MHB How Does the Poisson Process Model Customer Arrivals Over Time?

AI Thread Summary
The discussion centers on modeling customer arrivals using a Poisson process with parameter \(s\). Participants derive the probabilities for different arrival scenarios in a small time interval, concluding that the probability of one arrival is \(P(1 \text{ arrival}) = sh + o[h]\), the probability of more than one arrival is \(P(\text{more than one arrival}) = o[h]\), and the probability of no arrivals is \(P(\text{no arrival}) = 1 - sh + o[h]\). The remainder term \(R(sh)\) is explained as the truncation error of the series, which helps in understanding the approximations made. The conversation concludes with a participant expressing confidence in completing the calculations after clarifying the remainder term.
Poirot1
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Let customers arrive according to a poisson process with parameter st and let $X_{t}$ denote number of customers in the system by time t. Consider an interval [t,t+h] with h small.

Show that P(1 arrival)= sh + o[h], P(more than one arrival)=o[h] and P(no arrival)=1-sh+o[h].

I know P(1 arrival)=$she^{-sh}$ but how to get further?

Thanks
 
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Poirot said:
Let customers arrive according to a poisson process with parameter st and let $X_{t}$ denote number of customers in the system by time t. Consider an interval [t,t+h] with h small.

Show that P(1 arrival)= sh + o[h], P(more than one arrival)=o[h] and P(no arrival)=1-sh+o[h].

I know P(1 arrival)=$she^{-sh}$ but how to get further?

Thanks

We have a Poisson process with mean arrival rate \(s\), then the number of arrivals in an interval of length \(h\) is \(N\) and \( N \sim P(sh)\).

Then:

\( \displaystyle p(N=1)=sh\; e^{-sh} =sh \left( 1-sh + \frac{s^2h^2}{2}-... \right) = sh(1 + R(sh))=sh + shR(sh)\)

where the \(|R(sh)|\) for small positive \(h\) is bounded by \(sh\) (the truncation error for an alternating series of terms of decreasing absolute value is bounded by the absolute value of the first neglected term), so:

\( \displaystyle |sh \;R(sh)| < s^2h^2 = o ( h)\)

since \( \lim_{h \to 0} (s^2 h^2)/h = 0 \).

Now do the no arrivals case, then \(p(N>1)=1-(p(N=0)+p(N=1))\)

CB
 
Thanks, but can you explain what R(sh) is ?
 
Poirot said:
Thanks, but can you explain what R(sh) is ?

It is the remainder when the series is truncated after the first term, or the sum of the tail of the infinite series (they are the same thing).

CB
 
Ok I can easily do the rest now. Thanks
 
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