Probability of overlapping random pulses

Main Question or Discussion Point

I have a problem calculating the following probability.

There are two signals A and B each consisting of a series of "pulses" at times

{tA0, tA0+Δt, tA1, tA1+Δt, tA2,tA2+Δt, ...} and

{tB0, tB0+Δt, tB1, tB1+Δt, tB2, tB2+Δt,...}

The signal A is "on" in the time intervals [tAn, tAn+Δt], and it's off in the time intervals [tBn+Δt, tBn+1].

There is a given probability for the signal A to be "on" depending on the (random) times between the pulses; the same applies for the signal B.

How can one calculate the probability that for a certain time T both signals are "on"

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Stephen Tashi
You haven't stated a precise question.

What does "between the pulses mean?". What are the "the pulses"? Is signal A the only signal that is pulsing. What's the difference between being "on" and being in the state of emitting a pulse?

Do the times $t^B$ have anything to do with signal B?

What are the given probabilities and what events do they describe? Does the situation involving independent events of some kind?

Are you trying to find the probability that A and B are both "on" for an instant of time? For at least some interval of time?

Do you want the calculate that this ever happens once ( between time= 0 and "infinity"?) or do you want to calculate the mean number of time this events happens in an hour - or something like that?

You haven't stated a precise question.
Really? OK, not very precise, I agree.

What does "between the pulses mean?".
Between means
The signal A is ... off in the time intervals [tBn+Δt, tBn+1].
That's between the pulses [where it's on]

What are the "the pulses"?
A pulse is when a signal is "on"

Is signal A the only signal that is pulsing.
No; there are two sequences of pulses:
There are two signals A and B each consisting of a series of "pulses" at times

{tA0, tA0+Δt, tA1, tA1+Δt, tA2,tA2+Δt, ...} and

{tB0, tB0+Δt, tB1, tB1+Δt, tB2, tB2+Δt,...}
What's the difference between being "on" and being in the state of emitting a pulse?
Nothing; during a pulse the signal is “on”; between the pulses it’s "off".

Do the times $t^B$ have anything to do with signal B?
Of course, what else should the superscript B indicate??

What are the given probabilities and what events do they describe? Does the situation involving independent events of some kind?
I haven’t specified the probability for the times between the pulses. I hope one can find some kind of general ansatz. If not one may assume some kind of normal distribution. The different times where the signals are off are independent. So both two subsequent off-times of the same signal are independent as well as the two signals A and B.

Are you trying to find the probability that A and B are both "on" for an instant of time? For at least some interval of time?
I try to find the probability that that for some time t (picked randomly) both signals are “on”.

… do you want to calculate the mean number of time this events happens in an hour - or something like that?
yes.

Stephen Tashi
Of course, what else should the superscript B indicate??
You said:

The signal A is "on" in the time intervals [tAn, tAn+Δt], and it's off in the time intervals [tBn+Δt, tBn+1].
Did you mean to say A is off in the time intervals $[t_{A_n} + \triangle t, t_{A_{n+1}}]$?

I haven’t specified the probability for the times between the pulses. I hope one can find some kind of general ansatz. If not one may assume some kind of normal distribution.
You'd have to define the random variables that are going to have the distribution. What would they represent? Duration of pluse? Time between pulses? Time between change of state between pulse and non-pulse? Without knowing the physics of the the problem, it isn't clear what random variables to represent.

Normal distributions can produce values arbitrarily smaller than their mean so it isn't clear how to interpret these as durations of time, which are bounded below by 0.

I think this general type of problem has well-known solutions but the details are going to depend on how the phenomena of the pulses is modeled. For example, two computers trying to communicate on the same ethernet wire is one scenario. The overlap of two claps of thunder is another.

chiro
Hey tom.stoer.

For your problem, can we safely say that you want to find a distribution where you have two independent processes (corresponding to A and B) where you want to find P(A in a = X, B in b = Y) where a is an interval region in A (corresponding to a time interval) and b is the same for B where X corresponds to the number of pulses in that interval and B corresponds to the number of pulses in that respective interval?

Did you mean to say A is off in the time intervals $[t_{A_n} + \triangle t, t_{A_{n+1}}]$?
Oh sh..; of course I mean that!

You'd have to define the random variables that are going to have the distribution. What would they represent? ... Time between pulses?
Yes, time between the pulses. The duration of the pulses is Δt = const.

I think this general type of problem has well-known solutions but the details are going to depend on how the phenomena of the pulses is modeled. For example, two computers trying to communicate on the same ethernet wire is one scenario. The overlap of two claps of thunder is another.
Yes, something like that. Do you know any reference?

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For your problem, can we safely say that you want to find a distribution where you have two independent processes (corresponding to A and B) where you want to find P(A in a = X, B in b = Y) where a is an interval region in A (corresponding to a time interval) and b is the same for B where X corresponds to the number of pulses in that interval and B corresponds to the number of pulses in that respective interval?
As far as I understand - no.

The distribution of "off-times" between the pulses shall be some reasonable distribution (not a normal distribution - that's nonsense - I agree, but e.g. uniform distribution or Poisson distribution, ...).

Then I would like know a general ansatz how to calculate the probability that for some arbitrary time t both signals are "on".

Let's look at one signal. The probability for one signal to be "on" is just

$$P^A_\text{on} = \lim_{T\to\infty} \frac{T_\text{on}}{T}$$

Is it allowed to use a different limit, namely

$$T^A(n) = t^A_n$$
$$T^A_\text{on}(n) = n\,\Delta t$$
$$P^A_\text{on} = \lim_{n\to\infty} \frac{T^A_\text{on}(n)}{T^A(n)} = \lim_{n\to\infty} \frac{n\,\Delta t}{t^A_n}$$

That would mean that all one has to do is to calculate the times, i.e. to write down and evaluate a sum over random variables. Then my guess would be that the probability for both signals being "on" is just the product

$$P^{A\,\text{and}\,B}_\text{on} = P^A_\text{on} \cdot P^B_\text{on}$$

Stephen Tashi
or Poisson distribution
A Poission distribution is a distribution of "a number of counts", so you'd being using integer multiples of time if you did that.

$$P^A_\text{on} = \lim_{n\to\infty} \frac{T^A_\text{on}(n)}{T^A(n)} = \lim_{n\to\infty} \frac{n\,\Delta t}{t^A_n}$$
It isn't clear what expression means because ${t_A}_n$ is a random variable, not an ordinary variable. ${t_A}_n$ it isn't a deterministic function of $n$. If we think of $\{ {t_A}_n\}$ as a sequence of things, the things aren't single numbers. It's a sequence of random variables and the usual definition for such a sequence (if it has one) is that the limit is another random variable.

If you want a "ansatz" in the sense of a mere guess, you might try taking the limit, as time T, approaches infinity of a fraction involving the products: (the expected number of pulses of the signal that happen in time T)( delta T).

You haven't described any practical goal of you analysis. If you are trying to settle a academic controversy or a bet, then calculating the probability that both signal are on "at a randomly selected time" may answer that purpose. I don't see that this probabiliy has practical use otherwise. For example, you can't take that answer and formulate a distribution for the length of time intervals during which the signals overlap.

A guess about the general mathematics of your problem is that it is a "Markov renewal process". The four states of the process are: (A off, B off), (A on, B off), (A on, B off), (A on, B on). However, we'd have to think about it carefully to verify that.

A Poission distribution is a distribution of "a number of counts", so you'd being using integer multiples of time if you did that.
Not really. The question regarding pulses in a certain time intervall can be asked regarding arbitrary real values [t, t+Δt]; the Δt which is the duration of the pulse "on" has nothing to do with the Poisson distribution (or any other distributation).

It isn't clear what expression means ... It's a sequence of random variables and the usual definition for such a sequence ... is that the limit is another random variable.
I think you know what I have in mind; what would be the correct ansatz?

You haven't described any practical goal of you analysis.
Think about two signals A and B with fixed Δt and some distribution of the times {tA0, tA1, ...} and {tB0, tB1, ...}. What is the probability that for any time t two pulses from the signal A and B are overlapping?

A guess about the general mathematics of your problem is that it is a "Markov renewal process". The four states of the process are: (A off, B off), (A on, B off), (A on, B off), (A on, B on).
Thanks for the hint. I checked the properties of the Marnkov renewal process. It seems that what I have in mind is not one renewal process (with above mentioned states) but a pair of such processes.

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chiro
One question for the OP has to do with these pulses.

If these are pulses in the form of a Dirac-delta type pulse, then the only way they can over-lap is if they occupy the exact same position.

You have mentioned that you want to consider an interval with some delta t value, but if these delta's go to infinity and your pulse is basically a pure theoretical Dirac-delta t kind of event, then if you are considering a continuum for the domain, then the probability of getting a pulse in this Dirac-delta style event will be 0 anyway.

If you want to consider a non-zero length finite interval, then this is a little different because you can use something based on a Poisson process, or if you want to stick to the continuum, you use a distribution for the "waiting time" till an event takes place as opposed to using a rate parameter (like the Poisson does).

So if you consider modelling your pulse process as a "waiting time" model for both signals and then look at the relevant probabilities, that might serve your interest.

The problem is non-trivial but significantly easier if you can assume that the pulses follow a Poisson process - so that, conditional on the number of pulses, the pulse times are uniformly distributed within the interval (with a slight modification if the inter-pulse times follow a Poisson process, rather than the pulse start times). I'm not sure if this will lead to an analytic solution but it should at least be approximated with Mone Carlo methods.

If these are pulses in the form of a Dirac-delta type pulse, then the only way they can over-lap is if they occupy the exact same position.
It's about rectangular pulses which are "on" / "off".

You have mentioned that you want to consider an interval with some delta t value, but if these delta's go to infinity and your pulse is basically a pure theoretical Dirac-delta t kind of event, then if you are considering a continuum for the domain, then the probability of getting a pulse in this Dirac-delta style event will be 0 anyway.
Δt is finite, i.e. 0 < Δt < ∞, and const.

If you want to consider a non-zero length finite interval, then this is a little different because you can use something based on a Poisson process, or if you want to stick to the continuum, you use a distribution for the "waiting time" till an event takes place as opposed to using a rate parameter (like the Poisson does).
A Poisson process is fine.

So if you consider modelling your pulse process as a "waiting time" model for both signals and then look at the relevant probabilities, that might serve your interest.
Thanks, nice.