I am learning about the birth and death process, and I know an infinite M/M/1 queue with impatient customers can be described as a birth and death process, with arrival rate $\lambda$, service rate $\theta$ and the abandoning rate $r$, and the length of queue at time t is $X(t)=\frac{\lambda-\theta}{r}+c1*e^{-rt}$, where $c1$ is determined by the initial queue length.
However, my question is when $\lambda<\theta$, given the initial queue length, can we use $X(t)$ function to describe the queue length decreasing with $t$ and calculate the queue disappearing time $t'$ by taking queue length $X(t')=0$?