  We consider a renewal process that is conditioned on the number of events in
a fixed time horizon. We prove that a centered and scaled version of this
process converges to a Brownian bridge, as the number of events grows large,
which relies on first establishing a functional strong law of large numbers
result to determine the centering. These results are consistent with the
asymptotic behavior of a conditioned Poisson process. We prove the limit
theorems over triangular arrays of exchangeable random variables, obtained by
conditionning a sequence of independent and identically distributed renewal
processes. We construct martingale difference sequences with respect to these
triangular arrays, and use martingale convergence results in our proofs. To
illustrate how these results apply to performance analysis in queueing, we
prove that the workload process of a single server queue with conditioned
renewal arrival process can be approximated by a reflected diffusion having the
sum of a Brownian Bridge and Brownian motion as input to its regulator mapping.
