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A truck driver is waiting to join the traffic on the service road of a highway. The truck driver blocks the service road for four seconds when he joins the traffic. Suppose that vehicles arrive according to a Poisson process and that six seconds elapse, on average, between two consecutive vehicles. Answer the following questions by assuming that the truck driver makes sure that he has enough time to perform his maneuver before merging into the traffic. (a) What is the probability that the truck driver is able to join the traffic immediately on his arrival at the intersection with the service road? (b) What is the mathematical expectation of the gap (in seconds) between the truck and the nearest vehicle when the truck driver merges into the traffic? (c) Calculate the average number of vehicles that the truck driver must let go by before being able to merge into the traffic.

A truck driver is waiting to join the traffic on the service road of a highway. The truck driver blocks the service road for four seconds when he joins the traffic. Suppose that vehicles arrive according to a Poisson process and that six seconds elapse, on average, between two consecutive vehicles. Answer the following questions by assuming that the truck driver makes sure that he has enough time to perform his maneuver before merging into the traffic. (a) What is the probability that the truck driver is able to join the traffic immediately on his arrival at the intersection with the service road? (b) What is the mathematical expectation of the gap (in seconds) between the truck and the nearest vehicle when the truck driver merges into the traffic? (c) Calculate the average number of vehicles that the truck driver must let go by before being able to merge into the traffic.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 67E
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A truck driver is weiting to join the traffic on the service road of a highway. The truck driver blocks the service road for four seconds when be joins the traffic. Suppose that vehicles arrive according to a Poisson process and that six seconds elapse, on average, between two consecutive vehicles. Answer the following questions by assuming that the truck driver makes sure that he has enough time to perform his nmaneuver before merging into the traffic. (a) What is the probability that the truck driver is able to join the traffic immediately on his arrival at the intersection with the service road? (b) What is the mathematical expectation of the gap (in seconds) between the truck and the nearest vehicle when the truck driver merges into the traffic? (c) Calculate the average number of vehicles that the truck driver must let go by before being able to merge into the traffic.
Transcribed Image Text:A truck driver is weiting to join the traffic on the service road of a highway. The truck driver blocks the service road for four seconds when be joins the traffic. Suppose that vehicles arrive according to a Poisson process and that six seconds elapse, on average, between two consecutive vehicles. Answer the following questions by assuming that the truck driver makes sure that he has enough time to perform his nmaneuver before merging into the traffic. (a) What is the probability that the truck driver is able to join the traffic immediately on his arrival at the intersection with the service road? (b) What is the mathematical expectation of the gap (in seconds) between the truck and the nearest vehicle when the truck driver merges into the traffic? (c) Calculate the average number of vehicles that the truck driver must let go by before being able to merge into the traffic.
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