For a birth-death process that is M/M/1/4/$\infty$/G (as a queue system), how can I calculate the number of clients that can enter the system for each unit of time. Is this solving for $\lambda$ in $4 = L = \lambda W$?
What I doubt about my approach is that 4 is the maximum capacity of the system, and that doesn't tell us anything about the amount of clients at any given time?
I haven't seen the G used at the end to describe a queueing node before, but will assume that by M/M/1/4/∞/G we mean an M/M/1 queue which has maximum capacity 4, an unbounded potential calling population (our source will never run out of customers) and G here I assume denotes a general service discipline (e.g. FCFS, LCFS, SIRO).
This process can be described as continuous-time Markov chain with 5 states {0, 1, 2, 3, 4} describing the number of customers in the queue and transition rate matrix
$$Q = \begin{pmatrix} -\lambda & \lambda \\ \mu & -(\lambda + \mu) & \lambda \\ &\mu & -(\lambda + \mu) & \lambda \\ &&\mu & -(\lambda + \mu) & \lambda \\ &&&\mu & -\mu \end{pmatrix}.$$
I'm not entirely clear what you wish to compute as time isn't discrete in this model, so describing how many customers enter in each time unit would seem rather arbitrary. Perhaps you want to describe the long term average amount of time the queue spends in each of the 5 states. When it is in state 4 there are no arrivals (they are blocked because the system is at maximum capacity), but in all other states arrivals happen at rate $\lambda$. So if you compute the stationary probability distribution $\pi_i$ for each of the states then the long term average arrival rate is given by
$$\lambda(\pi_0+\pi_1+\pi_2+\pi_3) = \lambda(1-\pi_4).$$