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Question 5: An automated parking garage has a specific protocol for parking cars using a robotic system. The garage parks cars in batches of three. Once three cars are ready to be parked, the system moves them into parking spots simultaneously. New cars cannot enter the parking system until the current batch has been fully parked. Cars arrive at the garage entrance according to a Poisson process with a rate of $\lambda = 2$ cars per hour. The parking process for each batch of cars is exponentially distributed with a mean time of 1/3 hour. a. Develop a continuous time Markov chain model by defining the state, the state-space and drawing the transition diagram. b. Calculate the steady state distribution. c. Determine the probability that the parking system is idle in the long run d. Determine the fraction of time that there is at most two cars in the system waiting to be parked in the long run. e. What is the probability that a car is blocked from entering the garage on arrival because the garage is currently parking a batch in the long-run?

Name: question 5 an automated parking garage has a specific protocol for parking cars using a robotic system the garage parks cars in batches of three once three cars are ready to be parked the sy 86783
Uploaded: 2023-08-21T11:50:22-08:00
Duration: 3 min 32 s
Channel: Madhur L
Description: question 5 an automated parking garage has a specific protocol for parking cars using a robotic system the garage parks cars in batches of three once three cars are ready to be parked the sy 86783

          Question 5: An automated parking garage has a specific protocol for parking cars using a robotic
system. The garage parks cars in batches of three. Once three cars are ready to be parked, the system
moves them into parking spots simultaneously. New cars cannot enter the parking system until the
current batch has been fully parked. Cars arrive at the garage entrance according to a Poisson process
with a rate of $\lambda = 2$ cars per hour. The parking process for each batch of cars is exponentially distributed
with a mean time of 1/3 hour.
a. Develop a continuous time Markov chain model by defining the state, the state-space and
drawing the transition diagram.
b. Calculate the steady state distribution.
c. Determine the probability that the parking system is idle in the long run
d. Determine the fraction of time that there is at most two cars in the system waiting to be parked
in the long run.
e. What is the probability that a car is blocked from entering the garage on arrival because the
garage is currently parking a batch in the long-run?

question 5 an automated parking garage has a specific protocol for parking cars using a robotic system the garage parks cars in batches of three once three cars are ready to be parked the sy 86783

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