It is common convention to assume that for a Poisson process (or, in any counting process) with some rate, say $\lambda$, at any given exact point in time there may exist only one event. For instance, in a singular FIFO queue system, only one arrival may enter into the system at any given time. Further, if the queue has greater than or equal to 1 occupant currently being serviced, the system $X(t)$ at some time $T$ can only move to $X(T)+1, X(T), $ or $X(T)-1$ at particular time $T_{2} > T$ if the next arrival has not occurred yet until (potentially) $T_{2}$.
I have, however, not seen this ever proven to necessarily hold true in general for markovian counting processes -- only ever assumed. Does there exist a proof of this property?
This convention becomes important as, of course, one can then derive the Kolomogorov Forward/Backward Equations (calling all arrivals in infinitismal time $o(\Delta)$) and show that the infinitismal change in the probability distribution describing stochastic process $X(t)$ is of form (for transitions $i\rightarrow (j\not = i$)) $ P_{i,j}*$(rate of transition).