Analysis of general queuing system by selection functions

This work presents method for Lindley's equation spectral solution by using of selection functions for distribution approximation. Proposed method is based on approximation of “upstream” distribution spans by low order polynomials, while “downstream” distribution spans are approximated by sums of damped exponentials with low number of sum terms. Expected waiting time of demand in queue may be evaluated via numerical solution of linear algebraic equation. We demonstrated the effectiveness of proposed method by example of research of system W / P /1, where W and P are Weibull and Pareto distributions respectively. Solution accuracy depends on approximation accuracy for distributions performed during solving Lindley equation by spectral method. Method of selection functions provided to exchange Weibull distribution by two-part distribution containing “upstream” and “downstream” spans approximated by proposed approach. We showed that described approximation provides substantially smaller error in comparison with utilization unital approximation for whole distribution by the sum of damped exponentials. This cross-linking distribution makes able easy to produce its Laplace transform, that leads solution of considered complex problem to numerical solution of linear algebraic equation.