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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 \begin{equation} 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}. \end{equation} Now, Wikipedia says that the transition probability, assuming that the initial state is $0$ at time $0$ is given by \begin{equation} 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!} \end{equation} for any $j>0$. Actually, I really don't know how to find this expression. I was trying to write Kolomogorov backward equation \begin{equation} p_{0j}^\prime(t)=\sum_{z\in E}Q_{jz}p_{z0}(t) \end{equation} 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.

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  • $\begingroup$ 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.. $\endgroup$
    – rime
    Commented Jan 15, 2022 at 16:59
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    $\begingroup$ I tried to determine your Wikipedia links, and edited them in. Please feel free to correct if I got them wrong. $\endgroup$
    – LSpice
    Commented Jan 15, 2022 at 18:21
  • $\begingroup$ Thank you very much! $\endgroup$
    – rime
    Commented Jan 15, 2022 at 19:26
  • $\begingroup$ Have you checked any textbooks on queueing theory? $\endgroup$
    – user44143
    Commented Jan 15, 2022 at 21:20
  • $\begingroup$ 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? $\endgroup$
    – rime
    Commented Jan 15, 2022 at 21:27

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