Help with choosing method for duration estimation

[redford]

Hello everybody,

I am new here and I am not sure whether I put this question in a correct forum. If not I am sorry...anyway I hope you help me...

***** Before all: *****
TESTED: SSD METHOD DOES NOT WORK HERE SUFFICIENTLY RELIABLY.

Here it is:

***** Description: *****
Imagine we have set of N types of events. Each event, when it happens, has certain duration according to let's say lognorm distribution. The events happen after each other during certain period of time, then the period stops and next period starts. There is M periods of observations, during which the events are happening after each other. Each observation is fully or almost filled with event...but there could be some delays.

I know number of events of each type per each observation period and length of each observation period. I want to estimate the mean value of duration of each type of event.

***** Formulas: *****
index of type of event: i=1..N
index of observation: m=1..M
number of events of type i within observation m: X_im (I know)
index for event type i occurrence within observation period m: k=1..X_im
mean value of duration of event i: D_i (I want to find)
real duration of event k of type i within observation m:D_imk
sum of durations of events during observation period m: S_m
duration of observation period m: O_m (I know)
event duration ratio: real vs mean value: DR_imk (has probably lognorm distribution, definitely with mean value 1)
observation duration ratio: real vs mean value: OR_m (I don't know what distribution is best, but I know that OR_m<=1 and the values are mostly pretty clost to one...I guess that OR_m<=0.8 should be very rare, for simplification we can assume OR_m=0.95 or something like that)

SUM[D_imk; k=1..X_im, i=1..N]=S_m
D_imk=DR_imk*D_i (DR_imk ~ lognorm (1,sigma=?))
S_m=OR_m*O_m (OR_m <= 1 but close to 1)

***** Example: *****
let's have 5 days of observation: M=2
we have 3 types of events: N=3
First day happened 3 events type 1, 5 events type 2 and no event type 3; second day there are 2 occurences of each event...:
X_11=3, X_21=5, X_31=0,
X_12=2, X_22=2, X_32=2,
X_13=1, X_23=6, X_33=4,
X_14=0, X_24=0, X_34=100,
X_15=2, X_25=3, X_35=10
And here are the durations of the observations each day:
O_1=110, O_2=70, O_3=85, O_4=110, O_5=80
This is what I know and now I want to estimate D_i
(this example was prepared with D_1~20, D_2~10, D_3~1)
(in reality there may be N (number of event types) up to 10, 20 in extreme cases, M (number of observation periods) may be 100 or even more)

So, please help, how to solve it

Thanx lot
Redford

 

[fresh_42]

Looks like an ordinary queueing problem, probably FIFO. This is an entire branch in stochastics, see https://en.wikipedia.org/wiki/Queueing_theory and the links therein.