Erasure Probability Research Articles

Capacity of 3D Erasure Networks

In this paper, we introduce a large-scale 3D erasure network, where $n$ wireless nodes are randomly distributed in a cuboid of $n^{\lambda }\times n^{\mu }\times n^{\nu }$ with $\lambda +\mu +\nu =1$ for $\lambda ,\mu , \nu >0$ , and completely characterize its capacity scaling laws. Two fundamental path-loss attenuation models (i.e., exponential and polynomial power-law models) are used to suitably model an erasure probability for packet transmission. Then, under the two erasure models, we introduce a routing protocol using percolation highway in 3D space, and then analyze its achievable throughput scaling laws. It is shown that, under the two erasure models, the aggregate throughput scaling $n^{ \mathop{\mathrm{min}}\limits \{1-\lambda ,1-\mu ,1-\nu \}}$ can be achieved in the 3D erasure network. This implies that the aggregate throughput scaling $n^{2/3}$ can be achieved in 3D cubic erasure networks, while $\sqrt {n}$ can be achieved in 2D square erasure networks. The gain comes from the fact that, compared with 2D space, more geographic diversity can be exploited via 3D space, which means that generating more simultaneous percolation highways is possible. In addition, cut-set upper bounds on the capacity scaling are derived to verify that the achievable scheme based on the 3D percolation highway is order-optimal within a polylogarithmic factor under certain practical operating regimes on the decay parameters.

인프라구조 도움을 받는 소거 네트워크에서 용량에 대한 랜덤 노드 분포의 효과

ABSTRACT The nearest-neighbor multihop routing with/without infrastructure support is known to achieve the optimal capacity scaling in a large packet-erasure netw ork in which multiple wirele ss nodes and relay stations are regularly placed and packets are erased with a certain probability. In this paper, a throughput scaling law is shown for an infrastructure-supported erasure network where wireless nodes are randomly distributed, which is a more feasible scenario. We use an exponential decay model to suitably model an erasure probability. To achieve high throughput in hybrid random erasure networks, the multihop routing via highway using the percolation theory is proposed and the corresponding throughput scaling is derived. As a main result, the proposed percolation highway based routing scheme achieves the same throughput scaling as the nearest-neighbor multihop case in hybrid regular erasure networks. That is, it is shown that no performance loss occurs even when nodes are randomly distributed.

An LP Characterization of the Secret-message Capacity of Three Erasure Networks With Feedback

This paper presents exact capacity characterizations for the case, when a principal, Alice, wants to securely send a message to another principal, Bob, over three network configurations: the parallel edges network, the V-network, and the triangle network. We assume that: 1) a passive eavesdropper, Eve, overhears any one edge in the network; 2) each edge corresponds to an independent broadcast packet erasure channel with arbitrary erasure probabilities; and 3) all legitimate nodes can publicly but causally acknowledge whether they received each packet or not. We develop optimal achievability schemes that are expressed as linear programs (LPs) and share a two-phase structure, where at the first phase, we create secret keys, and at the second phase, we use them to encrypt the transmitted message. Our outer bounds are also expressed through LP formulations. We prove that our schemes are optimal by showing that the optimal solution of the outer bound LP and the optimal solution of the achievability scheme LP coincide.

Coding Schemes With Rate-Limited Feedback That Improve Over the No Feedback Capacity for a Large Class of Broadcast Channels

We propose two coding schemes for the two-receiver discrete memoryless broadcast channel (BC) with rate-limited feedback from one or both receivers. They improve over the no feedback capacity region for a large class of channels, including the class of strictly essentially less-noisy BCs that we introduce in this paper. Examples of strictly essentially less-noisy BCs are the binary symmetric BC or the binary erasure BC with unequal crossover or erasure probabilities at the two receivers. When the feedback rates are sufficiently large, our schemes recover all previously known capacity results for discrete memoryless BCs with feedback. In both our schemes, we let the receivers feedback quantization messages about their receive signals. In the first scheme, the transmitter simply relays the quantization information obtained from Receiver 1 to Receiver 2, and vice versa. This provides each receiver with a second observation of the input signal and can thus improve its decoding performance unless the BC is physically degraded. Moreover, each receiver uses its knowledge of the quantization message describing its own outputs so as to attain the same performance as if this message had not been transmitted at all. In our second scheme, the transmitter first reconstructs and processes the quantized output signals, and then sends the outcome as a common update information to both receivers. A special case of our second scheme also applies to memoryless BCs without feedback but with strictly causal state-information at the transmitter and causal state-information at the receivers. It recovers all previous achievable regions also for this setup with state-information.

Repair for Distributed Storage Systems With Packet Erasure Channels and Dedicated Nodes for Repair

We study the repair problem in distributed storage systems where storage nodes are connected through packet erasure channels and some nodes are dedicated to repair [termed as dedicated-for-repair (DR) storage nodes]. We first investigate the minimum required repair-bandwidth in an asymptotic setup, in which the stored file is assumed to have an infinite size. The result shows that the asymptotic repair-bandwidth over packet erasure channels with a fixed erasure probability has a closed-form relation to the repair-bandwidth in lossless networks. Next, we show the benefits of DR storage nodes in reducing the repair bandwidth, and then we derive the necessary minimal storage space of DR storage nodes. Finally, we study the repair in a nonasymptotic setup, where the stored file size is finite. We study the minimum practical-repair-bandwidth, i.e., the repair-bandwidth for achieving a given probability of successful repair. A combinatorial optimization problem is formulated to provide the optimal practical-repair-bandwidth for a given packet erasure probability. We show the gain of our proposed approaches in reducing the repair-bandwidth.

The Feedback Capacity of the Binary Erasure Channel With a No-Consecutive-Ones Input Constraint

The input-constrained erasure channel with feedback is considered, where the binary input sequence contains no consecutive ones, i.e., it satisfies the $(1,\infty )$ -RLL constraint. We derive the capacity for this setting, which can be expressed as $C_{\epsilon }=\max _{0\leq p\leq 0.5} \frac {(1-\epsilon )H_{b}(p)}{1+(1-\epsilon )p}$ , where $\epsilon $ is the erasure probability and $ H_{b}(\cdot )$ is the binary entropy function. Moreover, we prove that a priori knowledge of the erasure at the encoder does not increase the feedback capacity. The feedback capacity was calculated using an equivalent dynamic programming (DP) formulation with an optimal average-reward that is equal to the capacity. Furthermore, we obtained an optimal encoding procedure from the solution of the DP, leading to a capacity-achieving, zero-error coding scheme for our setting. DP is, thus, shown to be a tool not only for solving optimization problems, such as capacity calculation, but also for constructing optimal coding schemes. The derived capacity expression also serves as the only non-trivial upper bound known on the capacity of the input-constrained erasure channel without feedback, a problem that is still open.

Minimising the heat dissipation of quantum information erasure

Quantum state engineering and quantum computation rely on information erasure procedures that, up to some fidelity, prepare a quantum object in a pure state. Such processes occur within Landauer's framework if they rely on an interaction between the object and a thermal reservoir. Landauer's principle dictates that this must dissipate a minimum quantity of heat, proportional to the entropy reduction that is incurred by the object, to the thermal reservoir. However, this lower bound is only reachable for some specific physical situations, and it is not necessarily achievable for any given reservoir. The main task of our work can be stated as the minimisation of heat dissipation given probabilistic information erasure, i.e., minimising the amount of energy transferred to the thermal reservoir as heat if we require that the probability of preparing the object in a specific pure state be no smaller than . Here is the maximum probability of information erasure that is permissible by the physical context, and the error. To determine the achievable minimal heat dissipation of quantum information erasure within a given physical context, we explicitly optimise over all possible unitary operators that act on the composite system of object and reservoir. Specifically, we characterise the equivalence class of such optimal unitary operators, using tools from majorisation theory, when we are restricted to finite-dimensional Hilbert spaces. Furthermore, we discuss how pure state preparation processes could be achieved with a smaller heat cost than Landauer's limit, by operating outside of Landauer's framework.

Asymmetric Evaluations of Erasure and Undetected Error Probabilities

The problem of channel coding with the erasure option is revisited for discrete memoryless channels. The interplay between the code rate, the undetected and total error probabilities is characterized. Using the information spectrum method, a sequence of codes of increasing blocklengths $n$ is designed to illustrate this tradeoff. Furthermore, for additive discrete memoryless channels with uniform input distribution, we establish that our analysis is tight with respect to the ensemble average. This is done by analyzing the ensemble performance in terms of a tradeoff between the code rate, the undetected and total errors. This tradeoff is parameterized by the threshold in a generalized likelihood ratio test. Two asymptotic regimes are studied. First, the code rate tends to the capacity of the channel at a rate slower than $n^{-1/2}$ corresponding to the moderate deviations regime. In this case, both error probabilities decay subexponentially and asymmetrically. The precise decay rates are characterized. Second, the code rate tends to capacity at a rate of $n^{-1/2}$ . In this case, the total error probability is asymptotically a positive constant, while the undetected error probability decays as $\exp (- bn^{ 1/2})$ for some $b>0$ . The proof techniques involve the applications of a modified (or shifted) version of the Gartner–Ellis theorem and the type class enumerator method to characterize the asymptotic behavior of a sequence of cumulant generating functions.

Performance characterization and transmission schemes for instantly decodable network coding in wireless broadcast

We consider broadcasting a block of packets to multiple wireless receivers under random packet erasures using instantly decodable network coding (IDNC). The sender first broadcasts each packet uncoded once, then generates coded packets according to receivers' feedback about their missing packets. We focus on strict IDNC (S-IDNC), where each coded packet includes at most one missing packet of every receiver. But we will also compare it with general IDNC (G-IDNC), where this condition is relaxed. We characterize two fundamental performance limits of S-IDNC: 1) the number of transmissions to complete the broadcast, and 2) the average delay for a receiver to decode a packet. We derive a closed-form expression for the expected minimum number of transmissions in terms of the number of packets and receivers and the erasure probability. We prove that it is NP-hard to minimize the decoding delay of S-IDNC. We also derive achievable upper bounds on the above two performance limits. We show that G-IDNC can outperform S-IDNC %in terms of the number of transmissions without packet erasures, but not necessarily with packet erasures. Next, we design optimal and heuristic S-IDNC transmission schemes and coding algorithms with full/intermittent receiver feedback. We present simulation results to corroborate the developed theory and compare with existing schemes.

Capacity of large hybrid erasure networks with random node distribution

The Gupta---Kumar's nearest-neighbor multihop routing with/without infrastructure support achieves the optimal capacity scaling in a large erasure network in which n wireless nodes and m relay stations are regularly placed. In this paper, a capacity scaling law is completely characterized for an infrastructure-supported erasure network where n wireless nodes are randomly distributed, which is a more feasible scenario. We use two fundamental path-loss attenuation models (i.e., exponential and polynomial power-laws) to suitably model an erasure probability. To show our achievability result, the multihop routing via percolation highway is used and the corresponding lower bounds on the total capacity scaling are derived. Cut-set upper bounds on the capacity scaling are also derived. Our result indicates that, under the random erasure network model with infrastructure support, the achievable scheme based on the percolation highway routing is order-optimal within a polylogarithmic factor of n for all values of m.

Fountain Code Design for Broadcasting Systems With Intermediate-State Users

Several studies on fountain codes have proposed degree distribution optimization schemes to maximize symbol recovery rate. However, if the number of transmitted coded symbols is limited or the channel erasure probability is high, it may be impossible that a user recovers all of the data symbols regardless of degree distribution employed by the source. In this study, we focus on a new system model where one source transmits fountain-coded symbols to multiple users who already possess some data symbols and coded symbols. Assuming that each user can transmit a feedback packet containing its own state information before the source transmits coded symbols, we propose two types of degree distribution design schemes that are suitable for the system model. Simulation results demonstrate the efficiency of our proposed schemes by comparing with conventional methods in terms of symbol recovery rate and full recovery rate.

Stability of linear dynamic systems over the packet erasure channel: a co-design approach

This paper is concerned with the stability of linear time-invariant dynamic systems over the packet erasure channel subject to minimum bit rate constraint when an encoder and a decoder are unaware of the control signal. This assumption results in co-designing the encoder, decoder and controller. The encoder, decoder, controller and conditions relating transmission rate to packet erasure probability and eigenvalues of the system matrix A are presented for almost sure asymptotic stability of linear time-invariant dynamic systems over the packet erasure channel with feedback acknowledgment. When the eigenvalues of the system matrix A are real valued, it is shown that the obtained condition for stability is tight. Simulation result illustrates the satisfactory performance of the proposed encoder, decoder and controller for almost sure asymptotic stability.

Formulating the Net Gain of SISO-SFN in the Presence of Erasure Effect

In this paper, an analytical formula for the net gain of single-input single-output single frequency network (SISO-SFN) is derived. In order to formulate the net SISO-SFN gain (SISO-SFNG), we derive the average signal to noise ratio, where the SFN gain is calculated by the aggregate power sum as a function of the power imbalance, whereas the SFN loss is calculated by a two-term exponential model from a curve fitting as a function of the erasure probability, modulation order, and code rate. The accuracy and effectiveness of the derived formula are verified by comparing the measurement results with the analytical results. The derived formula helps to rationalize why the net gain is positive or negative under a given condition, e.g., a negative net gain is obtained if the power imbalance exceeds the erasure-free limit. The formula would be very useful to predict more realistic and accurate service coverage of SISO-SFN for various system configurations.

Faulty Successive Cancellation Decoding of Polar Codes for the Binary Erasure Channel

In this paper, faulty successive cancellation decoding of polar codes for the binary erasure channel is studied. To this end, a simple erasure-based fault model is introduced to represent errors in the decoder and it is shown that, under this model, polarization does not happen, meaning that fully reliable communication is not possible at any rate. Furthermore, a lower bound on the frame error rate of polar codes under faulty SC decoding is provided, which is then used, along with a well-known upper bound, in order to choose a blocklength that minimizes the erasure probability under faulty decoding. Finally, an unequal error protection scheme that can re-enable asymptotically erasure-free transmission at a small rate loss and by protecting only a constant fraction of the decoder is proposed. The same scheme is also shown to significantly improve the finite-length performance of the faulty successive cancellation decoder by protecting as little as 1.5% of the decoder.

On Optimal Policies for Network-Coded Cooperation: Theory and Implementation

Network-coded cooperative communication (NC-CC) has been proposed and evaluated as a powerful technology that can provide a better quality of service in the next-generation wireless systems, e.g., D2D communications. Previous contributions have focused on performance evaluation of NC-CC scenarios rather than searching for optimal policies that can minimize the total cost of reliable packet transmission. We break from this trend by initially analyzing the optimal design of NC-CC for a wireless network with one source, two receivers, and half-duplex erasure channels. The problem is modeled as a special case of Markov decision process (MDP), which is called stochastic shortest path (SSP), and is solved for any field size, arbitrary number of packets, and arbitrary erasure probabilities of the channels. The proposed MDP solution results in an optimal transmission policy per time slot, and we use it to design near-optimal heuristics for packet transmission in a network of one source and N ≥ 2 receivers. We also present numerical results that illustrate the performance of the proposed heuristics under a variety of scenarios. To complete our analysis, our heuristics are implemented in Aalborg University's Raspberry Pi testbed and compared with random linear network coding (RLNC) broadcast in terms of completion time, total number of required transmissions, and percentage of delivered generations. Our measurements show that enabling cooperation only among pairs of devices can decrease the completion time by up to 4.75 times, while delivering 100% of the 10000 generations transmitted, as compared to RLNC broadcast delivering only 88% of them in our tests.

On the Design of Fast Convergent LDPC Codes for the BEC: An Optimization Approach

The complexity-performance trade-off is a fundamental aspect of the design of low-density parity-check (LDPC) codes. In this paper, we consider LDPC codes for the binary erasure channel (BEC), use code rate for performance metric, and number of decoding iterations to achieve a certain residual erasure probability for complexity metric. We first propose a quite accurate approximation of the number of iterations for the BEC. Moreover, a simple but efficient utility function corresponding to the number of iterations is developed. Using the aforementioned approximation and the utility function, two optimization problems w.r.t. complexity are formulated to find the code degree distributions. We show that both optimization problems are convex. In particular, the problem with the proposed approximation belongs to the class of semi-infinite problems which are computationally challenging to be solved. However, the problem with the proposed utility function falls into the class of semi-definite programming (SDP) and thus, the global solution can be found efficiently using available SDP solvers. Numerical results reveal the superiority of the proposed code design compared to existing code designs from literature.

Modeling Network Coded TCP: Analysis of Throughput and Energy Cost

We analyze the performance of TCP and TCP with network coding (TCP/NC) in lossy networks. We build upon the framework introduced by Padhye et al. and characterize the throughput behavior of classical TCP and TCP/NC as a function of erasure probability, round-trip time, maximum window size, and duration of the connection. Our analytical results show that network coding masks random erasures from TCP, thus preventing TCP’s performance degradation in lossy networks. It is further seen that TCP/NC has significant throughput gains over TCP. In addition, we study the cost of improving the goodput per user in a wireless network. We measure the cost in terms of number of base stations, which is highly correlated to the energy cost of a network provider. We show that increasing the available bandwidth may not necessarily lead to increase in goodput, particularly in lossy wireless networks using TCP. We show that using protocols such as TCP/NC, which are more resilient to erasures in the network, may lead to a goodput commensurate with the bandwidth dedicated to each user. By increasing goodput, users’ transactions are completed faster; thus, the resources dedicated to these users can be released to serve other requests, consequently reducing the cost for the network providers.

<sc>xor</sc>-Based Encoding With Instantaneous Decoding for the Broadcast Erasure Channel With Feedback: The Three-User Case

We study the case of a three-user broadcast erasure channel with multiple unicast traffic sessions, where feedback from the users is fed back to the transmitter in the form of positive acknowledgment (ACK)/negative acknowledgment (NACK) messages. The capacity region of this system has been recently derived and two capacity-achieving coding algorithms employing intersession linear network coding have been proposed. Since these algorithms suffer from large computational complexity and decoding delay, our aim, in this paper, is to design a coding algorithm with reduced computational complexity and a low decoding delay that achieves a comparable rate region to the former algorithms. We exclusively consider algorithms that require no knowledge of channel statistics, only perform XOR operations among the packets, and allow for instantaneous decoding by any receiver that successfully receives a packet. We present such an algorithm, named IXOR, which operates on a specially constructed network of virtual queues and, through intelligent packet combining, achieves the capacity under a general condition, which is satisfied in the following settings: 1) spatially independent identically distributed erasure channels with arbitrary values of erasure probability, and 2) spatially independent erasure channels where the maximum erasure probability does not exceed 8/9.

Achieving strong security based on fountain code with coset pre‐coding

This study proposes an approach for achieving strong communication security based on explosive fountain code with coset pre-coding, where both main and wire-tap channels are memoryless binary erasure channels. Coset pre-coding is used to prevent the eavesdroppers from intercepting the confidential information from the leaked bits. Further, an explosive fountain code is designed to ensure the reliability and low information leakage. By this way, the proposed approach can keep strong security and reliability when the erasure probability of the main channel is slightly lower than that of the wire-tap channel. Extensive simulations are conducted to verify the correctness and effectiveness of the approach.

Fast decoding for RaptorQ codes using matrix dimensionality reduction

A very fast decoding algorithm using matrix dimensionality reduction for RaptorQ codes is proposed. The algorithm exploits a pre-calculated inverse matrix to achieve dimensionality reduction for the received code constraint matrix. As a result, the decoding complexity is decreased significantly, whereas the failure-overhead curve is still identical to that of conventional approaches. Simulations show that the decoding speed of the proposed algorithm can be as fast as 17.5 times the state-of-the-art algorithm when the erasure probability is relatively low.