Bounds on the rate of convergence for inhomogeneous $M/M/S$ systems with   either state-dependent transitions, or batch arrivals and service, or both
  In this paper one presents method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically one considers inhomogeneous $M/M/S$ queueing system with possibly state-dependent arrival and service intensities and additionally possible batch arrivals and batch service. The unified approach based on logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side result, one shows by virtue of numerical examples that the approach based on logarithmic norm can also be used for approximation of limiting characteristics (idle probability and mean number of customers in the system) of the considered systems with given approximation error. Extensive numerical examples are provided. 
Alexander Zeifman, Anna Korotysheva, Yacov Satin, Rostislav Razumchik,   Victor Korolev, Ksenia Kiseleva
math.PR