Investigation of almost deterministic queueing systems (Q1586602)

The deterministic queueing model \(\text{G}|\text{G}|1 |\infty\) under hugh loading is disturbed by a small random term. Namely, for waiting time \(w_{i+1} = \max \{ 0, w_i + a_i - b_i \}\), it is supposed \(a_i - b_i = -\varepsilon + {\varepsilon}^{\alpha} \Delta_i\), \(i \geq 0\), where \((\Delta_i)\) is a sequence of i.i.d. r.v., \({\mathbf E} \Delta_i = 0\), \({\mathbf D} \Delta_i = d\), and there exists \(\mu > 0\) such that \({\mathbf E} e^{\mu \Delta_i} < \infty\). Asymptotic behaviour of the stationary distribution for the Markov chain \(w_i , i\geq 0,\) is studied. It is given also an asymptotic analysis of the system G\(|\text{G}|1 |\infty\) with group service when the size of group \(n\) tends to infinity. The obtained results create a mathematical basis for the study of an insurance system with a small insurance percent, a small initial capital, and a small ruin probability.