Paradoxical behavior of infinite-server and loss systems having decreasing failure rate arrivals (Q2892435)

The main aim of this paper is to present a theoretical explanation of the paradoxical behaviour that if the inter-arrival time is a decreasing failure rate (DFR) random variable, then the stationary number of customers in the system \(N\) becomes stochastically less variable, as the service time becomes stochastically more variable. All the moments of \(N\) decrease except for the first moment. For the proof of this statement, the second law of thermodynamics, which states that the entropy of an isolated system is non-decreasing, is used. In other words, the entropy of the inter-departure time \(D\) in the infinite isolated system (here a \(GI/GI/\infty\) system, in which self-organization never occurs) is greater than or equal to the entropy of the inter-arrival time \(A\).NEWLINENEWLINENext, the obtained results show that, although the service time variability increases and the inter-arrival time distribution remains unchanged, the number of departures in \((0, t)\) written by \(N_{D}(t)\) becomes stochastically less variable. That gives the observed paradoxical behaviour for the stationary number of customers in the system. After that, the author proves that the blocking probability for the \(GI(DFR)/D(MFR)/c/c\) is largest in \(GI(DFR)/GI(MFR)/c/c\) loss system with the service time defined as a monotone failure rate (MFR) random variable. In particular, the \(GI(DFR)/GI/1/1\) system may be characterized by the paradoxical behaviour through the concavity of the DFR renewal function.NEWLINENEWLINEThe obtained results are applicable for any number of queueing models in many application areas including telephone, computer and telecommunication systems.