Queueing network model of a call center with customer retrials and impatient customers

The subject of mathematical study and modelling in this paper is an inbound call center that receives calls initiated by customers. A closed exponential queueing network with customer retrials and impatient customers is used as a stochastic model of call processing. A brief review of published results on the application of queueing models in the mathematical modeling of customer service processes in call centers is discussed. The network model is described. The possible customer states, customer routing, parameters, and customer service features are given. The allocation of customers by network nodes at a fixed time fully describes the situation in the call center at that time. The state of the network model under study is represented by a continuous-time Markov chain on finite state space. The model is studied in the asymptotic case under the critical assumption of a large number of customers in the queueing network. The mathematical approach used makes it possible to use the passage to the limit from a Markov chain to a continuous-state Markov process. It is proved that the probability density function of the model state process satisfies the Fokker – Planck – Kolmogorov equation. Using the drift coefficients of the Fokker – Planck – Kolmogorov equation, a system of ordinary differential equations for calculating the expected number of customers in each network node over time can be written. The solution of this system allows for predicting the dynamics of the expected number of customers at the model nodes and regulating the parameters of the call center operation. The asymptotic technique used is applicable both in transient and steady states. The areas of implementation of research results are the design of call centers and the analysis of their workload.