Timeline for Poisson counting process subinterval distribution
Current License: CC BY-SA 4.0
10 events
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| May 24, 2020 at 8:31 | comment | added | Dieter Kadelka | @Carlo Beenakker You are right. Much of the literature deals with average behaviour. The advantage to have a look into the queuing theory is that these formulate models which make questions as that from Hans tractable. The answer of Iosif Pinelis too can be formulated in this framework. I think that you do not find an explicite answer for Hans problem. | |
| May 24, 2020 at 7:33 | comment | added | Carlo Beenakker | I looked into this queueing literature. It seems to be very much focused on stationary distributions, so the limit $T\rightarrow\infty$ in which the probability in the OP is 1. | |
| May 22, 2020 at 17:29 | comment | added | Hans | Yes, I was sloppy. I have edited my notation in the question. I see the approach of Iosif Pinelis. Can you elaborate your approach in detail in your answer? Is it different from Iosif's? | |
| May 22, 2020 at 16:44 | comment | added | Dieter Kadelka | Hello @Hans, it would be much clearer if in addition you let $N(t)$ depend on some $\omega$, but I think this does not suffice. The left side of $\{N(t)|\ldots\}$ are elements of $\mathbb{N}_0$, so what is the union? Iosif Pinelis has shown a way to formalize your question. To solve your problem I would condition on $N(T) = k$, then you can assume that the times $S_1,\ldots,S_k$ are the ordered independent random variables $Y_1,\ldots,Y_k$, where each $Y_i \sim U(0,T)$ (uniform distribution on $[0,T]$). | |
| May 22, 2020 at 15:58 | comment | added | Hans | Why is Prob strange? I am specifically asking about the probability and I thought my notation was pretty clear. It is the probability of the union over $t$ of all the sample paths $N(t)$ that satisfies the condition that it counts exactly $n$ within the subinterval of length $\tau$ starting at $t$. I had a cursory search but have not found anything in that vein. | |
| May 22, 2020 at 7:16 | comment | added | Dieter Kadelka | @Hans Maybe you can ask your question (without the $Prob$ part, this looks strange) at or.stackexchange.com Unfortunately this seems to be not very active. And yes, "customer rejection in quenueing theory" seems to be a good question. At least you should find some similar problems. N.B.: The corresponding problem for $M/M$-queues is simple. | |
| May 22, 2020 at 2:58 | comment | added | Hans | By the way, it was not I who downvoted your answer. | |
| May 22, 2020 at 2:15 | comment | added | Hans | Your first paragraph is not while your second paragraph is indeed what the question asks. Is "customer rejection in quenueing theory" the right key phrase to Google? | |
| May 21, 2020 at 21:45 | comment | added | Dieter Kadelka | I'm not sure what the downvoting means. Is my model wrong? Intentionally I've not used the terminology of pure mathematics but of Operations Research. In OR, in particular in queueing theory, such sort of problems are investigated. | |
| May 21, 2020 at 20:51 | history | answered | Dieter Kadelka | CC BY-SA 4.0 |