I have the following queuing process with $X(t)$ people in a queue at time $t$.
$$X(t)=X(0)+N_1(t)-\int_0^t \text{min}(X(s-)\wedge m)N_2(ds)$$
The process describes a queue where according Poisson process $N_1$ with parameter $\lambda_1$ people arrive, and according to Poisson process $N_2$ with parameter $\lambda_2$, $m$ people are removed from the queue (if $m$ or more people are queuing). The process is assumed stationary.
I want to use Little's formula to compute the relation between people in the queue $X(t)$ and the average expected waiting time. The typical Palm calculus formula gives
$$E(X(t))=\lambda_1E_0(W_0),$$
where $E_0$ is the expected waiting time with respect to Palm measure $P_0$, (e.g. conditional probability on person arriving on the queue at time $0$). $W(0)$ is the waiting time of the person arriving at time $0$.
The problem is that there may be more than $m$ people queuing at the start. Say $n+1$ is just below $d$ multiples of $m$, then conditional on the queue being length $n$ before the person arrives, the wait time is clearly: $$E_0(W(0)|X(0)=n) =E(\tau_d)$$
Which is trivial to compute. The unconditional overall expectation of interest is then:
$$E_0(W(0))=\sum_{d=1}^\infty E(\tau_d)P((d-1)m \leq X(0)< d m)$$
However, the above seems quite useless, since there is no easy way to evaluate the probabilities. How to use the Little's formula correctly?