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If an process is called a Poisson process, it has small interval probabilities, stated as: For small \(\delta\), the probabilities \(P(k, \delta)\) satisfy
\[\begin{align*} P(0, \delta) &\approx 1 - \lambda\delta, \\ P(1, \delta) &\approx \lambda\delta, \\ P(k, \delta) &\approx 0, &\text{for } k=2, 3,\cdots, \end{align*}\]where \(P(k, \delta)\) is the prob. of \(k\) arrivals in interval of duration \(\delta\).
Let us approximate a Poisson process with a Bernoulli process. For a time interval \(\tau\), divide it into \(n\) slots such that each slot is of length \(\delta\). Thus \(n = \tau / \delta\). According to our definition, the prob. of an arrival within time \(\delta\) is \(P(1, \delta) \approx \lambda\delta = p\), where \(p\) is the parameter of a Bernoulli r.v. We know that the expected number of arrivals in \(n\) slots is \(np\), which in our case is \(\lambda\tau\). When \(n\to \infty\) and \(\delta\to 0\), \(np = \lambda\tau\) remains constant. From the discussion here, we see that a binomial PMF (\(n\) slots Bernoulli) converges to a Poisson PMF, here, with parameter \(\lambda\tau\). To conclude, we have
\[\begin{align*} P(k, \tau) = e^{-\lambda\tau}{(\lambda\tau)^k\over k!},\ k\in \Bbb N. \end{align*}\]Note that a Taylor series expansion of \(P(k, \tau)\) exactly yields the small interval probabilities, with a negligible \(O(\tau^2)\) term when \(\tau\) is small.
\(\lambda\) 被稱為 arrival rate。
Let \(Y_k\) be the r.v. of the \(k\)-th arrival. We have
\[\begin{align*} f_{Y_k}(y)\delta &\approx P(y\le Y_k \le y+\delta) \\ &\approx P(k-1, y) \cdot \lambda\delta. \end{align*}\]After cancelling \(\delta\) from both sides, we have
\[f_{Y_k}(y) = P(k-1, y)\lambda = {\lambda^ky^{k-1}e^{-\lambda y}\over (k-1)!},\]and this is called the Erlang distribution of order \(k\).
Let \(T_k = Y_k - Y_{k-1}\), which is the \(k\)-th inter-arrival time. Then \(T_1, T_2, \cdots\) are independent, identically distributed exponential r.v.s, and \(Y_k = T_1+\cdots T_k\).
We can also define a Poisson process as a sum of i.i.d exponential r.v.s, which is equivalent to our former definition.
新定義？具體一點？
The sum of two independent Poisson r.v.s, with parameter \(\lambda\) and \(\mu\) respectively, is a Poisson r.v. with parameter \(\lambda + \mu\).
從 convolution 或是 Poisson process（同一 process 切出兩獨立區間，並令 \(\lambda=1\)）來看都可以。
Since \(P(k, \tau) \sim \text{Poisson}(\lambda\tau)\), the concatenation of two independent Poisson processes, with arrival rate \(\lambda_1, \lambda_2\) and duration \(\tau_1, \tau_2\), is
\[\text{Poisson}(\lambda_1\tau_1 + \lambda_2\tau_2).\]Merge: merged process: \(\text{Poisson}(\lambda_1+\lambda_2)\)
Split: resulting streams are \(\text{Poisson}(\lambda q)\) and \(\text{Poisson}\lambda(1-q)\).
The two resulting streams are independent! Unlike those in the case of Bernoulli process.