Fluid approximation analysis of a call center model with time-varying arrivals and after-call work

Important features to be included in queueing-theoretic models of the call center operation are multiple servers, impatient customers, time-varying arrival process, and operator's after-call work (ACW). We propose a fluid approximation technique for the queueing model with these features by extending the analysis of a similar model without ACW recently developed by Liu and Whitt (2012). Our model assumes that the service for each quantum of fluid consists of a sequence of two stages, the first stage for the conversation with a customer and the second stage for the ACW. When the duration of each stage has exponential, hyperexponential or hypo-exponential distribution, we derive the time-dependent behavior of the content of fluid in each stage of service as well as that in the waiting room. Numerical examples are shown to illustrate the system performance for the cases in which the input rate and/or the number of servers vary in sinusoidal fashion as well as in adaptive ways and in stationary cases.