Estimation of the intensity of the flow of nonmonotone refusals in the queueing system \((\leq\lambda)/\text{G}/m\) (Q2722152)

The author studies the behaviour of queueing systems under the condition of light traffic. The \((\leq \lambda)/\text{G}/m\) queueing system, where the symbol \((\leq \lambda)\) means that, independently of the previous behavior, the probability of demand entry in the interval \(dt\) does not exceed \(\lambda dt,\) is considered. The case where the queue length attains the level \(r\geq m+1\) at the first time in the busy period is called a system failure. For the intensity \(\mu_{1}(t)\) of flow of homogeneous events associated with nonmonotone failure of the system, it is found a bound \(\mu_{1}(t)=O(\lambda^{r+1}\alpha_{1}^{m-1}\alpha_{r-m+1}),\) where \(\alpha_{k}\) is the \(k\)th moment of the service time distribution. The most difficult point in proving limit theorems and estimations in the case of light traffic is to obtain estimations for the probability of nonmonotone failures. Traditionally the busy period of the system \(\dots/\text{G}/m\) is majorized by the busy period of the system \(\dots/\text{G}/1.\) This leads to the overstated estimation for probability of nonmonotone failure on the busy period. In the present paper it is proposed an approach based on another interpretation of nonmonotone failures: these events are counted not from the beginning of the busy period, but from the beginning of the moment of the first block of claim in the busy period.