Stochastic nontransitivity effects in queueing systems

The paper continues a series of articles devoted to the nontransitivity of the stochastic precedence relation for triplets of independent random variables. Initially, this problem was posed in connection with the application in strength theory. With paired comparisons of iron bars from three factories, a paradoxical situation may arise that the bars from the first factory are "worse" than the bars from the second factory, the bars from the second factory are "worse" than the bars from the third factory, and the bars from the third factory are "worse" than the bars from the first factory. Further, the nontransitivity topic gained popularity for the example of the so-called nontransitive dice. In previous works of the cycle, on the one hand, it was proved that there can be no nontransitivity for many classical continuous distributions, on the other hand, examples of nontransitivity for distributions with polynomial density on a unit interval, as well as for mixtures of normal and exponential distributions of at most than two components. In this paper, we open the topic of the possible influence of nontransitivity on the behavior of stochastic systems. Namely, we study how the nontransitivity of service times relation in the three single-server queueing systems affects the sojourn times, and how in the infinite-server queueing systems it affects the maximum residual service times. The study uses the classic nontransitive triplet of random variables with the same means and variances. In the first case, simulation modeling is used; in the second case, the analytical approach is used.