(Solved by Humans)-A 4-stage buffer is operated as follows. At time t = n-, the

Paper Details

A 4-stage buffer is operated as follows. At time t = n-, the contents of the stages 3 and 4 are read out and replaced by what had been the contents of stages 1 and 2 respectively (t = n- is known as the buffer read-out time). Then at t = n+, a newly arriving message is read in (t = n+ is known as the buffer read-in time). The lengths of the arriving messages are independent and identically distributed random variables that are equally likely to fill either exactly 1, or exactly 2, or exactly 3 buffer stages. A message that fills exactly two buffer stages gets read into stages 1 and 2. A message that fills only one buffer stage gets read into stage 2 with stage 1 being left empty. A message of length 3 gets read into stages 1, 2, and 3 if stage 3 is empty when it arrives. If stage 3 is occupied when a length-3 message arrives, a buffer overflow occurs, in which case that length-3 message is routed to another buffer and stages 1 and 2 are left empty. Let Xn denote the number of buffer stages occupied at t = n (that is, after read-out and before readin).