Limit theorems on maximums of Poisson sequences and their applications to queuing theory

We consider a sequence of independent Poisson random variables $X_n$ with parameters $\lambda_n$. Asymptotic properties of maximums $Y_n=\max_{1\le i\le n}X_i$ under the dependence on properties of the sequence $\{\lambda_n\}$ are studied. The results obtained are applied to investigating the maximal number of claims in the system $M|G|\infty$ with parameters dependent on time.