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

In this technical report we establish fundamental limitations to the benefit of network coding in terms of energy and throughput in multihop wireless networks. Thereby we adopt two well accepted scenarios in the field: single multicast session and multiple unicast sessions. Most of our results apply to arbitrary wireless network and are, in particular, not asymptotic in kind. 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 in wireless networks. This result is in contrast to wired case where the gain of network coding can be an unboundedly large factor [13]. Also, we present a lower bound on the expected number of transmissions of multiple unicast sessions under an arbitrary network coding. We identify scenarios for which the network coding gain for energy saving becomes surprisingly close to 1, in some cases even exactly 1, corresponding to no benefit at all. Moreover, we prove that the gain of network coding in terms of transport capacity is bounded by a constant factor π in any arbitrary wireless network and for all traditional channel models. This shows that the traditional bounds on the transport capacity [1, 9, 10, 15] do not change more than constant factor π if we employ network coding. As a corollary, we find that the gain of network coding on the throughput of large scale homogeneous wireless networks is asymptotically bounded by a constant. Note that our result is more general than the previous work [25] and it is obtained by a different technique. Furthermore, we establish important theorems on employing network coding for multiple unicast sessions. These theorems relate a coding scheme to a simple routing scheme and they can be used as criteria for evaluating the potential gain of network coding in a given wired/wireless network. Then, we identify more scenarios where network coding has no gain on throughout or energy saving of the network. In conclusion, we show that in contrast to wired networks, the network coding gain in wireless networks is constraint by fundamental limitations. It is also important to mention that wireless channel model is different than wired network with bidirectional links.