I have been trying to prove null recurrence when $\lambda$ = s$\mu$ and transience when $\lambda$ = s$\mu$ for a M/M/s queue in regard to continuous time markov chains. the traffic intensity is $$p=\frac{\lambda}{\mu}$$
I have that my invariant measure is $$v_i=v_o \frac{p^i}{i!}; i<s$$ $$v_i=v_o \frac{p^i}{s!s^{i-s}}; i\ge s$$ I have that my invariant distribution $$\pi_i = e^{-p} \frac{p^i}{i!}$$
I understand that for null recurrence I need to show that both $$P_i[T_i<\infty]=1;E_i[T_i]=\infty $$ and for transience $$P_i[T_i<\infty]<1$$ I am really quite unsure how to go about this and any help would be greatly appreciated! Thank you.