Heavy traffic analysis of state-dependent fork-join queues with triggers

In this article we consider the approximation of the number of customers of a Poisson-type fork-join system of two queues by a reflected stochastic differential equation (RSDE). The model allows for state-dependent (service and arrival) rates. Also, a signal called ldquotriggerrdquo is considered. These signals are used to move a customer from one queue to the other. The model is applied to a web search system where these signals can be used to reduce imbalance (and as a consequence, the system response time). A stochastic optimal control problem is formulated in order to find the best routing policy of signals which reduces imbalance. Results obtained from a simulation indicate significant reduction in response time for the system using this optimal policy when compared to the system that does not use this strategy.