Consider an M/M/1 queue, where arrivals to the queue occur according to a Poisson process of rate $\lambda$. We assume FCFS (First Come First Serve) service discipline with i.i.d. exponential service times of rate $\mu$. For such a queue, we know that the stationary distribution of waiting times (queuing + service) of individual arrivals in the queue follow an exponential distribution of rate $\mu - \lambda$, i.e., the pdf of waiting time of $i^{th}$ arrival denoted by $W_i$ is given as $$ f(W_i = w_i) = (\mu - \lambda) e^{-(\mu-\lambda) w_i}, w_i > 0. $$
However, I am interested in finding the stationary distribution of two or multiple consecutive waiting times in the queue, i.e., $f(W_i, W_{i+1})$ or in specific $f(W_i, W_{i+1}, \ldots, W_{i+k})$ for some $k \geq 1$. Could someone guide me or refer me to some sources on how I can proceed with this.
Thank you.