There are two types of arriving requests to an object, either (S)hared or (E)xclusive. There can be multiple S locks at a given time on the item (i.e. S locks are compatible with each other), while there can be only one E lock and no S lock (i.e. exclusive locks are incompatible with both other exclusive and shared locks). Each request takes on average t seconds. When a new incoming request is received and it cannot be granted, it fails.
Let's assume that the assumptions for the Poisson distribution hold here and both request types are Poisson processes ($\lambda_S$ and $\lambda_E$ for the rate of shared and exclusive locks, respectively). How can I find the number of requests that fail?
(from this point on, things can be horribly wrong)
What I have so far:
$$P(\text{$k$ reqs of type $S$})=\frac{e^{-{\lambda_S} } {\lambda_S}^k }{k!}$$
$$P(\text{$k$ reqs of type $E$})=\frac{e^{-{\lambda_E} } {\lambda_E}^k }{k!}$$
So, we will have:
$$P(\text{an $S$ will be granted}) = P(\text{there exists $0$ of $E$} ) = e^{-{\lambda_E} }$$
$$P(\text{an $E$ will be granted}) = P(\text{there exists $0$ of $E$}) \cdot P(\text{there exists $0$ of $S$}) = e^{-{\lambda_E} } \cdot e^{-{\lambda_S} }$$
And then to get the expected value of success, we will weigh these two probabilities by their corresponding arrival rates.
I think my calculations are not correct, and I am confused how to use $t$, the average length of each request.