Waiting Lines CrossChek Sporting Goods’ Golf Division gets requests-for- quotes for custom-made golf clubs at a rate of about 4 per hour. Each of these.

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Presentation transcript:

Waiting Lines CrossChek Sporting Goods’ Golf Division gets requests-for- quotes for custom-made golf clubs at a rate of about 4 per hour. Each of these requests is individually assessed by a sales clerk, who then answers with a quote. It takes a clerk an average of 10 minutes to answer a request for a quote. What is the probability of 3 requests arriving in an hour? What is the probability of at least 2 requests arriving in an hour? What is the probability that it will take less than 10 minutes to process a quote request? What is the probability that it will take between 10 and 12 minutes to process a quote request?

Poisson Distribution Models distribution of numbers of arrivals during a specified period Few possible outcomes less than the mean, but higher likelihoods

Exponential Distribution Models distribution of service times Low outcomes more likely, very high outcomes still possible

Back to CrossChek’s Problem… CrossChek Sporting Goods’ Golf Division gets requests-for-quotes for custom-made golf clubs at a rate of about 4 per hour. Each of these requests is individually assessed by a sales clerk, who then answers with a quote. It takes a clerk an average of 10 minutes to answer a request for a quote. What is the probability of 3 requests arriving in an hour? What is the probability of at least 2 requests arriving in an hour? What is the probability that it will take less than 10 minutes to process a quote request? What is the probability that it will take between 10 and 12 minutes to process a quote request?

Back to CrossChek’s Problem… CrossChek Sporting Goods’ Golf Division gets requests-for-quotes for custom-made golf clubs at a rate of about 4 per hour. Each of these requests is individually assessed by a sales clerk, who then answers with a quote. It takes a clerk an average of 10 minutes to answer a request for a quote. What is the probability of 3 requests arriving in an hour? What is the probability of at least 2 requests arriving in an hour? What is the probability that it will take less than 10 minutes to process a quote request? 0.63 What is the probability that it will take between 10 and 12 minutes to process a quote request? 0.07

More Questions What is the average number of requests waiting to be processed? What is the average number of requests in the system? What is the average waiting time? What is the average time spent in the system? What is the probability that a request needs to wait before being processed? What is the probability that there are no requests in the system? What is the probability that there are 4 requests in the system? 7?

Fomulas QuestionNotationFormula Average number of customers in the waiting line LqLq L q = 2 /[µ(µ - )] Average number of customers in the system L L = L q + /  Average waiting timeWqWq W q = L q / Average time in the systemWW = W q + 1/µ Probability that a customer has to wait PwPw P w = /  Probability that there are no customers in the system P0P0 P 0 = 1 - /  Probability that there are n customers in the system PnPn P n = P 0 ( /  ) n

More Questions What is the average number of requests waiting to be processed? What is the average number of requests in the system? What is the average waiting time? What is the average time spent in the system? What is the probability that a request needs to wait before being processed? What is the probability that there are no requests in the system? What is the probability that there are 4 requests in the system? 7? QuestionNotationFormula Average number of customers in the waiting line LqLq L q = 2 /[µ(µ - )] Average number of customers in the system L L = L q + /  Average waiting timeWqWq W q = L q / Average time in the system WW = W q + 1/µ Probability that a customer has to wait PwPw P w = /  Probability that there are no customers in the system P0P0 P 0 = 1 - /  Probability that there are n customers in the system PnPn P n = P 0 ( /  ) n

More Questions What is the average number of requests waiting to be processed? 1.33 What is the average number of requests in the system? 2 What is the average waiting time? 20 min What is the average time spent in the system? 30 min What is the probability that a request needs to wait before being processed? 0.67 What is the probability that there are no requests in the system? 0.33 What is the probability that there are 4 requests in the system? 7? 0.066, 0.020

In-Store Service Suppose that CrossChek Sporting Goods’ uptown location receives customers at an average rate of 10 per hour It takes CrossChek an average of 5 minutes to serve each customer CrossChek has observed that if the line at the checkout has 4 or more people (including the one being served), new arrivals will walk out the door without purchasing anything with 0.3 probability CrossChek estimates the cost associated with a customer leaving to be $25 CrossChek is considering adding a new staff member at the cost of $28 per hour, that is expected to reduce the average service time to 3 minutes Should CrossChek hire?

In-Store Service Suppose that CrossChek Sporting Goods’ uptown location receives customers at an average rate of 10 per hour It takes CrossChek an average of 5 minutes to serve each customer CrossChek has observed that if the line at the checkout has 4 or more people (including the one being served), new arrivals will walk out the door without purchasing anything with 0.3 probability CrossChek estimates the cost associated with a customer leaving to be $25 CrossChek is considering adding a new staff member at the cost of $28 per hour, that is expected to reduce the average service time to 3 minutes Should CrossChek hire? Yes, costs are decreased by $31.25 per hour