  Let $P = \{p(i)\}$ be a measure of strictly positive probabilities on the set
of nonnegative integers. Although the countable number of inputs prevents usage
of the Huffman algorithm, there are nontrivial $P$ for which known methods find
a source code that is optimal in the sense of minimizing expected codeword
length. For some applications, however, a source code should instead minimize
one of a family of nonlinear objective functions, $\beta$-exponential means,
those of the form $\log_a \sum_i p(i) a^{n(i)}$, where $n(i)$ is the length of
the $i$th codeword and $a$ is a positive constant. Applications of such
minimizations include a problem of maximizing the chance of message receipt in
single-shot communications ($a<1$) and a problem of minimizing the chance of
buffer overflow in a queueing system ($a>1$). This paper introduces methods for
finding codes optimal for such exponential means. One method applies to
geometric distributions, while another applies to distributions with lighter
tails. The latter algorithm is applied to Poisson distributions. Both are
extended to minimizing maximum pointwise redundancy.
