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