# Birth and death process $M/M/\infty$

I was reading about continuous time Markov chains, when I met for the first time the theory of queue processes. In particular, I considered the following situation which I found on Wikipedia, called M/M/$$\infty$$ process. The model has transition matrix given by $$$$Q=\begin{pmatrix} -\lambda & \lambda &0 & 0 & 0& & 0&\ldots&0\\ \mu&-(\lambda+\mu) & \mu &0 &0 &&0 &\ldots &0\\ 0&2\mu & -(\lambda+2\mu)& \lambda & 0& &0 &\ldots & 0\\ 0 & 0 &3\mu & -(\lambda+3\mu) &\lambda &&0 &\ldots & 0 \\ \vdots &\vdots &\vdots&\vdots&\vdots &&\vdots &\ddots &\vdots \end{pmatrix}.$$$$ Now, Wikipedia says that the transition probability, assuming that the initial state is $$0$$ at time $$0$$ is given by $$$$p_{0j}(t)=\exp\left(-\frac{\lambda}{\mu}\left(1-e^{-\mu t}\right)\right)\frac{\left(\frac{\lambda}{\mu}\left(1-e^{-\mu t}\right)\right)^j}{j!}$$$$ for any $$j>0$$. Actually, I really don't know how to find this expression. I was trying to write Kolomogorov backward equation $$$$p_{0j}^\prime(t)=\sum_{z\in E}Q_{jz}p_{z0}(t)$$$$ where $$E$$ is the space of states. Obviously all the equations are coupled and I actually cannot find a plausible solution. How can I find that? I was also trying something different. If I write the embedded discrete time Markov chain $$Z_n=X_{T_n}$$ associated to the process, I could try to compute the same transition function for $$Z$$. The problem is that, also in that case I don't know how to procede in the computation, and I really don't see whether or not the two approaches give the same result.

• I also noticed that the limit of this distribution for $t\rightarrow+\infty$ is the Poisson distribution I found to be invariant for the Markov chain, but I do not figure why this is true..
– rime
Commented Jan 15, 2022 at 16:59
• I tried to determine your Wikipedia links, and edited them in. Please feel free to correct if I got them wrong. Commented Jan 15, 2022 at 18:21
• Thank you very much!
– rime
Commented Jan 15, 2022 at 19:26
• Have you checked any textbooks on queueing theory?
– user44143
Commented Jan 15, 2022 at 21:20
• I do not have any good reference about that! I'm only studying this via some lectures from Università degli Studi di Milano which are available online! Any suggestion?
– rime
Commented Jan 15, 2022 at 21:27