Takacs' asymptotic theorem and its applications: A survey

The book of Lajos Tak\'acs \emph{Combinatorial Methods in the Theory of Stochastic Processes} has been published in 1967. It discusses various problems associated with $$P_{k,i}=\mathrm{P}{\sup_{1\leq n\leq\rho(i)}(N_n-n)<k-i},\leqno(*)$$ where $N_n=\nu_1+\nu_2...+\nu_n$ is a sum of mutually independent, nonnegative integer and identically distributed random variables, $\pi_j=\mathrm{P}\{\nu_k=j\}$, $j\geq0$, $\pi_0>0$, and $\rho(i)$ is the smallest $n$ such that $N_n=n-i$, $i\geq1$. (If there is no such $n$, then $\rho(i)=\infty$.) (*) is a discrete generalization of the classic ruin probability, and its value is represented as $P_{k,i}={Q_{k-i}}/{Q_k}$, where the sequence $\{Q_k\}_{k\geq0}$ satisfies the recurrence relation of convolution type: $Q_0\neq0$ and $Q_k=\sum_{j=0}^k\pi_jQ_{k-j+1}$. Since 1967 there have been many papers related to applications of the generalized classic ruin probability. The present survey concerns only with one of the areas of application associated with asymptotic behavior of $Q_k$ as $k\to\infty$. The theorem on asymptotic behavior of $Q_k$ as $k\to\infty$ and further properties of that limiting sequence are given on pages 22-23 of the aforementioned book by Tak\'acs. In the present survey we discuss applications of Tak\'acs' asymptotic theorem and other related results in queueing theory, telecommunication systems and dams. Many of the results presented in this survey have appeared recently, and some of them are new. In addition, further applications of Tak\'acs' theorem are discussed.