ON SOME QUEUE LENGTH CONTROLLED STOCHASTIC PROCESSES

The authors study the input, output and queueing processes}n a gene-ral controlled sngle-server bulk queueng system. It s supposed that nter-ar-r}val time, service time, batch sze of arr}vng units and the capacity of the ser-ver depend on the queue length. The authors establ}sh an ergod}c}ty criterion for both the queueing pro-cess with continuous time parameter and the embedded process, study their transient and steady state behavior and prove ergodic theorems for some func-tionals of the nput, output and queue}ng processes. The following results are obtained: Invariant probability measure of the embedded process, stationary dstr}but}on of the process w}th continuous tme parameter, expected value of a busy period, rates of input and output processes and the relative speed of con-vergence of the expected queue length. Various examples (including an optimi-zation problem) llustrate methods developed}n the paper