Stochastic Approximation and Compound Delayed Observations with Independent Random Time Delay Distribution

The Robbins-Monro stochastic approximation procedure is modified for application in the presence of compound delayed observations with a time delay. This new case is investigated by considering the final result of components of a compound observation to be the delay of the observation. Under this assumption, the time delay distribution of the compound delayed observation is independent of the time delay distribution of its components. The modified procedure is applied to a multi-server delay-loss system with an approximated random time delay distribution. This approach is represented by considering a loss system with compound delayed observations arriving at each time-unit, where the service time is an integer-valued random variable, parallel inspection stations, and no waiting places. The asymptotic efficiency of the modified procedure is also investigated by studying the asymptotic properties of the procedure. In principle, our proposal can be applied to other stochastic approximation or recursive estimation procedures, where the results of experiments become known only after a random time delay.