Without data in the from of working data step code or expected results this is untested but may work if your data is sorted by group.
data want;
set have;
by group;
diftime= dif(time);
meanrate= mean(lag(rate),rate);
if not (first.group) then value=diftime*meanrate;
drop diftime meanrate;
run;
The DIF function is the difference of the current value of a variable from the previous observation in a data set on the Set statement. Lag returns the value from the previous observation.
Use of a BY statement creates automatic variables First. and Last. that indicate whether the current value is the first or last of a by group. These are numeric values of 1 (True) and 0 (False).
So this calculated the values needed using the current and last observation values of the rate and time but only calculated Value when it isn't the first row of a Group.