An increase in volumes of data and a shift towards live data enabled a stronger focus on resource-intensive tasks which run continuously over long periods. A Grid has potential to offer the required resources for these tasks, while considering a fair and balanced allocation of resources among multiple client agents. Taking this into account, a Grid might be unwilling to allocate its resources for long time, leading to task interruptions. This problem becomes even more serious if an interruption of one task may lead to the interruption of dependent tasks. Here, we discuss a new strategy for resource re-allocation which is utilized by a client with the aim to prevent too long interruptions by re-allocating resources between its own tasks. Those re-allocations are suggested by a client agent, but only a Grid can re-allocate resources if agreed. Our strategy was tested under the different Grid settings, accounting for the adjusted coefficients, and demonstrated noticeable improvements in client utilities as compared to when it is not considered. Our experiment was also extended to tests with environmental modelling and realistic Grid resource simulation, grounded in real-life Grid studies. These tests have also shown a useful application of our strategy.