Stochastic approximation with series of delayed observations

The stochastic approximation procedure with series of delayed observations is investigated. The procedure is formed by modifying the Robbins-Monro stochastic approximation procedure to be applicable in the presence of series of delayed observations. The modified procedure depends on a new base concerning the relation between service time of the series and service times of its components. Two loss systems are introduced for application to the proposed procedure. This new situation can be applied to increase the production of items in many fields such as biological, medical, life time experiments, and some industrial projects, where items are realized after random time delays. The efficiency of the procedure is computed. Our proposal is general and we expect that it can be applied to any other stochastic approximation procedure.