The influence of the memory for a special permutation channel

The author considers the following model of a permutation channel. In each time unit two sources produce one bit each (0 or 1 with probability P(0)=P(1)=0.5). These two bits arrive at an organizer, who in the same time unit has to output one bit. The other bit he may store in some memory device. If it is possible the output bit must be a 1. So if the arriving bits are 11, 10 or 01, then the organizer will send a 1 for sure. If both sources produce a 0, then the organizer may send a 1, which is stored in the memory device (and the size of the memory will be reduced by one bit in this case). If this is not possible, then the organizer must send a 0. A natural question is: how much influence does the size of the memory have on the behaviour of the sequence of bits transmitted by the organizer? As a simple measure for the influence of the memory the author considers the expected value of the first occurrence of a 0 in this sequence