On Improving the Balance between the Completion Time and Decoding Delay in Instantly Decodable Network Coded Systems
This paper studies the complicated interplay of the completion time (as a measure of throughput) and the decoding delay performance in instantly decodable network coded (IDNC) systems over wireless broadcast erasure channels with memory, and proposes two new algorithms that improve the balance between the completion time and decoding delay of broadcasting a block of packets. We first formulate the IDNC packet selection problem that provides joint control of the completion time and decoding delay as a statistical shortest path (SSP) problem. However, since finding the optimal packet selection policy using the SSP technique is computationally complex, we employ its geometric structure to find some guidelines and use them to propose two heuristic packet selection algorithms that can efficiently improve the balance between the completion time and decoding delay for broadcast erasure channels with a wide range of memory conditions. It is shown that each one of the two proposed algorithms is superior for a specific range of memory conditions. Furthermore, we show that the proposed algorithms achieve an improved fairness in terms of the decoding delay across all receivers.
Network coding (NC) [Yeung_flow, katti1;etal:2008, fragouli:widmer:boudec:2008] refers to mixing different information flows at the sender or intermediate nodes in a data communication network. It has been shown that NC can substantially improve the throughput of many wireless communication systems [katti1;etal:2008, fragouli:widmer:boudec:2008]. As a result, it has become a promising candidate for delivering high data rate content in future wireless communication networks. For example, NC has been considered for delivering high data rate multimedia broadcast or multicast services (MBMS) [nguyen:nguyen:yang:2007, Li:Wang:JSAC:11, sorour:valaee:2011, sorour:valaee:arxiv:2012]. In addition to being high data rate in nature, such applications also often have strict delay requirements. However, the higher throughput offered by NC does not necessarily translate into faster delivery of information to the application [fragouli:lun:medard:pakzad:2007, keller:drinea:fragouli:2008]. In general, the mixed information needs to be disentangled or network decoded first. Understanding the interplay between throughput and delay and devising NC schemes that strike a balance between the two are particularly important, which has proven to be challenging [fragouli:lun:medard:pakzad:2007, eryilmaz:ozdaglar:medard:2006, keller:drinea:fragouli:2008, costa:munaretto:widmer:baros:2008, drinea:fragouli:keller:2009, barros:costa:munaretto:widmer:2009, yeow:hoang:tham:2009, sundararajan:sadeghi:medard:2009, yazdi:sorour:valaee:kim:2009, sadeghi:shams:traskov:2010, sameh:valaee:globecom:2010, sorour:valaee:2010, swapna:eryilmaz:shroff:2010, sorour:valaee:2011, nistor:lucani:vinhoza:costa:barros:2011, Parastoo:Fisher:2012].
An important example that illustrates the tension between throughput and delay is random linear network coding (RLNC) [swapna:eryilmaz:shroff:2010, nistor:lucani:vinhoza:costa:barros:2011, ho:medard:koetter:karger:effros:2006] in broadcast erasure channels. In RLNC, the sender combines a frame or block of packets using random coefficients from a finite field and broadcasts different combinations until all receivers have received linearly independent coded packets. In this case, RLNC achieves the best throughput (block completion time) among block-based NC schemes [fragouli:lun:medard:pakzad:2007, eryilmaz:ozdaglar:medard:2006, swapna:eryilmaz:shroff:2010]. However, the delay performance may not be desirable, as decoding at the receivers is generally only possible after independent coded packets are successfully received.
In order to reduce the decoding delay in NC systems, an attractive strategy is to employ instantly decodable NC (IDNC). As the name suggests, IDNC aims to provide instant packet decoding at the receivers upon successful packet reception, a property that RLNC does not guarantee. A decoding delay occurs at a receiver when it is not targeted in an IDNC transmission. That is, it receives a packet that contains either no or more than one desired packets of that receiver. Compared to RLNC, IDNC in broadcast erasure channels can have a lower throughput. In other words, IDNC incurs a generally higher completion time for the broadcast of the same number of packets. However, it can provide a faster delivery of uncoded packets to the application layer, as required for MBMS. Therefore, similar tension between throughput and delay can also be observed in IDNC.
Inspired by the low-complexity XOR-based encoding and decoding process of IDNC and its potential application in MBMS and unicast settings [Li:Wang:JSAC:11, keller:drinea:fragouli:2008, sadeghi:shams:traskov:2010, sameh:valaee:globecom:2010, sorour:valaee:2010, sadeghi:traskov:koetter:2009, comm_letter_2013, Le:Tehrani:Dimakis:Markopoulou:2013], in this paper we are interested in understanding the interplay between its throughput and delay over broadcast erasure channels and proposing novel IDNC schemes that offer a better control of these performance metrics.
The problem of maximizing the throughput for a deadline-constrained video-streaming scenario is considered in [Li:Wang:JSAC:11], where each packet has a delivery deadline and has to be decoded before the deadline, otherwise it is expired. In this paper, however, we consider a block-based transmission, where all the packets in the block have to be received by all the receivers and there is no explicit packet deadline. Furthermore, in this paper, no new packet arrival is considered in the system while the transmission of a block is in progress. In addition, this study is applicable where partial decoding is beneficial and can result in lower delays irrespective of the order in which packets are being decoded. Examples of such applications can be found in sensor or emergency networks and multiple-description source coded systems [Multiple_Description_Coding_2005], in which every decoded packet brings new information to the destination, irrespective of its order.
In this context, the closest works to ours are [sameh:valaee:globecom:2010, sorour:valaee:arxiv:2012] and [sorour:valaee:2010]. In particular, the authors in [sameh:valaee:globecom:2010] aimed to improve the decoding delay of a generalized IDNC scheme. They showed that for a lower decoding delay, maximum number of receivers with the lowest packet erasure probabilities should be targeted in each IDNC transmission. In separate works [sorour:valaee:2010, sorour:valaee:arxiv:2012], the same authors aimed to improve the completion time of IDNC. They showed that for this purpose, the receivers with the maximum number of missing packets with the highest erasure probabilities should be targeted in each IDNC transmission.
A close study of [sameh:valaee:globecom:2010, sorour:valaee:2010, sorour:valaee:arxiv:2012] reveals that trying to improve either IDNC’s decoding delay or completion time on its own can result in undermining the other performance metric. In other words, while trying to improve the decoding delay, the receiver(s) with the maximum number of missing packets may remain untargeted, which can increase the completion time. Also trying to improve the completion time may limit the total number of receivers that can be targeted in each IDNC transmission, which can increase the decoding delay. To the best of our knowledge, there is no joint control of completion time and decoding delay for IDNC schemes in the literature. Thus, in this paper, our objective is to take a holistic approach, in which the completion time and decoding delay of IDNC are taken into account at the same time. In addition, we have observed that the decoding delay across various receivers in IDNC schemes of [sameh:valaee:globecom:2010, sorour:valaee:arxiv:2012] and [sorour:valaee:2010] can vary significantly. This may not be desirable in MBMS or other applications which should guarantee a certain quality of service across all receivers. These observations lead us to the following open problems:
Is there an IDNC scheme that can offer a balanced performance in terms of the completion time and decoding delay and can also provide a more uniform or fair decoding delay across all receivers for the broadcast of packets in erasure channels?
To address these questions in this paper, we propose a new IDNC transmission scheme which builds upon the contributions in [sameh:valaee:globecom:2010, sorour:valaee:arxiv:2012] and [sorour:valaee:2010]. At its core, our proposed scheme recognizes that 1) the completion time of each individual receiver is determined not only by the number of packets it is missing, but also by the number of IDNC transmissions in which it is not targeted (while still needing a packet(s)) and 2) the overall IDNC completion time is the maximum of individual completion times. Therefore, our IDNC transmission scheme gives priority to the receivers that have the highest expected completion time so far. More precisely, the priority of each receiver is the sum of two terms: The first term is its number of missing packets divided by its average packet reception probability. This is the expected number of transmissions to serve this receiver if it is targeted in all following transmissions. The second term is the decoding delay the receiver has experienced so far. Under this scheme, a receiver with a small number of missing packets which has remained untargeted in a number of previous transmissions may take precedence over other receivers. Hence, our scheme tends to equalize the decoding delay experience across the receivers. Furthermore, we will extend our proposed scheme to the case of broadcast erasure channels with memory [sadeghi:Kennedy:Rapajic:Shams:08], where the packet erasures occur in bursts, due to deep fading and shadowing. By following the proposed channel models in [sadeghi:shams:traskov:2010, sadeghi:Kennedy:Rapajic:Shams:08, MohammadKarim:Parastoo:PIMRC:2012, sameh:Neda:Parastoo:VTC:2013], we model the bursts of erasures (i.e. the memory of the channel) by a simple two-state Gilbert-Elliott channel (GEC) model and propose two algorithms that can offer an improved balance between the completion time and decoding delay of IDNC for different ranges of the channel memory.
With this introduction, we summarize the contributions and findings of our paper as follows: First, we present a holistic viewpoint of IDNC. We formulate the IDNC optimal packet selection that provides an improved balance between the completion time and decoding delay for broadcast transmission over memoryless channels as an SSP problem. However, since finding the optimal packet selection in the proposed SSP scheme is computationally complex, we use the SSP formulation and its geometric structure to find some guidelines that can be used to propose a new heuristic packet selection algorithm that efficiently improves the balance between the completion time and decoding delay in IDNC systems. Second, we extend the proposed packet selection algorithm to erasure channels with memory and propose two different variations of the algorithm that take into account the channel memory conditions and improve the balance between the completion time and decoding delay by selecting the packet combinations more effectively based on the channel memory conditions compared to the algorithms that are ignorant to the channel memory. Finally, by taking into account both the number of missing packets and the decoding delay of the receivers, the proposed algorithm provides a more uniform decoding delay experience across all receivers.
The rest of this paper is organized as follows. The system model is presented in Section II. The IDNC graph representation and packet generation is introduced in Section III. Section IV, presents the SSP problem formulation. In Section V, we present a geometric structure for the SSP problem that helps us to find the properties of the optimal packet selection policy. A heuristic algorithm for IDNC packet selection is proposed in Section VI. The proposed heuristic algorithm is then extended to erasure channels with memory in Section VII, where also a new layered algorithm is introduced. Section VIII presents the simulation results. Finally, Section IX concludes the paper.
Ii System Model
The system model consists of a wireless sender that is required to deliver a block (denoted by ) of source packets to a set (denoted by ) of receivers. Each receiver is interested in receiving all the packets of . The sender initially transmits the packets of the block uncoded in an initial transmission phase. Each sent packet is subject to erasure at receiver with the probability , which is assumed to be fixed during a block transmission period. Each receiver listens to all transmitted packets and feeds back a positive or negative acknowledgment (ACK or NAK) for each received or lost packet. At the end of the initial transmission phase, two “feedback sets” can be attributed to each receiver :
The Has set (denoted by ) is defined as the set of packets correctly received by receiver .
The Wants set (denoted by ) is defined as the set of packets that are missed at receiver in the initial transmission phase of the current block. In other words .
The senders then stores this information in the state feedback matrix (SFM) as:
An example of SFM with receivers and packets is given as follows:
In this example, denotes that packet 1 is missed at receiver 1, and denotes that packet 1 is correctly received at receiver 2.
After the initial transmission phase, a recovery transmission phase starts, in which the sender exploits the diversity of received and lost packets to transmit network coded combinations of the source packets. Note that we denote the Wants and Has sets of receiver at the start of the recovery transmission phase by and , respectively. After each transmission, for each received/lost packet, the receivers send ACK/NAK to the sender. This information is then used by the sender to update the SFM. This process is repeated until all receivers obtain all packets. Similar two-phase transmission schemes have been widely considered in the literature for IDNC schemes [sadeghi:shams:traskov:2010, sameh:valaee:globecom:2010, sorour:valaee:2010, sorour:valaee:arxiv:2012, comm_letter_2013, Le:Tehrani:Dimakis:Markopoulou:2013].
Based on the Wants and Has sets information, in the recovery transmission phase, the transmitted coded packets can be one of the following options for each receiver :
Non-innovative packet: A packet is non-innovative for receiver if it contains no source packets from .
Instantly decodable packet: A packet is instantly decodable for receiver if it contains only one source packet from . The set of receivers for which the transmitted packet is instantly decodable packet are referred to as the targeted receivers.
Non-instantly decodable packet: A packet is non-instantly decodable for receiver if it contains two or more source packets from .