Consider the problem of broadcasting an i.i.d, source sequence $X=\{X_{i}\}_{i=1}^{N}$ (possibly $N \rightarrow\infty$) to $n$ listeners over a discrete broadcast channel, consisting or $n$ channels with capacities $C_{1}=C_{max}\geq C_{2}\geq. \geq\ C_{n}=C_{min}$. Let the tuple ${\bf D}=(D_{1}, D_{2}, \ldots, D_{n})$ represent the average distortion in reconstructing sources at the $n$ listeners. The problem of characterizing all achievable tuples $D$ is till open for a general case. For a fairly general class of discrete channels, we prove the achievability of the tuple $\wp^{n}(\rho_{1}, \rho_{2}, \ldots, \rho_{n})=({\Bbb D}_{{X}}(\rho_{1}C_{{\rm I}}- \zeta), {\Bbb D}_{X}(\rho_{2}C_{2}-\zeta), \ldots, {\Bbb D}_{X}(\rho_{n}C_{{n}}-\zeta))$, provided that $\lambda_{i}= (\rho_{i}C_{i}-\rho_{i+1}C_{i+1})/C_{i} > 0$, for 1 $\leq i\leq n-1, \lambda_{n}=\rho_{{n}}$ and $\sum\nolimits_{i=1}^{n-1}\lambda_{{i}}\ \leq 1$, where ${\Bbb D}_{X}(R)$ is the distortion rate function of $X$. The penalty term $\zeta=1/2$ for a general source with real alphabets and is $\zeta=0$ if $X$ is progressively refinahle. The factor $0\leq\rho_{i}\leq 1$ is called the utilization of the $i^{th}$ channel. As an example, for $n=2$, we show that $\wp^{2}(1/(2-C_{2}/C_{1}), 1/(2-C_{1}/C_{2}))$ is achievable for any $C_{1}, C_{2}$. Furthermore, the common utilization of $\rho=(1+\ln(C_{max}/C_{min}))^{-1}$ is shown to be achievable for ${\bf all}$ channels. Conversely, we find examples of channels, namely erasure switch-to-talk channels, for which the proposed achievable utilizations are tight. In particular, while $\wp^{3}(2/3, 2/3, 2/3)$ is achievable for any compatible broadcast channel with capacities $C_{1}=2C_{2}=2C_{3}$, for any $\delta > 0$, we find examples of channels for which ${\wp}^{3}(2/3+\delta, 2/3+\delta, 2/3+ \delta)$ is not achievable.