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 0.65. 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.

Answer 1

a)0.75^ 2 = (3/4)^2 = 9/16 is the probability for delays in the next two periods; 0.5625

If the probability of not being in a deay state is p in two consecutive periods then we have

p = 0.85p +(1-0.75)(1-p) = 0.85p + 0.25 - 0.25p

p = 0.60p + 0.25

0.40p = 0.25

p = 0.25/0.40 = 5/8 = 0.625

b) Check the probability of no delay in the next period is 0.625(0.85) + 0.375(0.25)

which is 0.625 as the probability of no delay.

c) There is no way traffic conditions figure to be the same in the middle of the night as they are at rush hour. So no,the assumption of constant transition probabilitities would not be the same at all times.