Phone Strategy

Bengt is doing his exercise routine. It takes quite a while, mostly because he is too lazy to work very hard, so he doesn't get tired very quickly. In fact, his so called exercises mostly consist of posing, because he has seen that body builders do that a lot. It takes a predetermined time before Bengt gets tired, but there is no way to know in advance how long that is.
For some reason, Bengt has several friends. Now Sture wants to see him, but he knows that he has to wait until Bengt is finished with his exercise routine. It wouldn't do to disturb Bengt in person while he's busy with something so important, so Sture will phone him instead. The more often he calls, the sooner he will find out that Bengt has finished. Unfortunately, whenever he calls, Bengt has to take a break for one minute to answer the phone, so the time until he is finished is delayed by one minute. Sture now has to come up with the best strategy for getting a hold of Bengt as soon as possible.

1. Suppose Sture knows that Bengt always exercises for an hour, but he doesn't know how long it has been since he started. At which times should Sture call?
   
2. Suppose instead that Sture does know when Bengt started, but that the exercise varies. Sture has determined from experience that the exercising takes one hour on average, and that the probability that he will finish in the next short moment remains constant. If he doesn't call, after how long is the probability 1/2 that Bengt has finished?
   
3. With those conditions, at which times should he call?
   
4. Suppose instead that the probability density of finishing decreases, so that after a time t (not including phone pauses) the probability that he is still exercising is 1/2^t. What is then the probability density?
   
5. In that case, when should Sture call?
   
6. Suppose instead that Sture has already determined that the best times to call is once every hour. What is the probability density?
   
7. And suppose it is instead best to call after 2^n minutes, for non-negative integer n. What is the probability density?
   
8. After all, Bengt has more than one friend. With the conditions in question a, suppose Alban is also trying to call Bengt. Neither knows when Bengt started, nor when the other person called. Each wants to reach Bengt as soon as possible. As soon as one of them reaches Bengt, the other person has to wait another time p before Bengt is finished with the first person. What is the optimal phone strategy this time?
   
9. And what if we do the same thing with the conditions from question b?