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A traffic light is red for 20 seconds and green for 40 seconds. Cars arrive at the traffic light as a Poisson process with rate 2 per minute. They take exactly 20 seconds to go through the light. A car cannot start going through until the previous one has cleared the light. Assuming, as an initial condition, that there are no cars at the traffic light when it turns red:

A traffic light is red for 20 seconds and green for 40 seconds. Cars arrive at the traffic light as a Poisson process with rate 2 per minute. They take exactly 20 seconds to go through the light. A car cannot start going through until the previous one has cleared the light. Assuming, as an initial condition, that there are no cars at the traffic light when it turns red:

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter12: Queueing Models
Section12.5: Analytic Steady-state Queueing Models
Problem 21P
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A traffic light is red for 20 seconds and green for 40 seconds. Cars arrive at the traffic light as a Poisson process with rate 2 per minute. They take exactly 20 seconds to go through the light. A car cannot start going through until the previous one has cleared the light. Assuming, as an initial condition, that there are no cars at the traffic light when it turns red:
a. What is the probability that at least one car will arrive when the light is still red?
b. What is the probability that there will be no cars at the light when the light next turns red?

A traffic light is red for 20 seconds and green for 40 seconds. Cars arrive at the traffic light as a Poisson process with rate 2 per minute. They take exactly 20 seconds to go through the light. A car cannot start going through until the previous one has cleared the light. Assuming, as an initial condition, that there are no cars at the traffic light when it turns red: a. What is the probability that at least one car will arrive when the light is still red? b. What is the probability that there will be no cars at the light when the light next turns red?
Transcribed Image Text:A traffic light is red for 20 seconds and green for 40 seconds. Cars arrive at the traffic light as a Poisson process with rate 2 per minute. They take exactly 20 seconds to go through the light. A car cannot start going through until the previous one has cleared the light. Assuming, as an initial condition, that there are no cars at the traffic light when it turns red: a. What is the probability that at least one car will arrive when the light is still red? b. What is the probability that there will be no cars at the light when the light next turns red?
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