  In the classic gossip-based model of communication for disseminating
information in a network, in each time unit, every node $u$ is allowed to
contact a single random neighbor $v$. If $u$ knows the data (rumor) to be
disseminated, it disperses it to $v$ (known as PUSH) and if it does not, it
requests it from $v$ (known as PULL). While in the classic gossip model, each
node is only allowed to contact a single neighbor in each time unit, each node
can possibly be contacted by many neighboring nodes.
  In the present paper, we consider a restricted model where at each node only
one incoming request can be served. As long as only a single piece of
information needs to be disseminated, this does not make a difference for push
requests. It however has a significant effect on pull requests. In the paper,
we therefore concentrate on this weaker pull version, which we call 'restricted
pull'.
  We distinguish two versions of the restricted pull protocol depending on
whether the request to be served among a set of pull requests at a given node
is chosen adversarially or uniformly at random. As a first result, we prove an
exponential separation between the two variants. We show that there are
instances where if an adversary picks the request to be served, the restricted
pull protocol requires a polynomial number of rounds whereas if the winning
request is chosen uniformly at random, the restricted pull protocol only
requires a polylogarithmic number of rounds to inform the whole network.
Further, as the main technical contribution, we show that if the request to be
served is chosen randomly, the slowdown of using restricted pull versus using
the classic pull protocol can w.h.p. be upper bounded by $O(\Delta / \delta
\log n)$, where $\Delta$ and $\delta$ are the largest and smallest degree of
the network.
