Throughput-Optimal Broadcast on Directed Acyclic Graphs Abhishek Sinha, Georgios Paschos, Chih-ping Li, and Eytan Modiano Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139 Email: fsinhaa, gpasxos, cpli, [email protected] Abstract—We study the problem of broadcasting packets in design optimal wireless broadcast algorithms without the use wireless networks. At each time slot, a network controller acti- of spanning trees. To the best of our knowledge, there exists vates non-interfering links and forwards packets to all nodes at a no capacity-achieving scheduling policy for wireless broadcast common rate; the maximum rate is referred to as the broadcast capacity of the wireless network. Existing policies achieve the without spanning trees. broadcast capacity by balancing traffic over a set of spanning We begin by considering a rich class of scheduling policies trees, which are difficult to maintain in a large and time-varying Π that perform arbitrary link activations and packet forward- wireless network. We propose a new dynamic algorithm that ing, and characterize the broadcast capacity over this policy achieves the broadcast capacity when the underlying network class Π. We impose two additional constraints that improve topology is a directed acyclic graph (DAG). This algorithm utilizes local queue-length information, does not use any global the understanding of the problem. First, we consider the in-order topological structures such as spanning trees, and uses the idea subclass of policies Π ⊂ Π that enforce the in-order of in-order packet delivery to all network nodes. Although the delivery of packets. Second, we focus on the subset of policies in-order packet delivery constraint leads to degraded throughput Π∗ ⊂ Πin-order that allows the reception of a packet by a node in cyclic graphs, we show that it is throughput optimal in only if all its incoming neighbors have received the packet. DAGs and can be exploited to simplify the design and analysis of optimal algorithms. Our simulation results show that the It is intuitively clear that the policies in the more structured ∗ proposed algorithm has superior delay performance as compared class Π are easier to describe and analyze, but may yield to tree-based approaches. degraded throughput performance. We show the surprising result that when the underlying network topology is a directed I. INTRODUCTION acyclic graph (DAG), there is a control policy π∗ 2 Π∗ that Broadcast refers to the fundamental network functionality achieves the broadcast capacity. In contrast, there exists a of delivering data from a source node to all other nodes. It uses cyclic network in which no control policy in Π∗ or Πin-order packet replication and appropriate forwarding to eliminate un- can achieve the broadcast capacity. necessary packet retransmissions. This is especially important To enable the design of the optimal broadcast policy, we in power-constrained wireless systems which suffer from inter- establish a queue-like dynamics for the system state, which is ference and collisions. Broadcast applications include mission- represented by relative packet deficit. This is non-trivial for critical military communications [1], live video streaming [2], the broadcast problem because explicit queueing structure is and data dissemination in sensor networks [3]. difficult to maintain in the network due to packet replication. The design of efficient wireless broadcast algorithms faces As a result, achieving the broadcast capacity reduces to finding several challenges. Wireless channels suffer from interference, a scheduling policy that stabilizes the system states using the and a broadcast policy needs to activate non-interfering links drift analysis [5], [6]. at any time. Wireless network topologies undergo frequent In this paper, our contributions include: arXiv:1411.6172v1 [cs.NI] 22 Nov 2014 changes, so that packet forwarding decisions must be made • We define the broadcast capacity of a wireless network in an adaptive fashion. Existing dynamic multicast algorithms and show that it is characterized by an edge-capacitated that balance traffic over spanning trees [4] may be used for graph Gb that arises from optimizing the time-averages broadcasting, since broadcast is a special case of multicast. of link activations. For integral-capacitated DAGs, the These algorithms, however, are not suitable for wireless net- broadcast capacity is determined by the minimum in- works because enumerating all spanning trees is computation- degree of the graph Gb, which is equal to the maximal ally complex and needs to be performed repeatedly when the number of edge-disjoint spanning trees. network topology changes. • We design a dynamic algorithm that utilizes local queue- In this paper, we study the fundamental performance of length information to achieve the broadcast capacity of broadcasting packets in wireless networks. We consider a time- a wireless DAG network. This algorithm does not rely slotted system. At every slot, a scheduler decides which wire- on spanning trees, has small computational complexity, less links to activate and which packets to forward on activated and is suitable for mobile networks with time-varying links, so that all nodes receive packets at a common rate. topology. The broadcast capacity is the maximum common reception • We demonstrate the superior delay performance of our al- rate of distinct packets over all scheduling policies. We then gorithm, as compared to centralized tree-based algorithm 2 [4], via numerical simulations. In the literature, a simple method for wireless broadcast is to r λ r use packet flooding [7]. The flooding approach, however, leads to redundant transmissions and collisions, known as broadcast a b storm [8]. In the wired domain, it has been shown that a b forwarding useful packets at random is optimal for broadcast [9]; this approach does not extend to the wireless setting due c to interference and the need for scheduling [10]. Broadcast on c wired networks can also be done using network coding [11], [12]. However, efficient link activation under network coding (a) a wireless network (b) activation vector s1 remains an open problem. The rest of the paper is organized as follows. Section r II introduces the wireless network model. In Section III, r we define the broadcast capacity of a wireless network and provide a useful upper bound from a fundamental cut-set bound. In Section IV, we propose a dynamic broadcast policy a b a b that achieves the broadcast capacity in a DAG. Simulation results are presented in Section V. c c II. THE WIRELESS NETWORK MODEL We consider a wireless network that is represented by a (c) activation vector s2 (d) activation vector s3 directed graph G = (V; E; c; S), where V is the set of Fig. 1: A wireless network and its three feasible link activa- nodes, E is the set of directed links, c = (cij) denotes the capacity of links (i; j) 2 E, S is the set of all feasible tions under the primary interference constraint. link-activation vectors, and s = (se; e 2 E) 2 S is a binary vector indicating that the links e with s = 1 can e received by node i 2 V from the beginning of time up to time be activated simultaneously. The structure of the activation t, under a policy π 2 Π. The time average lim Rπ(T )=T set S depends on the interference model. Under the primary T !1 i is the rate of distinct packets received at node i. interference constraint, the set S consists of all binary vectors corresponding to matchings of the underlying graph G [13], Definition 1. A policy π is called a “broadcast policy of rate see Fig. 1. In the case of a wired network, S is the set of λ” if all nodes receive distinct packets at rate λ, i.e., all binary vectors since there is no interference. In this paper, 1 π we allow an arbitrary link-activation set S, which captures lim Ri (T ) = λ, for all i 2 V; w. p. 1, (1) T !1 T different wireless interference models. Let r 2 V be the source 1 node at which the broadcast traffic is generated. We consider a where λ is the packet arrival rate at the source node r. time-slotted system. In slot t, the number of packets generated Definition 2. The broadcast capacity λ∗ of a wireless network at the node r is denoted by A(t), where A(t) is i.i.d. over slots is the supremum of all arrival rates λ for which there exists with mean λ. These packets need to be delivered to all nodes a broadcast policy π of rate λ, π 2 Π. in the wireless network. A. An upper bound on broadcast capacity λ∗ III. WIRELESS BROADCAST CAPACITY We characterize the broadcast capacity λ∗ of a wireless Intuitively, the network supports a broadcast rate λ if there network by providing a useful upper bound. This upper bound exists a scheduling policy under which all network nodes is understood as a necessary cut-set bound of an associated can receive distinct packets at rate λ. The broadcast capacity edge-capacitated graph that reflects the time-average behavior is the maximally supportable broadcast rate in the network. of the wireless network. We provide an intuitive explanation Formally, we consider a class Π of scheduling policies where of the bound that will be formalized in Theorem 1 as follows. π each policy π 2 Π is a sequence of actions fπtgt≥1 taken in Fix a policy π 2 Π. Let βe be the fraction of time link every slot t. Each action πt consists of two operations: (i) the e 2 E is activated under π; that is, we define the vector scheduler activates a subset of links by choosing a feasible T activation vector s 2 S; (ii) each node i forwards a subset of π π 1 X π β = (βe ; e 2 E) = lim s (t); (3) T !1 T packets (possibly empty) to node j over an activated link (i; j), t=1 subject to the link capacity constraint.