There are 2 n possible sets which is far too large

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the set of buses that you want to take as soon as they arrive. There are 2 n possible sets, which is far too large for an efficient algorithm. Argue that you need only consider a small number of these sets.) Denote a strategy by S , the set of buses from which we will board the first bus that appears. An optimal strategy is one where the total expected time (wait time plus travel time) is minimal. Let X i exp (1 i ) as the problem states. Let the time of the first bus arrival be at time X = min { X i | i S } . We know from Lemma 8.5 that X exp ( i S 1 i ). After we have selected an arrival time for the first bus, then we can select bus i with probability 1 i P i S 1 i . Let T be the random variable for the travel time of that bus. For a given set S , the expected time is E S := E [ X + T ] = E [ X ] + E [ T ] = 1 i S 1 i + i S t i i i S 1 i . We want to choose the set S with the least expected time E S . Order the entire set of buses by their travel times in increasing order. We need only consider sets S that are a prefix of this list of buses. if bus i is not considered in set S , then all buses j with t j t i would also not be considered. Conversly, if bus i is considered in set S , then any bus j with less travel time t j t i would also be considered. Therefore, we want to find the set S ∈ {{ 1 } , { 1 , 2 } , ..., { 1 , 2 , ..., n }} results in the lowest expected wait time E S . Simply compute these values and pick the set S with the smallest E S . 6. (MU 8.21) The Ehrenfest model is a basic model used in physics. There are n particles moving randomly in a container. We consider the number of particles in the left and right halves of the container. A particle in one half of the container moves to the other half after an amount of time that is exponentially distributed with parameter 1, independently of all other particles. See Figure 8.6.
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