The construction and analysis of call-center model in overload traffic condition

Abstract

The functional and mathematical models of call center working in case of overload are constructed and analyzed. In the model the following features are considered: the possibility of serving of coming request by IVR (Interactive Voice Response); the option of waiting the beginning of service in case of blocking and the opportunity of request repetition in case of occupation of all waiting positions or unsuccessful finishing of waiting time. Markov process that describes model functioning is defined. Main performance measures of requests coming and serving are given with help of values of stationary probabilities of model's states. The values of performance measures are found after solving the system of state equations by Gauss-Zeidel iterative approach. Expressions that relates the model's main performance measures in form of local and global conservation laws are found. The obtained results can be used for indirect measurement of intensity of primary requests and the probability of call repetition. It is shown how to use the model and the derived results for reducing the negative effects of overload by filtering the input flows of primary and repeated attempts. The usage of the model for calculation of the numbers of operators and waiting places required to serve the incoming traffic flows with given value of probability of call losses and mean value of waiting the beginning of service is considered. Numerical results that illustrate the implementation of the derived expressions and algorithms are given.

Estimation of the Performance Measures of a Group of Servers Taking into Account Blocking and Call Repetition before and after Server Occupation

“…Relationship ( 5) can be used for the estimation of the error caused by truncation of the used state space (see Appendix A), for the indirect measurement of the intensity of primary calls and for solving a variety of other problems arising in the exact and approximate analysis of the introduced model [2,[18][19][20]. To find P(i) and J(i), it is necessary to solve the system of state Equation (1).…”

“…It allows use of Markov processes for model description but seriously complicates the numerical analysis. Performance measures can be found by solving the system of state equations either using the Gauss-Seidel iterative algorithm [2,[18][19][20] or by matrix-geometric methods, when the corresponding Markov process has a suitable structure [17,21,22]. The first approach is time-consuming, and the last is numerically unstable for real values of input parameters because of the necessity to convert a large matrix.…”

“…Furthermore, we refer to this model as a basic model with repeated calls. Performance measures of this model can be found using a recursive algorithm based on the concept of truncation of the model's state space [2,17,19,20,23]. For this model, the asymptotic formulae for the calculation of characteristics in the case of extreme load can be also found in explicit form [24].…”

“…The closest model with retrials to the model considered in this paper was analyzed in [2,19]. That model consists of a fully available group of servers with the possibility of waiting.…”

“…The values of the characteristics are obtained after solving the system of state equations by the Gauss-Seidel iterative algorithm. The distinguishing features of the model and results obtained in this paper, compared to [2,19], are as follows:…”