birth-death process in a two state system
$begingroup$
I'm trying to understand a stochastic process, and i'm not sure about a system that i'm studying.
Consider a two state system with a certain number $N$ of time-depending variables $x_i$ that can assume values 0,1.
At the initial time $t_0 = 0$ we have $$x_i(t_0) = 0 forall i=1,dots,N .$$ I assign to every variable a waiting time $t_i^{(lambda)}$, that is a random variate of an exponential distribution with parameter $lambda$; after its waiting time, the value of the variable $x_i$ switches from 0 to 1, so
$$ x_i(t_0 + t^{(lambda)}_i)=1 .$$
Now i assign a new waiting time $t^{(mu)}_i$, that is a random variate of a new exponential distribution with parameter $mu > lambda$, after which the value of $x_i$ switches back to the value 0. I let then the system evolve in time until the steady state is reached.
My question is: considering for example the arrivals from the state 0 to the state 1, this is NOT a Poisson process, right? In order to have it, the waiting time between arrivals should be exponentially distributed, and this is not the case. In other words, if i consider a Poisson process with parameter $lambda$ for the arrivals to the state 1, i should get a different process from the one that i have presented.
probability-theory stochastic-processes poisson-process birth-death-process
asked Nov 26 '18 at 12:39
mminitrmminitr
62
$endgroup$
add a comment |
$begingroup$
I'm trying to understand a stochastic process, and i'm not sure about a system that i'm studying.
Consider a two state system with a certain number $N$ of time-depending variables $x_i$ that can assume values 0,1.
At the initial time $t_0 = 0$ we have $$x_i(t_0) = 0 forall i=1,dots,N .$$ I assign to every variable a waiting time $t_i^{(lambda)}$, that is a random variate of an exponential distribution with parameter $lambda$; after its waiting time, the value of the variable $x_i$ switches from 0 to 1, so
$$ x_i(t_0 + t^{(lambda)}_i)=1 .$$
Now i assign a new waiting time $t^{(mu)}_i$, that is a random variate of a new exponential distribution with parameter $mu > lambda$, after which the value of $x_i$ switches back to the value 0. I let then the system evolve in time until the steady state is reached.
My question is: considering for example the arrivals from the state 0 to the state 1, this is NOT a Poisson process, right? In order to have it, the waiting time between arrivals should be exponentially distributed, and this is not the case. In other words, if i consider a Poisson process with parameter $lambda$ for the arrivals to the state 1, i should get a different process from the one that i have presented.
probability-theory stochastic-processes poisson-process birth-death-process
asked Nov 26 '18 at 12:39
mminitrmminitr
62
$endgroup$
add a comment |
$begingroup$
I'm trying to understand a stochastic process, and i'm not sure about a system that i'm studying.
Consider a two state system with a certain number $N$ of time-depending variables $x_i$ that can assume values 0,1.
At the initial time $t_0 = 0$ we have $$x_i(t_0) = 0 forall i=1,dots,N .$$ I assign to every variable a waiting time $t_i^{(lambda)}$, that is a random variate of an exponential distribution with parameter $lambda$; after its waiting time, the value of the variable $x_i$ switches from 0 to 1, so
$$ x_i(t_0 + t^{(lambda)}_i)=1 .$$
Now i assign a new waiting time $t^{(mu)}_i$, that is a random variate of a new exponential distribution with parameter $mu > lambda$, after which the value of $x_i$ switches back to the value 0. I let then the system evolve in time until the steady state is reached.
My question is: considering for example the arrivals from the state 0 to the state 1, this is NOT a Poisson process, right? In order to have it, the waiting time between arrivals should be exponentially distributed, and this is not the case. In other words, if i consider a Poisson process with parameter $lambda$ for the arrivals to the state 1, i should get a different process from the one that i have presented.
probability-theory stochastic-processes poisson-process birth-death-process
asked Nov 26 '18 at 12:39
mminitrmminitr
62
$endgroup$
I'm trying to understand a stochastic process, and i'm not sure about a system that i'm studying.
Consider a two state system with a certain number $N$ of time-depending variables $x_i$ that can assume values 0,1.
At the initial time $t_0 = 0$ we have $$x_i(t_0) = 0 forall i=1,dots,N .$$ I assign to every variable a waiting time $t_i^{(lambda)}$, that is a random variate of an exponential distribution with parameter $lambda$; after its waiting time, the value of the variable $x_i$ switches from 0 to 1, so
$$ x_i(t_0 + t^{(lambda)}_i)=1 .$$
Now i assign a new waiting time $t^{(mu)}_i$, that is a random variate of a new exponential distribution with parameter $mu > lambda$, after which the value of $x_i$ switches back to the value 0. I let then the system evolve in time until the steady state is reached.
My question is: considering for example the arrivals from the state 0 to the state 1, this is NOT a Poisson process, right? In order to have it, the waiting time between arrivals should be exponentially distributed, and this is not the case. In other words, if i consider a Poisson process with parameter $lambda$ for the arrivals to the state 1, i should get a different process from the one that i have presented.
probability-theory stochastic-processes poisson-process birth-death-process
probability-theory stochastic-processes poisson-process birth-death-process
asked Nov 26 '18 at 12:39
mminitrmminitr
62
asked Nov 26 '18 at 12:39
mminitrmminitr
62
edited Nov 26 '18 at 15:40
mminitr
asked Nov 26 '18 at 12:39
mminitrmminitr
62
asked Nov 26 '18 at 12:39
mminitrmminitr
62
asked Nov 26 '18 at 12:39
mminitrmminitr
62
62
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3014270%2fbirth-death-process-in-a-two-state-system%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3014270%2fbirth-death-process-in-a-two-state-system%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
