Question

Problem 16-05 (Algorithmic)

A major traffic problem in the Greater Cincinnati area involves traffic attempting to cross the Ohio River from Cincinnati to Kentucky using Interstate 75. Let us assume that the probability of no traffic delay in one period, given no traffic delay in the preceding period, is 0.8 and that the probability of finding a traffic delay in one period, given a delay in the preceding period, is 0.65. Traffic is classified as having either a delay or a no-delay state, and the period considered is 30 minutes.

1. Assume that you are a motorist entering the traffic system and receive a radio report of a traffic delay. What is the probability that for the next 60 minutes (two time periods) the system will be in the delay state? Note that this result is the probability of being in the delay state for two consecutive periods. If required, round your answer to three decimal places.
   
   
   
2. What is the probability that in the long run the traffic will not be in the delay state? If required, round your answers to three decimal places.
   
   
   
3. An important assumption of the Markov process model presented here has been the constant or stationary transition probabilities as the system operates in the future. Do you believe this assumption should be questioned for this traffic problem? Explain.
   
   The input in the box below will not be graded, but may be reviewed and considered by your instructor.
   

Answer 1

First let us define the below events
Delayt : Traffic delayed in current period
Delayt-1 : Traffic delayed in preceding period
No-Delayt : No traffic delayed in current period
No-Delayt-1 : No traffic delayed in preceding period
Based on the above definitions, define below given probabilities
P(No-Delayt/No-Delayt-1) = Probability of "no delay" in the current period given that "no delay" in preceding period
= 0.85
P(Delayt/Delayt-1) = Probability of "delay" in the current period given that "delay" in preceding period
= 0.65
and given that one period is equal to 30 minutes

a) Given that motorist has already knows that there is a traffic delay, probability of delay in next 60 minutes will be possible if both the consecutive periods is in the "delay" state.
Period A Period B

Since both delay events should occur together(consecutive) to get required probability, hence
Probability that next 60 minutes (two consecutive periods) will be in the delay state
= P(Delayt/Delayt-1) * P(Delayt/Delayt-1)
= 0.65 * 0.65
= 0.4225 [ANSWER]

(b) First of all calculate the probability of the current period in delay state given that preceding period in No-delay state
P(Delayt/No-Delayt-1) = 1 - P(No-Delayt/No-Delayt-1)
= 1 - 0.85
= 0.15
Now the required probability is,
P(All time No-Delay state) = 1 - P( one period in delay state)
= 1 - ( P(Delayt/Delayt-1) + P(Delayt/No-Delayt-1) )
= 1 - (0.65 + 0.15)
= 0.2  [ANSWER]

c) Yes. Since this suspicion expect that for 30 minutes traffic will be in delay state and soon after the 30 minutes, there will be No-postpone state which is a hypothetical circumstance. When all is said in done situation, traffic doesn't carry on like this. To make the model increasingly sensible, the probabilities may have been demonstrated as a component of time, rather than being steady for the 30 minutes