Bounds on the Benefit of Network Coding for Wireless Multicast and Unicast

In this paper, we explore fundamental limitations of the benefit of network coding in multihop wireless networks. We study two well-accepted scenarios in the field: single multicast session and multiple unicast sessions. We assume arbitrary but fixed topology and traffic patterns for the wireless network. We prove that the gain of network coding in terms of throughput and energy saving of a single multicast session is at most a constant factor. Also, we present a lower bound on the average number of transmissions of multiple unicast sessions under any arbitrary network coding. We identify scenarios under which network coding provides no gain at all, in the sense that there exists a simple flow scheme that achieves the same performance. Moreover, we prove that the gain of network coding in terms of the maximum transport capacity is bounded by a constant factor of at most $(pi)$ in any arbitrary wireless network under all traditional Gaussian channel models. As a corollary, we find that the gain of network coding on the throughput of large homogeneous wireless networks is asymptotically bounded by a constant. Furthermore, we establish theorems which relate a network coding scheme to a simple routing scheme for multiple unicast sessions. The theorems can be used as criteria for evaluating the potential gain of network coding in a given wired or wireless network. Based on these criteria, we find more scenarios where network coding has no gain on throughput or energy saving.