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.

Check the explanation

Step-by-step explanation:

First define the below events

Delayt : Traffic delay in current period

Delayt-1 : Traffic delay in preceding period

No-Delayt : No traffic delay in current period

No-Delayt-1 : No traffic delay in preceding period

Based on above definitions, define below given probabilities

P(No-Delayt/No-Delayt-1) = Probability of "no delay" in current period given that "no delay" in preceding period

= 0.85

P(Delayt/Delayt-1) = Probability of "delay" in current period given that "delay" in preceding period

= 0.75

and given that one period = 30 minutes

a) Given that motorist 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 "delay" state.

Period A Period B

Delay > Delay > Delay

Since both the delay events should occur together(consecutive) to get the required probability, hence

Probability that next 60 minutes(two consecutive periods) will be in delay state

= P(Delayt/Delayt-1) * P(Delayt/Delayt-1)

= 0.75 * 0.75

= 0.5625 [ANSWER]

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

First calculate the probability of currren 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,

  P(All the 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. Because this assumption assumes that for the 30 minutes traffic will be in delay state and just after 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 modeled as a function of time, instead of being constant for 30 minutes.