  Standing sausage modes in flare loops are important for interpreting
quasi-periodic pulsations (QPPs) in solar flare lightcurves. We propose an
inversion scheme that consistently uses their periods $P$ and damping times
$\tau$ to diagnose flare loop parameters. We derive a generic dispersion
relation governing linear sausage waves in pressure-less straight tubes, for
which the transverse density inhomogeneity takes place in a layer of arbitrary
width $l$ and is of arbitrary form. We find that $P$ and $\tau$ depend on the
combination of $[R/v_{\rm Ai}, L/R, l/R, \rho_{\rm i}/\rho_{\rm e}]$, where $R$
is the loop radius, $L$ is the looplength, $v_{\rm Ai}$ is the internal
Alfv\'en speed, and $\rho_{\rm i}/\rho_{\rm e}$ is the density contrast. For
all the density profiles examined, $P$ and $\tau$ experience saturation when
$L/R \gg 1$, yielding an inversion curve in the $[R/v_{\rm Ai}, l/R, \rho_{\rm
i}/\rho_{\rm e}]$ space with a specific density profile when $L/R$ is
sufficiently large. When applied to a spatially unresolved QPP event, the
scheme yields that $R/v_{\rm Ai}$ is the best constrained, whereas $l/R$
corresponds to the other extreme. For spatially resolved QPPs, while $L/R \gg
1$ cannot be assumed beforehand, an inversion curve remains possible due to
additional geometrical constraints. When a spatially resolved QPP event
involves another mode, as is the case for a recent event, the full set of
$[v_{\rm Ai}, l, \rho_{\rm i}/\rho_{\rm e}]$ can be inferred. We conclude that
the proposed scheme provides a useful tool for magneto-seismologically
exploiting QPPs.
