  In this paper, we dynamically select the transmission rate and design
wireless network coding to improve the quality of services such as delay for
time critical applications. In a network coded system, with low transmission
rate and hence longer transmission range, more packets may be encoded, which
increases the coding opportunity. However, low transmission rate may incur
extra transmission delay, which is intolerable for time critical applications.
We design a novel joint rate selection and wireless network coding (RSNC)
scheme with delay constraint, so as to maximize the total benefit (where we can
define the benefit based on the priority or importance of a packet for example)
of the packets that are successfully received at the destinations without
missing their deadlines. We prove that the proposed problem is NP-hard, and
propose a novel graph model to mathematically formulate the problem. For the
general case, we propose a transmission metric and design an efficient
algorithm to determine the transmission rate and coding strategy for each
transmission. For a special case when all delay constraints are the same, we
study the pairwise coding and present a polynomial time pairwise coding
algorithm that achieves an approximation ratio of 1 - 1/e to the optimal
pairwise coding solution, where e is the base of the natural logarithm.
Finally, simulation results demonstrate the superiority of the proposed RSNC
scheme.
