Assume $\ell$ independent lines of customers arrive at a single $M/M/1$ First-In-First-Out (FIFO) queue. The rate of arrival of the line $L_i$ is $\lambda_i$, and all of the arrival processes are Poisson point processes (exponential inter-arrival time). We also assume that the processing rate of the queue is $\mu>\sum_{i=1}^{\ell}\lambda_i$. How does the first line $L_1$ observe the queue?
My intuition tells me that $L_1$ observes the queue as an $M/M/1$ queue with service rate $\mu-\lambda_2-\lambda_3\ldots-\lambda_{\ell}$. Is that right? If yes, how can I prove this? Is that in the literature?
Please let me know any helpful comments.