For a M/M/1 queue let $N_q(t) = (Q(t)-1)^{+}$ be the number of customers in the queue except the one being served. We have to show that $N_q(t)$ is not a continuous-time Markov chain. [src: Sidney Resnick, Adventures in Stochastic Processes Pg-460, Problem 5.28]
My intuition: Let us take a trajectory of the process, assume that three arrivals occur at t = 1, 2, 3 and service time of the first customer is 5, Then values of $N_q(t)$ is 0, 1, 2 for the given time instant now if we want the value of $N_q(5)$, then it will depend on number of arrivals happening between t=4 and t=5 and $N_q(4)$ which seems to violate the Markov property.
I believe that we can mathematically prove this, but I do not seem to see an obvious way, so I would like some hints if possible. Thanks in advance.