how to find the appropriate math function for series with random numbers in Excel

I think to have understood your problem, but for what you said the second sequence you write is wrong. From time 45 min and 50 min there is an increment of 3 orders in 5 minutes, that's impossible since you can process maximum 5/3,9 = 1,282 orders per 5 minutes.
I don't know if there is a formula for calculate what you need but take a look at my table.

And consider this formula for the first column (the function SE() is IF() in my language):

for B3 -> =SE(B4=0;0;SE(B4=1;1;5/3,9))
for B4 -> =B1
for B5 -> =B4-B3
for B7 -> =INT(B3)  --  for C7 -> INT(C3)  --  and so on for D7, F7, ...

for column C, D and the others use this formula in C and expand it to the others columns:

for C3 -> =SE(C1=0;SE(B5=0;B3;SE(B5<(5/3,9);B3+B5;B3+(5/3,9)));SE((C1+B5)<(5/3,9);B3+C1+B5;B3+5/3,9))
for C4 -> =B4+C1
for C5 -> =C4-C3

You have to consider the numbers in every column as the state of the odrers/queue at the end of the 5 min time slot. You can see the total orders recieved, the queue of the orders that are not yet done, the orders done and the orders finished (in fact in 5 min if you recieved 2 orders you have finished one but you have processed more than one...).

This works assuming that the orders arrive at the beginning of the 5 minute time slotss…if the orders arrive at the third minute of the five considered as "time slot" you can't calculate how many orders you can process unless you specify when this orders arrive during the 5 minutes slot.

About the formulas...well it's just evaluating the different cases that can verify:

• if there are no orders and no queued orders there is no work to do, so the orders processed will be the ones already processed
• if there are no orders and the queued orders are less than 5/3,9 (the maximum orders that can be processed in 5 min) the total orders processed will be the ones already done plus the queued
• if there are no orders and the queued ones are more than 5/3,9 the total orders processed will be the ones already done plus 5/3,9 (all the work that can be done)
• if there are orders and the sum of the orders arrived plus the queued ones in the previous time slot is less than 5/3,9 the total orders processed will be the ones already done plus the sum of the ones arrived and the ones queued
• if there are orders and the sum of the orders arrived plus the queued ones is grater than 5/3,9 will do the max work (5/3,9) and the total orders will be the ones already done plus 5/3,9

Hope this is the solution of your problem ;)

Ettore

Edit:

Just realized that are not needed all those tests, this is enough for C3 end extended to D3, E3, ...

=SE((C1+B5)<(5/3,9);B3+C1+B5;B3+5/3,9)

;)