Network coding delay: A brute-force analysis

Physical Communication, 2013

We consider the problem of minimizing delay when broadcasting over erasure channels with feedback. A sender wishes to communicate the same set of µ messages to several receivers. The sender can broadcast a single message or a combination (encoding) of messages to all receivers at each timestep, through separate erasure channels. Receivers provide feedback as to whether the transmission was received. If, at some time step, a receiver cannot identify a new message, delay is incurred. Our notion of delay is motivated by real-time applications that request progressively refined input, such as the successive refinement of an image encoded using multiple description coding. Our setup is novel because it combines coding techniques with feedback information to the end of minimizing delay. Uncoded scheduling or use of erasure correction coding, such as maximum distance separable (MDS) codes, has been well-studied in the literature. We show that our setup allows Θ(µ) benefits as compared to both previous approaches for offline algorithms, while feedback allows online algorithms to achieve smaller delay compared to online algorithms without feedback. Our main complexity results are that the offline minimization problem is N P-hard when the sender only schedules single messages and that the general problem remains N P-hard even when coding is allowed. However we show that coding does offer complexity gains by exhibiting specific classes of erasure instances that become trivial under coding schemes. We also discuss online heuristics and evaluate their performance through simulations.

In this paper, we investigate the throughput and decoding-delay performance of random linear network coding as a function of the coding window size and the network size in an unreliable single-hop broadcast network setting. Our model consists of a source transmitting packets of a single flow to a set of N receivers over independent erasure channels. The source performs random linear network coding (RLNC) over K (coding window size) packets and broadcasts them to the receivers. We note that the broadcast throughput of RLNC must vanish with increasing N , for any fixed K. Hence, in contrast to other works in the literature, we investigate how the coding window size K must scale for increasing N . By appealing to the Central Limit Theorem, we approximate the Negative Binomial random variable arising in our analysis by a Gaussian random variable. We then obtain tight upper and lower bounds on the mean decoding delay and throughput in terms of K and N . Our analysis reveals that the coding window size of ln(N ) represents a phase transition rate below which the throughput converges to zero, and above which it converges to the broadcast capacity. Our numerical investigations show that the bounds obtained using the Gaussian approximation also apply to the real system performance, thus illustrating the accuracy of the analysis.

IEEE/ACM Transactions on Networking

In this paper, we introduce construction techniques for network coding in bidirectional networks with arbitrary transmission delays. These coding schemes reduce the number of transmissions and achieve the optimal rate region in the corresponding broadcast model for both multiple unicast and multicast cases with up to three users, under the equal rate constraint. The coding schemes are presented in two phases; first, coding schemes for line, star and line-star topologies with arbitrary transmission delays are provided and second, any general topology with multiple bidirectional unicast and multicast sessions is shown to be decomposable into these canonical topologies to reduce the number of transmissions. As a result, the coding schemes developed for the line, star, and line-star topologies serve as building blocks for the construction of more general coding schemes for all networks. The proposed schemes are proved to be real time in the sense that they achieve the minimum decoding delay. With a negligible size header, these coding schemes are shown to be applicable to unsynchronized networks, i.e., networks with arbitrary transmission delays. Finally, we demonstrate the applicability of these schemes by extensive simulations. The implementation of such coding schemes on a wireless network with arbitrary transmission delays can improve performance and power efficiency.

2009 IEEE International Symposium on Information Theory, 2009

We consider the problem of minimizing delay when broadcasting over erasure channels with feedback. A sender wishes to communicate the same set of µ messages to several receivers over separate erasure channels. The sender can broadcast a single message or a combination (encoding) of messages at each timestep. Receivers provide feedback as to whether the transmission was received. If at some time step a receiver cannot identify a new message, delay is incurred. Our notion of delay is motivated by real-time applications that request progressively refined input, such as the successive refinement of an image encoded using multiple description coding. Our setup is novel because it combines coding techniques with feedback information to the end of minimizing delay. It allows Θ(µ) benefits as compared to previous approaches for offline algorithms, while feedback allows online algorithms to achieve smaller delay than online algorithms without feedback. Our main complexity results are that the offline minimization problem is N P-hard when the sender only schedules single messages and that the general problem remains N P-hard even when coding is allowed. However we show that coding does offer delay and complexity gains over scheduling. We also discuss online heuristics and evaluate their performance through simulations.

IEEE Transactions on Communications, 2000

Intra-session network coding has been shown to offer significant gains in terms of achievable throughput and delay in settings where one source multicasts data to several clients. In this paper, we consider a more general scenario where multiple sources transmit data to sets of clients and study the benefits of inter-session network coding, when network nodes have the opportunity to combine packets from different sources. In particular, we propose a novel framework for optimal rate allocation in inter-session network coding systems. We formulate the problem as the minimization of the average decoding delay in the client population and solve it with a gradient-based stochastic algorithm. Our optimized inter-session network coding solution is evaluated in different network topologies and compared with basic intra-session network coding solutions. Our results show the benefits of proper coding decisions and effective rate allocation for lowering the decoding delay when the network is used by concurrent multicast sessions.

IEEE Communications Letters, 2018

Sparse Network Coding (SNC) is a promising technique for reducing the complexity of Random Linear Network Coding (RLNC), by selecting a sparse coefficient matrix to code the packets. However, the performance of SNC for the Average Decoding Delay (ADD) of the packets is still unknown. In this paper, we study the performance of ADD and propose a Markov Chain Model to analyze this SNC metric. This model provides a lower bound for decoding delay of a generation as well as a lower bound for decoding delay of a portion of a generation. Our results show that although RLNC provides a better decoding delay of an entire generation, SNC outperforms RLNC in terms of ADD per packet. Sparsity of the coefficient matrix is a key parameter for ADD per packet to transmit stream data. The proposed model enables us to select the appropriate degree of sparsity based on the required ADD. Numerical results validate that the proposed model would enable a precise evaluation of SNC technique behavior.

2006

Jurnal Teknologi

Network coding is a technique known to efficiently utilize the bandwidth by exploiting the broadcast nature of the wireless medium. Network coding reduces the number of retransmissions by allowing the relay not only to forward the packets, but to do some logic operation. However, considering the randomness and the asymmetric nature of the traffic in the wireless medium, it is usually very challenging for the relay to predict when the next packet is coming, thus the main question for the relay when receives a packet is whether to hold the packet in order to obtain a network coding opportunity or to rebroadcast the packet directly and eliminate the delay. In this paper, we address this challenge by introducing two schemes; Bandwidth Consideration Scheme (BCS) which considers pure network coding to achieve the maximum improvement in network throughput, and Time Limited Scheme (TLS), which uses the network coding but considers the imposed delay. The results show that, BCS can lead to up...

2011

We resolve the question of optimality for a wellstudied packetized implementation of random linear network coding, called PNC. In PNC, in contrast to the classical memoryless setting, nodes store received information in memory to later produce coded packets that reflect this information. PNC is known to achieve order optimal stopping times for the manyto-all multicast problem in many settings.

IEEE Transactions on Communications, 2000

We study joint network and channel code design to optimize delay performance. Here the delay is the transmission time of information packets from a source to sinks without considering queuing effects. In our systems, network codes (network layer) are on top of channel codes (physical layer) which are disturbed by noise. Network codes run in a rateless random method, and thus have erasure-correction capability. For the constraint of finite transmission time, transmission errors are inevitable in the physical layer. A detection error in the physical layer means an erasure of network codewords. For the analysis, we model the delay of each information generation in the network layer as independent, identically distributed random variables. The calculation approaches for delay measures are investigated for coded erasure networks. We show how to evaluate the rate and erasure probability of a set of channels belonging to one cut. We also show that the min-cut determines the decoding error probability in the sinks if the number of information packets is large. We observe that for a given amount of source information, larger packet length leads to fewer packets to be transmitted but higher physical-layer detection error probabilities. Further, longer transmission time (delay) in the physical-layer causes smaller detection error probability at the physical layer. Thus, both parameters have opposite impacts on the physical and network layer, considering delay. We should find the optimal values of them in a cross-layer approach. We then formulate the problems of optimizing delay performance, and discuss solutions for them.