Question

The emergency telephone (911) center in a large city receives an average of 120 calls per hour during a typical day. On average, each call requires about 121 seconds for a dispatcher to receive the emergency call, determine the nature and location of the problem, and send the required individuals (police, firefighters, or ambulance) to the scene. The center is currently staffed by 4 dispatchers a shift but must have an adequate number of dispatchers on duty and it has asked a consultant to determine a proper level of staffing. The consultant assumes that the calls come at random over time, so that the Poisson probabilities apply.

How many calls should the emergency call center expect in a 10-minute period? Please show how you arrived at your answer.                             [2 points]

Define X to be the number of telephone calls to the emergency call center in a 10-minute period and assume that X has a Poisson Probability distribution. Please calculate the probability that the call center will receive 20 calls in the 10-minute period.                                                                          [2 points]

Please calculate the probability that the call center will receive fewer than 15 calls in the 10-minute period. Show your work.                        [2 points]

Please calculate the probability that the call center will receive more than 28 calls in the 10-minute period. Show your work.                        [2 points]

Please calculate the probability that the call center will receive not more than 30 calls in the 10-minute period. Show your work.                        [2 points]

Please calculate the probability that the call center will receive at least 25 calls in the 10-minute period. Show your work.                                [2 points]

Calculate and interpret the standard deviation of X.                  [2 points]   

For what purpose can the emergency call center use the kind of information you have calculated above? If the call center wants to limit the mean caller waiting time to less than 2 seconds, do you think they should add a fifth dispatcher to each shift? Please show how you arrived at your answer. [Note: the caller waiting time in this problem is the time between the emergency telephone starts ringing and the time a dispatcher picks it up.]                              [4 points]   

Answer 1

1 hour = 60 minutes

The emergency call center expect an average of 120 calls per hour during a typical day so number of call expected in 10 minutes period is

The  probability that the call center will receive 20 calls in the 10-minute period is

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The probability that the call center will receive fewer than 15 calls in the 10-minute period is

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The probability that the call center will receive more than 28 calls in the 10-minute period is

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The probability that the call center will receive not more than 30 calls in the 10-minute period is

The probability that the call center will receive at least 25 calls in the 10-minute period is

The standard deviation is: