$\textbf{Q}: $Suppose you are helping out in a disaster relief, at a relief station. Assume that every morning, supplies will be delivered to the station, and that the amount of supplies delivered is a Poisson distributed r.v. with parameter $\lambda$. Furthermore, each supply (here we consider each supply as a single object) in the morning is drawn and distributed to affected populations from the relief station with probability $p$, before the replenishing of supplies the next morning.
Note that the storage capacity of the station is finite, of value $N$, and that should there be a case where the sum of the amount of supplies delivered every morning and that of the number of present supplies in the station exceeds $N$, the amount of supplies in the station will be capped at $N$.
So given the above information, formulate a stochastic model that describes the amount of supplies in the station every morning, just before the resupply that same morning.
$\textbf{Edit/ Addition:}$ I will like to formulate parameters for the model, that allows me to compute the mean number of stock that can be found in the station's storage, and also the expected amount of supplies being drawn from the station, which is not destroyed due to lack of storage.
$\textbf{My thoughts:}$ My approach is to view this as a question on birth and death processes, but part of me strongly tells me to approach this question as a queueing model. With regards to the birth and death process approach, I came up with the birth and death parameters to be $\lambda$ and $pN$ respectively. Is my approach and conceptual understanding wrong? Some clarification will be deeply appreciated as I am still new to $\textit{birth and death processes}$ as well as $\textit{queueing theory}$, having just studied topics on Poisson processes.