Calculate average life time in system with inputs and observed items inside system in a given time

Question

In a given time (t) i observe the inputs that enter a system and the total items that are the sistem, i.e

time    inputs  items in system  
t1         i1       N1   
t2         i2       N2   
t3         i3       N3 
....   

how can i calculate the average life of an item in the system in a given time? (no exponential decay assumption)

Answer 1

Say there are $i_t$ inputs at time $t$, $t=0,\ldots,M$, and $N_t$ items in the system at that time (including those $i_t$).
The number of items entering the system from time $0$ to $M$ is $E_M = \sum_{t=0}^M i_t$, and the total time spent in the system for all items up to time $M$ is $T_M = \sum_{t=0}^M N_t$. This will include any items that were already there before time $0$. If we could only include in these totals the items that both entered and left in the time interval, $T_M/E_M$ would give the average time in the system per item. We can't do that, but hopefully if $M$ is large the number of items that entered and left will be large compared to those that were there before $t=0$ or were still there at $t=M$. Thus for large $M$, $T_M/E_M$ should be a good approximation to the average time in the system per item.