This paper is focused on stability conditions of a multi-server queueing system with regenerative input flow where a random number of servers is simultaneously required for each customer and each server's completion time is constant. It turns out that the stability condition depends on the rate of the input flow and not on its structure.
We consider a multichannel queuing system with heterogeneous servers and regenerative input flow. The necessary and sufficient condition is established and functional limit theorems are proved in overloaded and critically loaded regimes. The ergodicity condition is obtained for multichannel system with unreliable servers. Some approaches for ergodicity of systems with abandonment are discussed.
The order statistics and the empirical mathematical expectation (also called the estimate of mathematical expectation in the literature) are considered in the case of infinitely increasing random variables. The Kolmogorov concept which he used in the theory of complexity and the relationship with thermodynamics which was pointed out already by Poincar\'e are considered. We compare the mathematical expectation (which is a generalization of the notion of arithmetical mean, and is generally equal to infinity for any increasing sequence of random variables) with the notion of temperature in thermodynamics similarly to nonstandard analysis. It is shown that there is a relationship with the Van-der-Waals law of corresponding states. A number of applications of this concept in economics, in internet information network, and self-teaching systems are also considered.