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?

  • $\begingroup$ Please define the quantity $\rho_{x0}$ and expand on why you cannot find this probability. $\endgroup$ – Xi'an Apr 22 at 13:29
  • $\begingroup$ @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? $\endgroup$ – Jin Yu Li Apr 22 at 22:14

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