Dynamics of birth-death process with discouraged arrivals (alternatively, M/M/1 queue with balking customers)

Take a continuous-time birth-death process, where $k \in \{0,1,\ldots\}$ is the count and

• the arrival rate of death is $\mu \geq 0$ for $k = 1, 2, \ldots$
• the arrival rate of births is $\alpha_k > 0$ for $k = 0,1,\ldots$. For extra simplicity, assume that there is "harmonic discouragement" for some $\alpha > 0$, $$\alpha_k = \frac{\alpha}{k+1}$$

(This follows Kleinrock Queueing Systems volume 1, page 99 section 3.3, also discussed in http://irh.inf.unideb.hu/~jsztrik/education/16/SOR_Main_Angol.pdf, though both only seem to deal with the stationary results )

The stationary distribution of this process comes directly from Kleinrock, with pdf $$\tilde{p}_k = \left(\frac{\alpha}{\mu}\right)^k\frac{1}{k!}e^{-\alpha/\mu}$$

The infinitesimal generator of this process is something like $$\mathbb{Q} \equiv \begin{bmatrix} -\alpha &\alpha& 0 & \ldots \\ \mu & -(\mu + \frac{\alpha}{2}) & \frac{\alpha}{2} & \ldots \\ \ldots & \ldots& \ldots& \ldots \end{bmatrix}$$

Of course, stacking the pdf as $p(t) \equiv \begin{bmatrix}p_0(t) & p_1(t) & \ldots \end{bmatrix}$ the solution to this comes from the KFE, $$p'(t) = p(t) \cdot \mathbb{Q},\quad s.t.\, p(0) = \begin{bmatrix}1 & 0 & \ldots \end{bmatrix}$$

Alternatively, if you prefer looking at the ODEs directly for $k > 1$ $$p'_k(t) = -\left(\mu + \frac{\alpha}{k+1}\right) p_k(t) + \frac{\alpha}{k}p_{k-1}(t) + \mu p_{k+1}(t)$$ and, $$p'_0(t) = -\alpha p_0(t) + \mu p_1(t)$$

QUESTION: What is the evolution of the density over time (from a $0$ count initial condition)? Alternatively, what is the evolution of the moment generating function over time?

• In solving this, I imagine you would need to find a PDE in the moment generating function of $p_k(t)$, call it $G(t,z)$. Any ideas on how to derive it?
• Is there a known "closed form" solution for $p(t)$? (I use closed form loosely here as infinite sums are perfectly reasonable)?

• Is there is closed form for the evolution of the expectation, $E_t(k)$?

• You probably won't have much success finding a (practically usable) closed form for the transient probabilities. Even the $M/M/1$ has a rather nasty expression for $\mathbb P(X_t=j\mid X_0=0)$ (cf. Leguesdron et al. 1992, equation $(13)$). Sep 3 '16 at 1:07
• @Math1000 thanks. I suspect you are right. If you post an answer, I would accept it. Sep 3 '16 at 13:29