X(t) ~= (number of people who have arrived between 0 and t) - (number
   of people who have been fully served between 0 and t)

   Both of these values are calculable.

   S_n is the number of minutes that pass after person (n-1) arrives,
   before person n arrives.

   We can likely write both of these in abstract parts, as X(t) is written
   above.

   Number of people who have arrived = sum of 1 for each person whose
   arrival time is less than or equal to t
   Number of people who have been fully served = sum of 1 for each person
   whose serving conditions have been met

   Serving conditions.  The time at which a person is completely served is
   the maximum of one minute after they arrive, and two minutes after the
   previous person arrived.

   The time at which a person arrives, assuming person 0 arrives at time
   0, is likely Sum[j=1...infinity, S_j * I{j <= n}]