We present an elementary model of random size varying population given
  by a stationary continuous state branching process.  For this model we
  compute the joint distribution of:  the time to the most recent common
  ancestor,  the size  of the  current population  and the  size  of the
  population  just before  the most  recent common  ancestor  (MRCA). In
  particular  we show a  natural mild bottleneck  effect as  the size  of the
  population  just before the  MRCA is  stochastically smaller  than the
  size of  the current  population.  We also  compute the number  of old
  families which  corresponds to the  number of individuals  involved in
  the last coalescent event of  the genealogical tree.  By studying more
  precisely  the genealogical structure  of the  population, we  get
  asymptotics for  the number  of ancestors just  before the
  current  time.  We  give  explicit  computations in  the  case of  the
  quadratic  branching  mechanism.   In  this  case,  the  size  of  the
  population  at the MRCA  is, in  mean, less  by 1/3  than size  of the
  current population size.  We also provide in this case the fluctuations for
  the renormalized number of ancestors.