On the lifetime of a random binary sequence

Consider a system with m elements which is used to fulfill tasks. Each task is sent to one element which fulfills a task and the outcome is either fulfillment of the task (''1'') or the failure of the element (''0''). Initially, m tasks are sent to the system. At the second step, a complex of length m"1 is formed and sent to the system, where m"1 is the number of tasks fulfilled at the first step, and so on. The process continues until all elements fail and the corresponding waiting time defines the lifetime of the binary sequence which consists of ''1'' or ''0''. We obtain a recursive equation for the expected value of this waiting time random variable.