or less that can be in the system with 95% (or as close to 95%). For the Multi-server problem you will need to use a similar process.

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