 |
|
| Science Forum Index » Statistics - Math Forum » Question on Poisson Process... |
|
Page 1 of 1 |
|
| Author |
Message |
| bill3... |
 Posted: Thu Oct 29, 2009 9:05 am |
|
|
|
Guest
|
Hello, I have a question on whether a process is Poisson or not.
Suppose we have a room with x people, and occasionally people leave
from the room.
Can we model the number of people that leave from the room at a given
small time period S as a poisson process, so P(one person leaves) =
lamda*S?
I am asking because I think that the number of people that leave
during S' depends on the number of people that are still in the room,
so it depends on the number of people that leave during S. Thus we
cannot model it as a poisson process that demands independent events.
Is my rationale correct?
Thank you. |
|
|
| Back to top |
|
|
|
| Paul... |
| Posted: Fri Oct 30, 2009 4:58 am |
|
|
|
Guest
|
If nobody ever enters the room, your logic is correct, and in
particular at some point the population will be zero, after which the
probability of a departure in any time interval is zero.
/Paul
bill3 wrote:
[quote]Hello, I have a question on whether a process is Poisson or not.
Suppose we have a room with x people, and occasionally people leave
from the room.
Can we model the number of people that leave from the room at a given
small time period S as a poisson process, so P(one person leaves) =
lamda*S?
I am asking because I think that the number of people that leave
during S' depends on the number of people that are still in the room,
so it depends on the number of people that leave during S. Thus we
cannot model it as a poisson process that demands independent events.
Is my rationale correct?
Thank you.[/quote] |
|
|
| Back to top |
|
|
|
| bill3... |
| Posted: Fri Oct 30, 2009 6:54 am |
|
|
|
Guest
|
Thank you very much for your reply.
If the number of the people in the room is significantly larger than
the departure rate can we approximate the system as a Poisson process?
-Vasilis
On Oct 30, 2:58 pm, Paul <ru... at (no spam) msu.edu> wrote:
[quote]If nobody ever enters the room, your logic is correct, and in
particular at some point the population will be zero, after which the
probability of a departure in any time interval is zero.
/Paul
bill3 wrote:
Hello, I have a question on whether a process is Poisson or not.
Suppose we have a room with x people, and occasionally people leave
from the room.
Can we model the number of people that leave from the room at a given
small time period S as a poisson process, so P(one person leaves) > > lamda*S?
I am asking because I think that the number of people that leave
during S' depends on the number of people that are still in the room,
so it depends on the number of people that leave during S. Thus we
cannot model it as a poisson process that demands independent events.
Is my rationale correct?
Thank you.
[/quote] |
|
|
| Back to top |
|
|
|
| Rich Ulrich... |
| Posted: Fri Oct 30, 2009 2:08 pm |
|
|
|
Guest
|
On Thu, 29 Oct 2009 12:05:45 -0700 (PDT), bill3 <giotsas at (no spam) gmail.com>
wrote:
[quote]Hello, I have a question on whether a process is Poisson or not.
Suppose we have a room with x people, and occasionally people leave
from the room.
Can we model the number of people that leave from the room at a given
small time period S as a poisson process, so P(one person leaves) =
lamda*S?
I am asking because I think that the number of people that leave
during S' depends on the number of people that are still in the room,
so it depends on the number of people that leave during S. Thus we
cannot model it as a poisson process that demands independent events.
Is my rationale correct?
[/quote]
If one person leaving does not encourage other people
to leave or to stay, then this is a simple survivorship curve,
which is modeled on the Poisson. So your rationale is
incomplete.
The usual survivorship model takes into account that the
N is decreasing.
A model would be wrong that does not take the N into
account. If the N is large enough, the model might be
"right" enough to be useful.
--
Rich Ulrich |
|
|
| Back to top |
|
|
|
| Paul... |
| Posted: Sat Oct 31, 2009 5:13 am |
|
|
|
Guest
|
bill3 wrote:
[quote]
If the number of the people in the room is significantly larger than
the departure rate can we approximate the system as a Poisson process?
[/quote]
Comparing the initial number of people to the departure rate is not
quite right -- I think the right comparison is the initial number of
people to the expected number of departures over whatever (finite)
time horizon you're interested in. If the number of people left at
the end of the horizon is "fairly certain" to be "large" (and assuming
that people depart singly, and do not influence other people to
depart), then a Poisson process seems like a reasonable approximation.
/Paul |
|
|
| Back to top |
|
|
|
| bill3... |
| Posted: Sat Oct 31, 2009 12:35 pm |
|
|
|
Guest
|
On Oct 30, 8:08 pm, Rich Ulrich <rich.ulr... at (no spam) comcast.net> wrote:
[quote]On Thu, 29 Oct 2009 12:05:45 -0700 (PDT), bill3 <giot... at (no spam) gmail.com
wrote:
Hello, I have a question on whether a process is Poisson or not.
Suppose we have a room with x people, and occasionally people leave
from the room.
Can we model the number of people that leave from the room at a given
small time period S as a poisson process, so P(one person leaves) > >lamda*S?
I am asking because I think that the number of people that leave
during S' depends on the number of people that are still in the room,
so it depends on the number of people that leave during S. Thus we
cannot model it as a poisson process that demands independent events.
Is my rationale correct?
If one person leaving does not encourage other people
to leave or to stay, then this is a simple survivorship curve,
which is modeled on the Poisson. So your rationale is
incomplete.
The usual survivorship model takes into account that the
N is decreasing.
A model would be wrong that does not take the N into
account. If the N is large enough, the model might be
"right" enough to be useful.
--
Rich Ulrich
[/quote]
Thank you a lot for your answer.
To give more details about the model, I assume N is large enough and
apart from the departures there are also births (with rate b) and
deaths (with rate d) but b and d are (almost) equal.
From your answer I understand that If I also assume that the departure
of a person may result in the *simultaneous* departure of other
persons, then I break the Poisson process.
-Vasilis |
|
|
| Back to top |
|
|
|
| danheyman at (no spam) yahoo.com... |
| Posted: Sat Oct 31, 2009 12:55 pm |
|
|
|
Guest
|
On Oct 31, 6:35 pm, bill3 <giot... at (no spam) gmail.com> wrote:
[quote]On Oct 30, 8:08 pm, Rich Ulrich <rich.ulr... at (no spam) comcast.net> wrote:
On Thu, 29 Oct 2009 12:05:45 -0700 (PDT), bill3 <giot... at (no spam) gmail.com
wrote:
Hello, I have a question on whether a process is Poisson or not.
Suppose we have a room with x people, and occasionally people leave
from the room.
Can we model the number of people that leave from the room at a given
small time period S as a poisson process, so P(one person leaves) > > >lamda*S?
I am asking because I think that the number of people that leave
during S' depends on the number of people that are still in the room,
so it depends on the number of people that leave during S. Thus we
cannot model it as a poisson process that demands independent events.
Is my rationale correct?
If one person leaving does not encourage other people
to leave or to stay, then this is a simple survivorship curve,
which is modeled on the Poisson. So your rationale is
incomplete.
The usual survivorship model takes into account that the
N is decreasing.
A model would be wrong that does not take the N into
account. If the N is large enough, the model might be
"right" enough to be useful.
--
Rich Ulrich
Thank you a lot for your answer.
To give more details about the model, I assume N is large enough and
apart from the departures there are also births (with rate b) and
deaths (with rate d) but b and d are (almost) equal.
From your answer I understand that If I also assume that the departure
of a person may result in the *simultaneous* departure of other
persons, then I break the Poisson process.
-Vasilis
[/quote]
Multiple departures makes the departure process a "batch-Poisson"
process; the departure epochs form a Poisson process and the number of
departures at each epoch can have any distribution over the non-
negative integers.
If this is your model, with single departures the population size is a
"birth-death" process, which is covered in most stochastic process
texts, and in detail in queueing theory texts. In particular, for one-
at-a-time departures, Burke's theorem asserts that in the steady state
(i.e. when the effects of the initial conditions go away) the
departures form a Poisson process. This has no need of assuming the
current population size in large.
Dan Heyman |
|
|
| Back to top |
|
|
|
| bill3... |
| Posted: Sat Oct 31, 2009 3:02 pm |
|
|
|
Guest
|
On Oct 31, 10:55 pm, "danhey... at (no spam) yahoo.com" <danhey... at (no spam) yahoo.com>
wrote:
[quote]On Oct 31, 6:35 pm, bill3 <giot... at (no spam) gmail.com> wrote:
On Oct 30, 8:08 pm, Rich Ulrich <rich.ulr... at (no spam) comcast.net> wrote:
On Thu, 29 Oct 2009 12:05:45 -0700 (PDT), bill3 <giot... at (no spam) gmail.com
wrote:
Hello, I have a question on whether a process is Poisson or not.
Suppose we have a room with x people, and occasionally people leave
from the room.
Can we model the number of people that leave from the room at a given
small time period S as a poisson process, so P(one person leaves) > > > >lamda*S?
I am asking because I think that the number of people that leave
during S' depends on the number of people that are still in the room,
so it depends on the number of people that leave during S. Thus we
cannot model it as a poisson process that demands independent events..
Is my rationale correct?
If one person leaving does not encourage other people
to leave or to stay, then this is a simple survivorship curve,
which is modeled on the Poisson. So your rationale is
incomplete.
The usual survivorship model takes into account that the
N is decreasing.
A model would be wrong that does not take the N into
account. If the N is large enough, the model might be
"right" enough to be useful.
--
Rich Ulrich
Thank you a lot for your answer.
To give more details about the model, I assume N is large enough and
apart from the departures there are also births (with rate b) and
deaths (with rate d) but b and d are (almost) equal.
From your answer I understand that If I also assume that the departure
of a person may result in the *simultaneous* departure of other
persons, then I break the Poisson process.
-Vasilis
Multiple departures makes the departure process a "batch-Poisson"
process; the departure epochs form a Poisson process and the number of
departures at each epoch can have any distribution over the non-
negative integers.
If this is your model, with single departures the population size is a
"birth-death" process, which is covered in most stochastic process
texts, and in detail in queueing theory texts. In particular, for one-
at-a-time departures, Burke's theorem asserts that in the steady state
(i.e. when the effects of the initial conditions go away) the
departures form a Poisson process. This has no need of assuming the
current population size in large.
Dan Heyman
[/quote]
Thank you for your reply, it is very illuminating.
To sum up, since there may be simultaneous departures I have a batch
Poisson process.
Otherwise it would be a birth-death process, where the number of the
population does not affect the model - even though the birth and death
rates are equal (the departure rate is different from the death
rate)?.
The number of population would affect only a pure-death process.
-Vasilis |
|
|
| Back to top |
|
|
|
|
|
All times are GMT - 5 Hours
The time now is Tue Nov 24, 2009 9:45 am
|
|