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- January 23rd 2009, 12:29 PM #1

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- Jan 2009
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## the chairlift problem

There is a chairlift with seats capable of carrying up to 10 people.

Every 50 seconds a new chair starts from the bottom of the chairlift.

There are four different group of skiers that are skiing on the slope served by this chairlift.

The size of each group (number of skiers) can vary from time to time since some of the skiers of each group may stop at the bar and drinking something hot.

At each ride a group can have a size between 1 and 10 skiers.

The groups want always to sit all together on the chair, if there is not enough space for the entire group to sit they will wait for the next seat and allow another group (if any is waiting) to take the chairlift.

Two or more groups can sit on the same chair if the sum of their sizes is less or equal 10.

The group which is waiting for a longer time and which size is less than the capacity of the chair (taking into account previous groups sitting on the same chair) will sit on the chair with an higher priority (FIFO).

Assume that each group arrives at the entrance of the chairlift with an uniformly-distributed size and a known probability.

What is the mean time that a group of 5 people must wait before sitting on a chair?

What is the maximum time?

Example

at time t1 there are G1(4) G2(8) G3(2) waiting (G1 is waiting for the longest time while G3 for the shortest), the number in brackets represents the size of the group.

at time t1+dt a chair will start: on this chair will sit G1 and G3.

I've received this in a job interview and I had about 30 minutes to solve it, I'm quite confident about my computation of the maximum time while I don't know how to compute the mean time. Excuse me in advance for the English but I'm writing what I remember since they didn't give me a copy of the text..

Anyone knows a possible solution?