Discrete Time Multi-servers Loss Systems and Stochastic Approximation with Delayed Groups of Customers

The stochastic approximation procedure with delayed groups of delayed customers is investigated. The Robbins-Monro stochastic approximation procedure is adjusted to be usable in the presence of delayed groups of delayed customers. Two loss systems are introduced to get an accurate description of the proposed procedure. Each customer comes after fixed time-intervals with the stage of the following customer is accurate according to the outcome of the preceding one, where the serving time of a customer is assumed to be discrete random variable. Some applications of the procedure are given where the analysis of their results is obtained. The analysis shows that efficiencies of the procedure can be increased by minimizing the number of customers of a group irrespective of their service times that may take maximum values. Efficiencies depend on the maximum service time of the customer and on the number of customers of the group. The most important result is that efficiencies of the procedure are increased by increasing the service time distributions as well as service times of customers .This new situation can be applied to increase the number of served customers where the number of served groups will also be increased. The results obtained seem to be acceptable. In general, our proposal can be utilized to other stochastic approximation procedures to increase the production in many fields such as medicine, computer sciences, industry, and applied sciences.

Insensitivity of the mean field limit of loss systems under SQ(d) routeing

In this paper, we study a large multi-server loss model under the SQ(d) routeing scheme when the service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity, giving rise to a mean field model. The fixed point of the limiting mean field equations (MFEs) was seen to be insensitive to the service time distribution in simulations, but no proof was available. While insensitivity is well known for loss systems, the models, even with state-dependent inputs, belong to the class of linear Markov models. In the context of SQ(d) routeing, the resulting model belongs to the class of nonlinear Markov processes (processes whose generator itself depends on the distribution) for which traditional arguments do not directly apply. Showing insensitivity to the general service time distributions has thus remained an open problem. Obtaining the MFEs in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFEs and then show that the MFEs have a unique fixed point that coincides with the fixed point in the exponential case, thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean field limit.