Broadcast Packet Erasure Channels Research Articles

Movable Antenna-Aided Broadcast Packet Erasure Channels: Capacity with Dynamic Position Plan

Movable Antenna-Aided Broadcast Packet Erasure Channels: Capacity with Dynamic Position Plan

Capacity of Broadcast Packet Erasure Channels With Single-User Delayed CSI

We characterize the capacity region of the two-user broadcast packet erasure channel (PEC) with single-user delayed channel state information (CSI). More precisely, we assume one receiver does not provide its channel state to the other two nodes (the other receiver and the transmitter), while the other receiver reveals its state globally with unit delay. This is a hybrid CSI at the transmitter (CSIT) setting where the transmitter has the delayed CSI of one user but not the other. Previous results developed opportunistic network coding schemes for this setting, which strictly enlarge the achievable rate region compared to the no-CSIT baseline. Characterization of the capacity region with single-user delayed CSI, however, remained open. In this work, we develop an improved achievability strategy and show that the capacity region, surprisingly, matches that of the broadcast PEC with global delayed CSI of both users. The key to such improvement over previous results is a new precoding strategy for the retransmission phase of the opportunistic network coding scheme. It harnesses the single-user delayed CSI in the retransmission phase, so that interference from the feedback receiver can be aligned at the other receiver. Besides the broadcast PEC with two private messages, an extension to a model with an additional common message is also provided and the corresponding capacity region with single-user CSI also matches that with global delayed CSI. Finally, further extensions to three-user cases are also provided.

Energy-Aware Scheduling for Broadcast Erasure Channels With Two Energy Harvesting Receivers

A feedback-based coding algorithm is studied for broadcast packet erasure channels (BPECs) with energy harvesting receivers. A probabilistic algorithm is outlined, which is able to achieve the capacity region of two-user BPECs. The probabilistic algorithm is carefully studied from the optimization perspective. In this paper, we take into account the energy levels of receivers. Two new feedback schemes are proposed to reduce the energy consumption of receivers and we explain the advantage of each feedback scheme. We formulate the problem of maximizing the minimum weighted throughput of receivers for each feedback scheme. After several convexifications, we make the formulated optimizations tractable and propose an algorithm to solve the problems globally without compromising optimality. Once optimization problems are solved, we propose an energy-aware scheduling for packet broadcasting. Finally, a packet header design is proposed to enable utilization of the energy-aware scheduling. Simulation results verify that to achieve the optimal throughput, network-coding is necessary.

A New Capacity-Approaching Scheme for General 1-to-K Broadcast Packet Erasure Channels With ACK/NACK

The capacity region of 1-to- K broadcast packet erasure channels with ACK/NACK is well known for some scenarios, e.g., $ {K}\leq 3$ , etc. However, existing achievability schemes either require knowing the target rate $\vec {{R}}$ in advance, and/or have a complicated description of the achievable rate region that is difficult to prove whether it matches the capacity or not. This work proposes a new network coding scheme with the following features: (i) its achievable rate region is identical to the capacity region for all the scenarios in which the capacity is known; (ii) its achievable rate region is much more tractable and has been used to derive new capacity rate vectors; (iii) it employs sequential encoding that naturally handles dynamic packet arrivals; (iv) it automatically adapts to unknown packet arrival rates $\vec {{R}}$ ; (v) it is based on $\mathsf {GF}({q})$ with ${q}\geq {K}$ . In addition to analytically characterizing the achievable rate region of the proposed scheme, numerical simulation has been used to verify its queue length and delay performance.

Capacity Regions of Two-Receiver Broadcast Erasure Channels With Feedback and Memory

The two-receiver broadcast packet erasure channel with feedback and memory is studied. Memory is modeled using a finite-state Markov chain representing a channel state. Two scenarios are considered: 1) when the transmitter has causal knowledge of the channel state (i.e., the state is visible) and 2) when the channel state is unknown at the transmitter, but observations of it are available at the transmitter through feedback (i.e., the state is hidden). In both scenarios, matching outer and inner bounds on the rates of communication are derived and the capacity region is determined. It is shown that similar results carry over to channels with memory and delayed feedback and memoryless compound channels with feedback. When the state is visible, the capacity region has a single-letter characterization and is in terms of a linear program. Two optimal coding schemes are devised that use feedback to keep track of the sent/received packets via a network of queues: a probabilistic scheme and a deterministic backpressure-like algorithm. The former bases its decisions solely on the past channel state information and the latter follows a max-weight queue-based policy. The performance of the algorithms is analyzed using the frameworks of rate stability in networks of queues, max-flow min-cut duality in networks, and finite-horizon Lyapunov drift analysis. When the state is hidden, the capacity region does not have a single-letter characterization and is, in this sense, uncomputable. Approximations of the capacity region are provided and two optimal coding algorithms are outlined. The first algorithm is a probabilistic coding scheme that bases its decisions on the past $L$ acknowledgments and its achievable rate region approaches the capacity region exponentially fast in $L$ . The second algorithm is a backpressure-like algorithm that performs optimally in the long run.

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.

The Capacity Region of Two-Receiver Multiple-Input Broadcast Packet Erasure Channels With Channel Output Feedback

This paper studies the capacity of the two-receiver multiple-input broadcast packet erasure channels (PECs) with channel output feedback, which is in contrast with the single-input setting of the existing works. Motivated by the immense success of linear network coding (LNC) in theory and in practice, this paper first focuses on LNC schemes and characterizes the LNC feedback capacity region of two-receiver multiple-input broadcast PECs. A new linear-space-based approach is proposed, which unifies the problems of finding a capacity outer bound and devising the achievability scheme into a single linear programming problem. In particular, an LP solver is used to exhaustively search for the LNC scheme(s) with the best possible throughput, the result of which is thus guaranteed to attain the LNC feedback capacity. It is then proven by pure algebraic arguments that the LNC capacity region matches a simple capacity region outer bound, which proves that the derived LNC capacity region is indeed the true capacity. A byproduct of the above results is a complete LNC capacity region characterization for two-receiver partially Markovian and partially controllable broadcast PECs.

Two-Flow Capacity Region of the COPE Principle for Wireless Butterfly Networks With Broadcast Erasure Channels

This paper characterizes the full capacity region of the COPE principle for 2-flow wireless butterfly networks with broadcast packet erasure channels (PECs). The capacity results in this paper allow for random overhearing with arbitrary overhearing probabilities, arbitrary scheduling policies, network-wide channel state information (CSI) feedback after each transmission, and potential use of nonlinear network codes. An information-theoretic outer bound is derived that takes into account the delayed CSI feedback of the underlying broadcast packet erasure channels. For the achievability, this paper proposes a new class of linear network codes, named as the space-based linear network coding (SBLNC), that achieves the capacity outer bound. Further, the proposed outer and inner bounds are later generalized for the setting in which a transmission may be heard by its 2-hop neighbor(s), the so-called opportunistic routing scenario. When allowing the possibility of opportunistic routing, the proposed inner and outer bounds do not always meet. Numerical experiments, however, show that the relative gap of the two bounds is less than 0.08% in average. The proposed bounds thus tightly bracket the capacity region even when combining the COPE principle with opportunistic routing.

On the Capacity of 1-to-$K$ Broadcast Packet Erasure Channels With Channel Output Feedback

This paper focuses on the 1-to- <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$K$</tex></formula> broadcast packet erasure channel (PEC), a generalization of the broadcast binary erasure channel from the binary symbol to a finite field <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mathop{\ssr GF}(q)$</tex></formula> with sufficiently large <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$q$</tex></formula> . We consider the setting in which the source node has instant feedback of the channel outputs of the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$K$</tex> </formula> receivers after each transmission. The main results of this paper are: (i) The capacity region for general 1-to-3 broadcast PECs and (ii) The capacity region for two types of 1-to- <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$K$</tex></formula> broadcast PECs: the symmetric PECs, and the spatially independent PECs with one-sided fairness constraints. This paper also develops (iii) A pair of outer and inner bounds of the capacity region for arbitrary 1-to- <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$K$</tex></formula> broadcast PECs, which can be easily evaluated by any linear programming solver. The proposed inner bound is proven by a new class of intersession network coding schemes, termed the packet evolution schemes, which is based on the concept of code alignment in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mathop{\ssr GF}(q)$</tex></formula> that is in parallel with the interference alignment techniques for the Euclidean space. Extensive numerical experiments show that the outer and inner bounds meet for almost all broadcast PECs encountered in practical scenarios and thus effectively bracket the capacity of general 1-to- <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$K$</tex> </formula> broadcast PECs with COF.

On the Capacity of Wireless 1-Hop Intersession Network Coding—A Broadcast Packet Erasure Channel Approach

Motivated by practical wireless network protocols, this paper focuses on wireless intersession network coding (INC) over a 1-hop neighborhood, of which the exact capacity region remains an open problem. Towards better understanding of the capacity, this work first models the wireless overhearing events by broadcast packet erasure channels that are memoryless and stationary. Since most INC gain is resulted from destinations overhearing packets transmitted by other sources, this work then focuses exclusively on the 2-staged INC schemes, which fully capture the throughput benefits of overheard message side information (MSI) through the use of one-time feedback but refrain from exploiting the broadcast spatial diversity gain of channel output feedback. Under this setting, a capacity outer bound is provided for any number of M coexisting unicast sessions. For the special cases of M ≤ 3, it is shown that the outer bound can be achieved and is indeed the capacity. To quantify the tightness of the outer bound for M ≥ 4, a capacity inner bound for general M is provided. Both the outer and inner bounds can be evaluated by any linear programming solver. Numeric results show that for 4 ≤ M ≤ 5 with randomly chosen channel parameters, the difference between the outer and inner bounds is within 1% for 96.7% of the times. Focusing exclusively on the benefits of MSI, the results in this paper can also be viewed as the generalization of index-coding capacity from wireline broadcast with binary alphabets to wireless broadcast with high-order alphabets.