Generally, In a M/M/1/K system, the incoming rate is $\lambda$, effective incoming rate $\lambda_e$ is equal to $\lambda(1-P_k)$, where $P_k$ is the probability that queue waiting space is full. This is also the outgoing rate.
My question is, if the incoming rate is a combination of two rates, i.e. $\lambda=\lambda_1+\lambda_2$, then can I assume that both the rates will drop some incomings(due to only K space being available) at the same ratio?
I.e. can I assume that outgoing rate in this case is $\lambda_1(1-P_k) + \lambda_2(1-P_k)$ and both the incoming rates have lost jobs in equal proportion?
For a simpler scenario without effective incoming rate, suppose there is a queue where customers arrive at rate $\lambda$ and they go into either of two lines with equal probability $\dfrac{1}{2}$. There is one server who serves the customer at rate $\lambda$. Regardless of the number of different lines, the one server can only serve one customer at a time so the exit rate is still $\lambda$.
If I am reading the question correctly, your outgoing rate is correct because $\lambda_1(1-P_k)+\lambda_2(1-P_k)=(1-P_k)(\lambda_1+\lambda_2)=\lambda(1-P_k)=\lambda_e$