Maxima in the system \(\operatorname {M}|\operatorname {G}|\infty\) with the input flow intensity depending on state (Q1406383)

The paper deals with the system of commercial service of \(\text{M}|\text{G}|\infty\) type with the input flow intensity depending on the system state. Let \(M_n\) be the maximum number of orders on the \(n\) periods of regeneration. The author cites a number of assertions of the type: if \( 0<r<1\), then \(\limsup_{n\to\infty} M_n/\log_a n\leq 1\), where \(r = \mu \limsup_{k\to\infty} \lambda_k/k\), \( \lambda_k\to 0\), is the order of intensity and \(\mu\) is the average service time.