  Stochastic dominance is a technique for evaluating the performance of online
algorithms that provides an intuitive, yet powerful stochastic order between
the compared algorithms. Accordingly this holds for bijective analysis, which
can be interpreted as stochastic dominance assuming the uniform distribution
over requests. These techniques have been applied to some online problems, and
have provided a clear separation between algorithms whose performance varies
significantly in practice. However, there are situations in which they are not
readily applicable due to the fact that they stipulate a stringent relation
between the compared algorithms.
  In this paper, we propose remedies for these shortcomings. First, we
establish sufficient conditions that allow us to prove the bijective optimality
of a certain class of algorithms for a wide range of problems; we demonstrate
this approach in the context of some well-studied online problems. Second, to
account for situations in which two algorithms are incomparable or there is no
clear optimum, we introduce the bijective ratio as a natural extension of
(exact) bijective analysis. Our definition readily generalizes to stochastic
dominance. This renders the concept of bijective analysis (and that of
stochastic dominance) applicable to all online problems, and allows for the
incorporation of other useful techniques such as amortized analysis. We
demonstrate the applicability of the bijective ratio to one of the fundamental
online problems, namely the continuous $k$-server problem on metrics such as
the line, the circle, and the star. Among other results, we show that the
greedy algorithm attains bijective ratios of $O(k)$ consistently across these
metrics. These results confirm extensive previous studies that gave evidence of
the efficiency of this algorithm on said metrics in practice, which, however,
is not reflected in competitive analysis.
