The paper under review considers the \(M/M/N/N+R\) queueing system, in which the rate of arrival is \(\lambda\) and the reciprocal of the mean service time in each server is \(\mu\). Let \(\lambda_j=\lambda\) and \(\mu_j=\mu\min\{j,N\}\), and let \(p_j(t)=\operatorname{P}\{X(t)=j\}\), where \(X(t)\) is a birth-and-death process with the birth and death rates \(\lambda_j\) and \(\mu_j\), respectively, \(0<j\leq N+R\). Denote \(a=\frac{\lambda}{\mu}\). Then, for \(0\leq j\leq N\), the stationary probabilities are \[ \pi_j=c\frac{a^j}{j!}, \] and, for \(N<j\leq N+R\), they are \[ \pi_j=c\frac{a^j}{N!N^{j-N}}, \] where \(c\) is the normalisation constant.
The paper studies bounds for the rate of convergence, as \(t\to\infty\), of the probabilities \(p_j(t)\) to the stationary probabilities \(\pi_j\), and study the behaviour of \(p_j(t)\) as a function of \(R\), \(N\) and arrival rate \(\lambda\), allowing the parameter \(\lambda\) to be dependent on \(N\).