# probabilities related to a transient single-server queue

Consider an $$N = 1$$ server queue with arrival rate $$\lambda > 0$$ and service rate $$\mu = 1$$. If the process is transient, find $$\rho{_{x0}}$$ for $$x ≥ 1$$.

My attempt: The process is transient if $$\sum_{x=1}^{\infty}\frac{\mu_{1}...\mu_{x}}{\lambda_{1}...\lambda_{x}}<\infty$$, which simplifies to $$\sum_{x=0}^{\infty}(\frac{N\mu}{\lambda})^{x}<\infty$$. Since $$N=1$$ and $$\mu=1$$, it will be transient when $$\lambda>1$$. Then what do I do from here?

• Please define the quantity $\rho_{x0}$ and expand on why you cannot find this probability. – Xi'an Apr 22 at 13:29
• @Xi'an $\rho _{x0}=P_{x}(T_{0}<\infty)$, which is the probability the process eventually visits state 0 from state x. Is there a method to figure out this probability? – Jin Yu Li Apr 22 at 22:14