Hi all, I was wondering whether anyone could explain to me how to build the transition matrix for this question. Abit confused by it? Cheers
Hello You'll need to use the probabilities calculated in part (i). For example, the probability of transitioning from 0 passengers waiting to 1 passenger waiting is just the probability of 1 passenger arriving, which is 0.2707. As pods can take up to 2 passengers, the first three rows are just the arrival probabilities calculated in part (i). If there are 2 passengers or fewer, all of them are taken by the pod. So, the number of new passengers waiting when the next pod arrives is therefore just how many passengers appear in the 2 minute period before the next pod. The probabilities associated with the number of arrivals in the 2 minute window are exactly those from part (i). Similarly, if there are 4+ passengers, then all of them are taken by the pod and / or taxis. So, again, the number of passengers waiting for the next pod is the number of arrivals in the 2 minutes before it arrives. The probabilities here are therefore the same as the first three rows. If there are exactly 3 passengers waiting when a pod arrives, then 2 of the passengers get taken and 1 remains. So, the probability that there are 0 passengers waiting when the next pod arrives is 0 (as there is at least 1). The probability that there are i passengers waiting is the probability that there are i-1 arrivals (as, again, there is already one passenger waiting). So, for example, the probability that there are 3 passengers waiting is the probability that there are 2 arrivals, which is 0.2707. Hope that helps Andy
