The “Sensitive” Markovian queueing system and its application for a call center problem

In this paper, we study the $$M_n/M_n/c/K+M_n$$ M n / M n / c / K + M n queueing system where customers arrive according to a Poisson process with state-dependent rates. Moreover, the rates of the exponential service times and times to abandonment of the queued customers can also change whenever the system size changes. This implies that a customer may experience different service rates throughout the time she is being served. Similarly, a queued customer can change her patience time limits while waiting in the queue. Thus, we refer to the analyzed system as the “sensitive” Markovian queue. We conduct an exact analysis of this system and obtain its steady-state performance measures. The steady-state system size distribution yields itself via a birth–death process. The times spent in the queue by an arbitrary or an eventually served customer are represented as the times until absorption in two continuous-time Markov chains and follow Phase-type distributions with which the queueing time distributions and moments are obtained. Then, we demonstrate how the $$M_n/M_n/c/K+M_n$$ M n / M n / c / K + M n queue can be employed to approximately yet accurately estimate the performance measures of the $$M_n/GI/c/K+GI$$ M n / G I / c / K + G I type call center.