Problems 1. Consider an database server designed to handle a maximum of 15 transactions per second. During the peak hour of its activity, transactions arrive at the average rate of 10 per second (10 s ¹). Assuming that the number of transactions arriving per second follows a Poisson distribution, compute the probability that the server will be overloaded during a peak hour.
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- What is the total effect on the economy of a government tax rebate of $500 to each household in order to stimulate the economy if each household will spend of the rebate in goods and services?A mechanic who works in the maintenance center of a large company and requires an average time of 4 hours to repair a machine, distributed exponentially. Damaged machines arrive in a queue based on a Poisson check-in process at an average rate of 0.25 per hour. If the mechanic used a more modern tool, the average repair time would be reduced to 2.5 hours (also exponentially distributed). If an hour of standby for a machine costs USD25, calculate the expected hourly savings that would result from using the latest tool.One of the important statistics of interest to all 4-year colleges and Universities is student retention rate. It is defined as “the percentage of a school's first-time, first-year undergraduate students who continue at that school the next year”. A very small 4-year college in the Midwest claims to have a retention rate of 85% and admitted 60 Freshmen in Fall 2020. Assuming the claim is true: After doing some research, you find out that this college admitted 70 freshmen in Fall 2019 and 65 of them continued on to their second year in Fall 2020. Are you inclined to believe their 85%retention claim? Justify with appropriate calculations.
- In this problem, assume Poisson arrivals and exponential service times.Customers of Plaka Record rental outlet arrive at the store to rent CDs with an arrivalrate of 1.25 customers per minute. The store has only 1 counter with a service rate of 2 customers per minute. Suppose that Plaka Record rental outlet adds another counter to serve thecustomers simultaneously.1. The average number of customers in the waiting line2. The average number of customers in the system3. The average time a customer spends in the waiting line. The average time a customer spends in the systemFrom 3:00 P.M. to 8:00 P.M. the local K-Star Supermarket has a steady stream of customers. Customers finish shopping and arrive at the checkout area at a rate of 70 per hour (Poisson distributed).It is assumed that when customers arrive at the cash registers, they will divide themselves relatively evenly so that all the checkout lines contain an equal number. The average checkout time at a register is 7 minutes (exponentially distributed). The store manager’s service goal is for customers to be out of the store within 12 minutes (on average) after they complete their shopping and arrive at a cash register. How many cash registers must the store have open to achieve the manager’s service goal?The problem of traffic delay in a highway can be studied in the following way. Let us assume that the probability of no traffic delay in one period, given no traffic delay in previous period, is 0.85 and that the probability of finding a traffic delay in one period, given a delay in the previous period is 0.75. The period of time is 30 minutes. Define the approach that should be used to solve the problem and identify all your inputs of the model.
- Suppose a group of consumers spend 30% of their disposable income on food, 20% on clothing, and 50% on rent. If over the course of a year the price of food rises 10%, the price of clothing drops 5% and rent rises 15%, what is the average price increase experienced by these consumers?According to scientists, the cockroach has had 300 million years to develop a resistance to destruction. In a study conducted by researchers, 5,000 roaches (the expected number in a roach-infested house) were released in the test kitchen. One week later, the kitchen was fumigated and 15,693 dead roaches were counted, a gain of 10,693 roaches for the 1-week period. Assume that none of the original roaches died during the 1-week period and that the standard deviation of x, the number of roaches produced per roach in a 1-week period, is 1.6. Use the number of roaches produced by the sample of 5,000 roaches to find a 95% confidence interval for the mean number of roaches produced per week for each roach in a typical roach-infested house. Find a 95% confidence interval for the mean number of roaches produced per week for each roach in a typical roach-infested house.According to scientists, the cockroach has had 300 million years to develop a resistance to destruction. In a study conducted by researchers, 5,000 roaches (the expected number in a roach-infested house) were released in the test kitchen. One week later, the kitchen was fumigated and 18,857 dead roaches were counted, a gain of 13,857 roaches for the 1-week period. Assume that none of the original roaches died during the 1-week period and that the standard deviation of x, the number of roaches produced per roach in a 1-week period, is 1.4. Use the number of roaches produced by the sample of 5,000 roaches to find a 90% confidence interval for the mean number of roaches produced per week for each roach in a typical roach-infested house. Find a 90% confidence interval for the mean number of roaches produced per week for each roach in a typical roach-infested house. (Round to three decimal places as needed.)
- A firm that operates a large, direct-to-consumer sales force would like to implement a system to monitor the progress of new agents. A key task for agents is to open new accounts; an account is a new customer to the business. To build such a system, the firm has been monitoring sales of new agents over the past 2 years. The response of interest is the profit to the firm (in dollars) of contracts sold by agents over their first year. Among the possible predictors of this performance is the number of new accounts developed by the agent during the first 3 months of work.Two geysers that are near to each other are labelled as Geyser A and Geyser B. The time between eruptions for a given geyser follows an exponential distribution. The eruption times of the two geysers are independent of one another. The mean amount of time between eruptions of Geyser A is 39 minutes, and the mean amount of time between eruptions of Geyser B is 66 minutes. Given that exactly one of the geysers erupts within the next 49 minutes, determine the probability that it will be Geyser A. O 0.6654 0.7850 O 0.7252 O 0.6953 O 0.7551Vehicles arrive at a car detailing service at a rate of 10 per minute according to a Poissondistribution. For simplicity, assume that there is only one lane and one worker who can serve atan average rate of 12 vehicles per minute. Service times are exponentially distributed. Part A: What is the average length of the queue? Part B: What is the average time a vehicle must spend to get through the system? Part C: What is the utilization rate of the worker? Part D: What is the probability that when you arrive at the shop, there will be three ormore vehicles ahead of you?
