Probability of repeated events with fixed delay times co-occuring
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
I am trying to estimate the probability of signal collisions for a research project I am working on. The project uses acoustic emitters, with each emitter encoding a unique identification signal to tell the emitters apart. I would like to estimate the probability that signals from these emitters overlap at the receiver. Because signal overlap will result in spurious data, I need an idea of how many receivers I can use and as well as realistic values for signal lengths and the time delay between emissions.
Therefore, I would like to derive a general expression for the probability of signal collisions in terms of the number of emitters $N$, signal length $t_sig$, and time delay between emissions $T$.
$N$ will be ~ 10 - 100, $t_sig$ will be on the order of milliseconds, and $T$ will be between 10s and 60s.
Each emitter will have a random start time uniformly distributed in the interval $[0,T]$. Each emitter will have equal values for $t_sig$ and $T$. The start time of signal emissions from different emitters are independent of one another.
probability statistics probability-distributions
asked Aug 9 at 18:19
Patrick Barnes
11
 |Â
show 1 more comment
up vote
0
down vote
favorite
I am trying to estimate the probability of signal collisions for a research project I am working on. The project uses acoustic emitters, with each emitter encoding a unique identification signal to tell the emitters apart. I would like to estimate the probability that signals from these emitters overlap at the receiver. Because signal overlap will result in spurious data, I need an idea of how many receivers I can use and as well as realistic values for signal lengths and the time delay between emissions.
Therefore, I would like to derive a general expression for the probability of signal collisions in terms of the number of emitters $N$, signal length $t_sig$, and time delay between emissions $T$.
$N$ will be ~ 10 - 100, $t_sig$ will be on the order of milliseconds, and $T$ will be between 10s and 60s.
Each emitter will have a random start time uniformly distributed in the interval $[0,T]$. Each emitter will have equal values for $t_sig$ and $T$. The start time of signal emissions from different emitters are independent of one another.
probability statistics probability-distributions
asked Aug 9 at 18:19
Patrick Barnes
11
I don't understand -- you want to derive a probability, but I don't see any random elements in your description. The only time you mention something (pseudo)random, you say that it should be ignored. So how is this not a deterministic process?
– joriki
Aug 9 at 20:36
It is deterministic, but as the number of emitters increases, so does the probability of collisions. The random element is when each individual emitter begins sending signals. For example, if there is only one emitter, the probability of collision is zero. If there are two emitters, they probably won't collide, but there is a random chance that their signals will overlap if they start at the same time. The start time of each emitter is independent of all other emitters, so as the number of emitters increases, the chance of signal collisions also increases. Does that clarify?
– Patrick Barnes
Aug 10 at 0:52
It does, almost. You also need to specify a distribution for the start times of the emitters. Perhaps you have something in mind like each start time being independently uniformly distributed over $[0,T]$? Please add all this information to the question; the question should be self-contained without the comments. Currently it doesn't mention that the emitters start at random times.
– joriki
Aug 10 at 5:43
Given your clarifications of the question, it seems to me that it's a duplicate of this question that you linked to (with $Z=2$). Could you point out either where you see differences in the questions or where you had trouble applying the answer to that question to yours?
– joriki
Aug 10 at 5:56
@joriki: I updated and condensed the question. I think the primary reason I can't apply the answer the previously asked question is because I am very rusty with math and do not fully understand that answer. I am unsure what is meant by "only the leading term in an expansion in q:=Y/T will be relevant." Is the expansion referred to a Taylor series? I do not know what q is, and I do not completely understand what is meant by "$Z$-fold overlap".
– Patrick Barnes
Aug 12 at 0:12
 |Â
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to estimate the probability of signal collisions for a research project I am working on. The project uses acoustic emitters, with each emitter encoding a unique identification signal to tell the emitters apart. I would like to estimate the probability that signals from these emitters overlap at the receiver. Because signal overlap will result in spurious data, I need an idea of how many receivers I can use and as well as realistic values for signal lengths and the time delay between emissions.
Therefore, I would like to derive a general expression for the probability of signal collisions in terms of the number of emitters $N$, signal length $t_sig$, and time delay between emissions $T$.
$N$ will be ~ 10 - 100, $t_sig$ will be on the order of milliseconds, and $T$ will be between 10s and 60s.
Each emitter will have a random start time uniformly distributed in the interval $[0,T]$. Each emitter will have equal values for $t_sig$ and $T$. The start time of signal emissions from different emitters are independent of one another.
probability statistics probability-distributions
asked Aug 9 at 18:19
Patrick Barnes
11
I am trying to estimate the probability of signal collisions for a research project I am working on. The project uses acoustic emitters, with each emitter encoding a unique identification signal to tell the emitters apart. I would like to estimate the probability that signals from these emitters overlap at the receiver. Because signal overlap will result in spurious data, I need an idea of how many receivers I can use and as well as realistic values for signal lengths and the time delay between emissions.
Therefore, I would like to derive a general expression for the probability of signal collisions in terms of the number of emitters $N$, signal length $t_sig$, and time delay between emissions $T$.
$N$ will be ~ 10 - 100, $t_sig$ will be on the order of milliseconds, and $T$ will be between 10s and 60s.
Each emitter will have a random start time uniformly distributed in the interval $[0,T]$. Each emitter will have equal values for $t_sig$ and $T$. The start time of signal emissions from different emitters are independent of one another.
probability statistics probability-distributions
asked Aug 9 at 18:19
Patrick Barnes
11
edited Aug 12 at 0:10
asked Aug 9 at 18:19
Patrick Barnes
11
asked Aug 9 at 18:19
Patrick Barnes
11
asked Aug 9 at 18:19
Patrick Barnes
11
11
I don't understand -- you want to derive a probability, but I don't see any random elements in your description. The only time you mention something (pseudo)random, you say that it should be ignored. So how is this not a deterministic process?
– joriki
Aug 9 at 20:36
It is deterministic, but as the number of emitters increases, so does the probability of collisions. The random element is when each individual emitter begins sending signals. For example, if there is only one emitter, the probability of collision is zero. If there are two emitters, they probably won't collide, but there is a random chance that their signals will overlap if they start at the same time. The start time of each emitter is independent of all other emitters, so as the number of emitters increases, the chance of signal collisions also increases. Does that clarify?
– Patrick Barnes
Aug 10 at 0:52
It does, almost. You also need to specify a distribution for the start times of the emitters. Perhaps you have something in mind like each start time being independently uniformly distributed over $[0,T]$? Please add all this information to the question; the question should be self-contained without the comments. Currently it doesn't mention that the emitters start at random times.
– joriki
Aug 10 at 5:43
Given your clarifications of the question, it seems to me that it's a duplicate of this question that you linked to (with $Z=2$). Could you point out either where you see differences in the questions or where you had trouble applying the answer to that question to yours?
– joriki
Aug 10 at 5:56
@joriki: I updated and condensed the question. I think the primary reason I can't apply the answer the previously asked question is because I am very rusty with math and do not fully understand that answer. I am unsure what is meant by "only the leading term in an expansion in q:=Y/T will be relevant." Is the expansion referred to a Taylor series? I do not know what q is, and I do not completely understand what is meant by "$Z$-fold overlap".
– Patrick Barnes
Aug 12 at 0:12
 |Â
show 1 more comment
I don't understand -- you want to derive a probability, but I don't see any random elements in your description. The only time you mention something (pseudo)random, you say that it should be ignored. So how is this not a deterministic process?
– joriki
Aug 9 at 20:36
It is deterministic, but as the number of emitters increases, so does the probability of collisions. The random element is when each individual emitter begins sending signals. For example, if there is only one emitter, the probability of collision is zero. If there are two emitters, they probably won't collide, but there is a random chance that their signals will overlap if they start at the same time. The start time of each emitter is independent of all other emitters, so as the number of emitters increases, the chance of signal collisions also increases. Does that clarify?
– Patrick Barnes
Aug 10 at 0:52
It does, almost. You also need to specify a distribution for the start times of the emitters. Perhaps you have something in mind like each start time being independently uniformly distributed over $[0,T]$? Please add all this information to the question; the question should be self-contained without the comments. Currently it doesn't mention that the emitters start at random times.
– joriki
Aug 10 at 5:43
Given your clarifications of the question, it seems to me that it's a duplicate of this question that you linked to (with $Z=2$). Could you point out either where you see differences in the questions or where you had trouble applying the answer to that question to yours?
– joriki
Aug 10 at 5:56
@joriki: I updated and condensed the question. I think the primary reason I can't apply the answer the previously asked question is because I am very rusty with math and do not fully understand that answer. I am unsure what is meant by "only the leading term in an expansion in q:=Y/T will be relevant." Is the expansion referred to a Taylor series? I do not know what q is, and I do not completely understand what is meant by "$Z$-fold overlap".
– Patrick Barnes
Aug 12 at 0:12
I don't understand -- you want to derive a probability, but I don't see any random elements in your description. The only time you mention something (pseudo)random, you say that it should be ignored. So how is this not a deterministic process?
– joriki
Aug 9 at 20:36
I don't understand -- you want to derive a probability, but I don't see any random elements in your description. The only time you mention something (pseudo)random, you say that it should be ignored. So how is this not a deterministic process?
– joriki
Aug 9 at 20:36
It is deterministic, but as the number of emitters increases, so does the probability of collisions. The random element is when each individual emitter begins sending signals. For example, if there is only one emitter, the probability of collision is zero. If there are two emitters, they probably won't collide, but there is a random chance that their signals will overlap if they start at the same time. The start time of each emitter is independent of all other emitters, so as the number of emitters increases, the chance of signal collisions also increases. Does that clarify?
– Patrick Barnes
Aug 10 at 0:52
It is deterministic, but as the number of emitters increases, so does the probability of collisions. The random element is when each individual emitter begins sending signals. For example, if there is only one emitter, the probability of collision is zero. If there are two emitters, they probably won't collide, but there is a random chance that their signals will overlap if they start at the same time. The start time of each emitter is independent of all other emitters, so as the number of emitters increases, the chance of signal collisions also increases. Does that clarify?
– Patrick Barnes
Aug 10 at 0:52
It does, almost. You also need to specify a distribution for the start times of the emitters. Perhaps you have something in mind like each start time being independently uniformly distributed over $[0,T]$? Please add all this information to the question; the question should be self-contained without the comments. Currently it doesn't mention that the emitters start at random times.
– joriki
Aug 10 at 5:43
It does, almost. You also need to specify a distribution for the start times of the emitters. Perhaps you have something in mind like each start time being independently uniformly distributed over $[0,T]$? Please add all this information to the question; the question should be self-contained without the comments. Currently it doesn't mention that the emitters start at random times.
– joriki
Aug 10 at 5:43
Given your clarifications of the question, it seems to me that it's a duplicate of this question that you linked to (with $Z=2$). Could you point out either where you see differences in the questions or where you had trouble applying the answer to that question to yours?
– joriki
Aug 10 at 5:56
Given your clarifications of the question, it seems to me that it's a duplicate of this question that you linked to (with $Z=2$). Could you point out either where you see differences in the questions or where you had trouble applying the answer to that question to yours?
– joriki
Aug 10 at 5:56
@joriki: I updated and condensed the question. I think the primary reason I can't apply the answer the previously asked question is because I am very rusty with math and do not fully understand that answer. I am unsure what is meant by "only the leading term in an expansion in q:=Y/T will be relevant." Is the expansion referred to a Taylor series? I do not know what q is, and I do not completely understand what is meant by "$Z$-fold overlap".
– Patrick Barnes
Aug 12 at 0:12
@joriki: I updated and condensed the question. I think the primary reason I can't apply the answer the previously asked question is because I am very rusty with math and do not fully understand that answer. I am unsure what is meant by "only the leading term in an expansion in q:=Y/T will be relevant." Is the expansion referred to a Taylor series? I do not know what q is, and I do not completely understand what is meant by "$Z$-fold overlap".
– Patrick Barnes
Aug 12 at 0:12
 |Â
show 1 more comment
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2877546%2fprobability-of-repeated-events-with-fixed-delay-times-co-occuring%23new-answer', 'question_page');
);
Post as a guest
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
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
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
I don't understand -- you want to derive a probability, but I don't see any random elements in your description. The only time you mention something (pseudo)random, you say that it should be ignored. So how is this not a deterministic process?
– joriki
Aug 9 at 20:36
It is deterministic, but as the number of emitters increases, so does the probability of collisions. The random element is when each individual emitter begins sending signals. For example, if there is only one emitter, the probability of collision is zero. If there are two emitters, they probably won't collide, but there is a random chance that their signals will overlap if they start at the same time. The start time of each emitter is independent of all other emitters, so as the number of emitters increases, the chance of signal collisions also increases. Does that clarify?
– Patrick Barnes
Aug 10 at 0:52
It does, almost. You also need to specify a distribution for the start times of the emitters. Perhaps you have something in mind like each start time being independently uniformly distributed over $[0,T]$? Please add all this information to the question; the question should be self-contained without the comments. Currently it doesn't mention that the emitters start at random times.
– joriki
Aug 10 at 5:43
Given your clarifications of the question, it seems to me that it's a duplicate of this question that you linked to (with $Z=2$). Could you point out either where you see differences in the questions or where you had trouble applying the answer to that question to yours?
– joriki
Aug 10 at 5:56
@joriki: I updated and condensed the question. I think the primary reason I can't apply the answer the previously asked question is because I am very rusty with math and do not fully understand that answer. I am unsure what is meant by "only the leading term in an expansion in q:=Y/T will be relevant." Is the expansion referred to a Taylor series? I do not know what q is, and I do not completely understand what is meant by "$Z$-fold overlap".
– Patrick Barnes
Aug 12 at 0:12