I was thinking of a birth-death style stochastic process, but I'm not quite sure how to model it or classify it.
It is defined as follows. At each step in the process:
- An individual is born/immigrates with probability drawn from an exponential distribution
- An individual dies/emigrates with probability drawn from a Gaussian distribution
Is there any way to write this process compactly? Maybe in the form of a master equation? For example:
$$P_n(t) = \lambda P_{n-1}(t)+\mu P_{n+1}(t)-(\lambda + \mu)P_n(t)$$
Where $\lambda \sim Exponential(1)$ and $\mu \sim \mathcal{N}(0,1)$. Although I'm not sure that's the solution since I can't enforce the condition $\lambda<\mu$ to prevent blowup. Any help or direction would be appreciated. Thank you!
Let $\lambda$ be the arrival rate and $F$ the cdf for the death times. Let $T_0=0$ and $T_n, n\geqslant 1$ the times immediately after deaths; i.e. if $\{X(t):t\geqslant 0\}$ is the process in question, then $$ T(1) = \inf\{t>0: X(t)< X(t)^-\}, $$ where $X(t)^-:=\lim_{s\uparrow t}X(s)$. Then we have the embedded Markov chain $$ P=\begin{pmatrix} a_0 & a_1 & a_2 & a_3 & a_4 & \cdots\\ a_0 & a_1 & a_2 & a_3 & a_4 & \cdots\\ 0 & a_0 & a_1 & a_2 & a_3 & \cdots\\ 0 & 0 & a_0 & a_1 & a_2 & \cdots\\ 0 & 0 &0 & a_0 & a_1 & \cdots\\ \vdots &\vdots & \vdots &\vdots &\vdots &\ddots \end{pmatrix} $$ where $$ a_v = \int_0^\infty e^{-\lambda u}\frac{(\lambda u)^v}{v!}\ \mathsf d F(u). $$
This is the usual method of analyzing $M/G/1$ queues, although some thought should be given to the use of normally-distributed death times, since normal distributions take negative values...