It is shown that efficient reversible variable length codes (RVLCs) with numerous codewords can be obtained if suboptimal RVLCs for the average distributions of monotone sources with relatively small alphabet sizes are multiplied by some fixed-length codes. Employing these RVLCs, the best known upper bounds on the average redundancy of optimal RVLC are almost halved. In particular, it is proved that the average redundancy of optimal RVLC for sources withnsymbols is less than 11/n+0.11345 bits forn> 32. Some other upper bounds are also derived which are either suboptimal in some sense for smallnor easily computable for largenor descriptive of asymptotic behavior. Moreover, we prove that the penalty of using optimal RVLC instead of Huffman code is less than 0.0848 bits for almost all sources with sufficiently large alphabet size. These results provide stronger evidence that, overall, the cost of benefiting from the desired properties of RVLC is not significant in terms of the redundancy.

The problem of designing a fixed code for encoding all monotone sources with a large alphabet size n is studied. It is proved that if the code-lengths of a code sequence are of order O(log n), then the redundancy for almost all monotone sources with n symbols is very close to that for a specific distribution, so-called average distribution. Noting this point, for any large alphabet size n, a new code with a very simple structure is proposed whose redundancy is close to that of the Huffman code for almost all monotone sources with alphabet size n.

The performance of optimal prefix-free encoding for memoryless sources with a large alphabet size is studied. It is shown that the redundancy of the Huffman code for almost all sources with a large alphabet size$n$is very close to that of the average distribution of the monotone sources with$n$symbols. This value lies between 0.02873 and 0.02877 bit for sufficiently large$n$.

In order to determine how suboptimal the Shannon code is, one should compare its performance with that of the optimal code, i.e., the corresponding Huffman code, in some sense. It is well known that in the worst case the redundancy of both the Shannon and Huffman codes can be arbitrarily close to 1. Beyond this worst case viewpoint, very little is known. In this paper, we compare the performance of these codes from an average point of view. The redundancy is considered as a random variable on the set of all sources with n symbols and its average is evaluated. It is shown that the average redundancy of the Shannon code is very close to 0.5 bits, whereas the average redundancy of the Huffman code is less than n-1(1+ln n)+0.086 bits . It is also proven that the variance of the redundancy of the Shannon code tends to zero as n increases. Therefore, for sources with alphabet size n, the redundancy of the Shannon code is approximately 0.5 bits with probability approaching 1 as n→ ∞.

Consider a smart grid system in which every user may or may not choose to participate in Demand Side Management (DSM). This will lead to a general co-existence problem between participant and non-participant users. To gain insights, first, we show that some existing electricity billing mechanisms suffer from severe fairness and co-existence defects. Next, we propose an alternative billing mechanism that can tackle the coexistence and fairness problems by taking into account not only the users' total load, but also the exact shape of their load profiles. Our analytical results provide mild sufficient conditions on the choice of system parameters to assure fairness. Furthermore, our simulation results confirm that the proposed billing mechanism significantly improves the fairness index of the DSM system.

The Shannon code is a very simple suboptimal scheme for computing prefix-free codelengths for a given memoryless source. A recursive version of the well-known Shannon code (RSh) is investigated in this paper. It has a redundancy which never exceeds that of the Shannon code. Unlike the Huffman code, the RSh code and its variations can be directly applied to sources with an infinitely countable alphabet. The average redundancy, taken over the set of all n-tuple distributions, is considered as a criterion for code comparisons. For n>;40, the average redundancy is approximately 0.14 bits for the RSh code, compared to approximately 0.5 and 0.03 bits for the Shannon and Huffman codes, respectively. For large n, a constrained version of the RSh code, called CRSh, has an average redundancy of only 0.07 bits for unsorted symbol probabilities. If the symbol probabilities are sorted in descending order, then a very simple modification of the RSh algorithm results in a code which is near-optimal from the average redundancy point of view.

For a given memoryless information source, the Huffman code is the optimal prefix-free code in the sense of redundancy. Generally, the length of each codeword in the Huffman code is a function of all symbol probabilities p1, p2, ..., pn. In contrast, with the best known suboptimal code, i.e., the Shannon code, the length of the i-th codeword (i.e. [- log pi]) is a function of only pi. In this paper, two recursive versions of the Shannon code (RYY and RSh) are proposed which have redundancy which lies between that of the Huffman code and the Shannon code. In particular, the redundancy is not greater than that of the Shannon code and the i-th codeword length does not depend on pi+1, pi+2, ..., pn. In order to evaluate the overall performance of the proposed codes, their redundancy is considered as a random variable on the set of all sources with n symbols. An algorithm for generating random n-tuple distributions is derived and the expected value of the redundancy of the resulting codes is estimated. Recently, it was proven that the average redundancy of the Shannon code is around 0.5 bits. Simulation shows that for n > 20 the average redundancy of the proposed codes are about 0.1 and 0.06, while it is approximately 0.03 for the Huffman code.