A major traffic problem in the Greater Cincinnati area involves traffic attempti

Question

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 0.85 and that the probability of finding a traffic delay in one period, given a delay in the preceding period, is 0.75. Traffic is classified as having either a delay or a no-delay state, and the period considered is 30 minutes.

a. 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.

b. What is the probability that in the long run the traffic will not be in the delay state?

c. An important assumption of the Markov process models presented in this chapter 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

Explanation / Answer

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

= 0.75

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

---------------- -----------------

Delay -------> Delay ------> Delay

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)

= 0.75 * 0.75

= 0.5625 [ANSWER]

b) Probability that in long run the traffic will not be in the delay state is equivalent to probability that all the time in No-delay state.

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 - (0.75 + 0.15)

= 0.10 [ANSWER]

c) Yes. Since this assumption assumes that for 30 minutes traffic will be in delay state and just after the 30 minutes, there will be No-delay state which is a hypothetical situation. In general scenario, traffic doesn't behave like this. To make the model more realistic, the probabilities might have been modelled as a function of time, instead of being constant for the 30 minutes.

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