An infinite queue single server model is considered where requests arrive from independent Poisson streams and demand service according to arbitrary distribution functions which may be different for different requests. Associated with each request is an urgency number which, together with request's time of arrival, defines a deadline for beginning its service. This relative urgency discipline has at its two limiting case the fint-come first-serve and head of the line discipline. In [1] the mean waiting time is computed approximately and dose bounds are derived there. Here we present simulation results, derive close approximations for the tails of the waiting-time distribution functions and compare them to those of the two limiting cases.