Task allocation in a multi-server system

We consider a slotted queueing system with $C$ servers (processors) that can handle tasks (jobs). Tasks arrive in batches of random size at the start of every slot. Any task can be executed by any server in one slot with success probability $alpha$. If a task execution fails, then the task must be handled in some later time slot until it has been completed successfully. Tasks may be processed by several servers simultaneously. In that case, the task is completed successfully if the task execution is successful on at least one of the servers. We determine the distribution of the number of tasks in the system for a broad class of task allocation strategies. Subsequently, we examine the impact of various allocation strategies on the mean number of tasks in the system and the mean response time of tasks. It is proven that both these performance measures are minimized by the strategy which always distributes the tasks over the servers as evenly as possible. Some numerical experiments are performed to illustrate the performance characteristics of the various strategies for a wide range of scenarios.