Comparison of different approaches for estimating the average waiting time in the queue for queuing system Н2/Н2/1

The article discusses two different approaches for estimating the average waiting time of a request in a queue for queuing systems with second order hyperexponential distribution of interarrival and service time. The first approach involves solving the Lindley integral equation using the spectral method and reduces to finding an expression for the spectral decomposition as a composition of two multipliers that would give a rational function of an argument. The second approach is based on the assumption of ergodicity of the initial sequence of waiting time for a request in a queue considering the rational form of the Laplace transform from the exponent. The key point of this approach is the usage of the characteristic function defined by the Laplace transform for the probability density function of the sum of the random variables. To calculate the average delay time in the queue, the parameters of hyperexponential distributions are determined with the approximation of random variables defining the interarrival and service time at the level of three moment characteristics. This approach is designed to improve the adequacy and reliability of queuing models. In addition, only for the input distributions of the queuing system described by the hyperexponential distribution of the second order we can get an analytical solution for average waiting time.