Tails of waiting times and their bounds

Tails of distributions having the form of the geometric convolution are considered. In the case of light-tailed summands, a simple proof of the famous Cramér asymptotic formula is given via the change of probability measure. Some related results are obtained, namely, bounds of the tails of geometric convolutions, expressions for the distribution of the 1st failure time and failure rate in regenerative systems, and others. In the case of heavy-tailed summands, two-sided bounds of the tail of the geometric convolution are given in the cases where the summands have either Pareto or Weibull distributions. The results obtained have the property that the corresponding lower and upper bounds are tailed-equivalent. This revised version was published online in June 2006 with corrections to the Cover Date.     

Regions of stability of queuing systems

A method of stability analysis is considered for the solution of the inverse queuing problem, and quantitative estimates are obtained.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei — Trudy Seminara, pp. 107–115, 1980.The author expresses his gratitude to V. V. Kalashnikov for a useful discussion of this paper.     

Multicriterion commutative effects in simplest models of queueing systems

We investigate the effect of two types of communication of simplest queueing systems on the mean queue length and the relaxation time of the composite system. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.