Performance Trade-offs in a Network Coding Router Omer H. Abdelrahman and Erol Gelenbe, Fellow, IEEE Dept. of Electrical & Electronic Engineering Imperial College London, UK Email:{o.abd06, e.gelenbe}@imperial.ac.uk Abstract—We consider the problem of optimizing the perfor- mance of a network coding router with two stochastic flows. We develop a queueing model which accounts for the fact that coding is not performed when packets are transmitted, but is done by a separate program or hardware which operates independently of the hardware that sends packets out over links. We formulate and solve a constrained optimization problem which provides the optimal time that the router should wait before sending the information that it has uncoded, so that the average response time of the system is minimized. The trade-offs between delay and bandwidth or energy associated with the choice of the waiting time are also investigated, and the results indicate that network coding offers significant performance gains in a moderate to heavily loaded system. I. I NTRODUCTION In the emerging field of network coding [1], routers are allowed to process and mix information within packets before forwarding them towards their destinations. This approach has the potential of increasing throughput and improving robustness of communication networks. However, its impact on packet delay is not yet fully understood. Indeed, although a lower traffic rate per link will necessarily reduce the link delay, and thus the overall delay that a given packet travelling through a network will experience, network coding can also increase delay in several ways. The need for combining packets at nodes may force packets to wait for the arrival of other packets with which they will be combined, introducing a potential synchronization delay. Also, although individual link delays will be reduced, node delays may in fact not be affected because in order to reconstitute the packet streams at output nodes, the nodes will have to carry on the average the same amount of traffic if no information is to be lost, so that congestion would not be reduced or may even be increased by network coding. Finally, the need to decode packets at output nodes implies a further delay for the “right” combination of packets to arrive before a given packet can be decoded and forwarded to the receiver. The trade-off in network coding between delay and trans- mission costs (bandwidth and energy) under stochastic packet arrivals has been considered previously. In [2], the energy delay trade-off for a two-way relay network is analyzed assum- ing that the relay accumulates packets from one direction and sends them either after packets from the other direction arrive or the number of packets waiting exceeds the buffer capacity. Packet transmission is then assumed to occur instantaneously. The analysis indicates that in the case of even traffic load, the average delay must tend to infinity in order to achieve minimum energy consumption. A discrete time analysis of this scenario under probabilistic network coding is presented in [3]. The two-way relay network has also been studied for slotted Aloha [4], and it was shown that the delay throughput trade-off depends on the transmission probability of the relay. In [5], the stability and energy consumption of network coding in a wireless tandem network with slotted transmis- sion are considered; intermediate nodes can either transmit self-generated packets or encode two relay flows received from neighboring nodes. The results obtained suggest that immediate transmission of first available packets yields higher throughput as compared to waiting for additional packets to arrive before coding, but this gain comes at the expense of reducing energy efficiency. Conversely, we show that, under stochastic arrivals, immediate transmission of packets cannot offer the opportunity for coding. In [6], the multicast delay and throughput trade-off with intra-flow coding is considered for a slotted-time collision- based wireless network, showing that coding improves throughput and energy costs at the expense of higher packet delays as compared to plain routing. However, the results were obtained under the unrealistic assumptions of one-bit packet lengths and saturated queues at source and relay nodes. There has been considerable work on evaluating the per- formance of network coding in different mathematical frame- works. In [7], the achievable rate regions under quality of service (QoS) constraints are computed for a butterfly network with and without network coding. However, the analysis is based on a fluid flow model which does not capture the bursty nature of packet arrivals which is essential for understand- ing network coding gains. End-to-end QoS bounds for both network coding and plain forwarding have been derived in [8] using deterministic network calculus; the results show that coding can improve the worst case delays even in topologies where no throughput gains are expected. Network calculus, however, can only provide bounds that may not be tight in practice. The contributions of the present paper are twofold. First, we decouple the different stages of service in a network coding router in order to address the fact that coding is not performed when packets are transmitted, but is performed by a separate program or hardware which operates independently of the hardware that forwards packets. In contrast, existing theoretical work on network coding has assumed either zero transmission time [2], slotted time [3]–[5] or packet length based service time [9]–[11]. Second, our model gives rise 978-1-4244-7115-7/10/$26.00 ©2010 IEEE