Average waiting time?

I want to calculate the mean waiting time due to aggregation of 'm' arrivals following poisson process with rate lambda. Arrivals are served in chunk i.e., if $m$ arrivals reach buffer, OR deterministic timer 'T' expires. The timer ALWAYS starts at the first arrival. So, first arrival reach always at 't=0'. Can some one explain me how to find the mean waiting time due to 'k' arrivals?

Hint: In my opinion, it is a truncated poisson process? and we can assume the uniform distribution between $m$ arrivals, thus it can be something like [meanWaitingTime of(first arrival)+meanWaitingTime of(last arrival) ]/2?

Is it right? if yes, how to calculate these two mean waiting time considering both cases of aggregation i.e., maximum buffer size 'r' or timer expiry 'T'?

Or there is another simple way to calculate it?