Q: Stochastic Process

[Tony Peluso & Ella Heyder]

Hi;

I posted this question to sci.math with no response yet.

I have a steady-state system with "N" people in it. At the beginning of
each period, "n" people enter the system and "n" people leave it. Say
the average lifetime of a person in the system is "a". (I’ll assume
that lifetimes are Poisson, though I’m not sure this is strictly
necessary - others may have different ideas.) Given "a" and "N", is
there any way of estimating "n" ? Do I need other parameters?

Thanks,

Tony Peluso

[Armann Ingolfsson]

The process is not very well specified (e.g., steady state in what
sense, is N constant or random, is n the same every period), but
assuming that Little's Law holds, then N is the average number in
system, n is the arrival rate, and a is the average time in system,
so

N = (n)(a),

so

a = N/n.

------------------------------------------------------------------
Armann Ingolfsson
Department of Finance and Management Science, Faculty of Business
University of Alberta
Edmonton, AB T6G 2R6, CANADA

phone: 1 403 492 7982
fax: 1 403 492 3325
http://www.ualberta.ca/~aingolfs/

[Rodney Beard]

Tony Peluso & Ella Heyder <pel...@magi.com> wrote:
>Hi;
>
>I posted this question to sci.math with no response yet.
>
>I have a steady-state system with "N" people in it. At the beginning of
>each period, "n" people enter the system and "n" people leave it. Say
>the average lifetime of a person in the system is "a". (I’ll assume
>that lifetimes are Poisson, though I’m not sure this is strictly
>necessary - others may have different ideas.) Given "a" and "N", is
>there any way of estimating "n" ? Do I need other parameters?
>

>Thanks,
>
>Tony Peluso

Doesn't it go something like this or am I missing something:

The total time available T=aN, if you measure the lenght of time the system is
in use or if this isn't random, set the length of time arbitrarily and
calculate n for any given lenght of time. So, t=an (I assume a is the same for
all individuals not just those arriving and departing) is the length of time of
interest either measured or arbitrary calculate (t/T)*N to obtain the number n
that pass through the system in the time interval t.

You don't need the Poisson assumption unless you want to generalize the model
to include more ``systems''. Then you could view the problem as a queue with
arrivals and departues. But I don't think that's necessary. In queueing models
you would normally know n - the arrival and departure rate and be interessted
in the probability of how many are in the system. But you want to go the other
way, so the problem is different.

Rodney Beard
University of Queensland