On observable chaotic maps for queueing analysis
A queueing model based on chaotic mapping offers a number of distinct advantages over both stochastic and static deterministic models. Depending on the type of chaotic map used, such a queue can capture transient behavior, intermittency, steady state behavior, and complex distributions in arrival rates. These characteristics are especially desirable in many queueing applications in transportation. Earlier studies resulted in chaotic queueing models that cannot be estimated using observed arrivals. An alternative queueing model is presented along with methods to specify the model, interpret its results, and estimate its parameters. The proposed queueing model uses chaotic maps of inter-arrival times to generate arrivals so that parameters can be calibrated with observable data. A sample queue based on the ergodic logistic map is presented. To calibrate the mapping based on observed data, a joint parameter and state estimation algorithm is presented using the method of successive averages. An illustration is made with two connected queues to show how a purely deterministic queueing network can still result in a joint invariant distribution. The results offer a positive view of this method and its applicability to queueing problems, particularly in the field of transportation and dynamic network loading.