The study of the optimality of low-complexity greedy scheduling techniques in wireless communications networks is a very complex problem. The Local Pooling (LoP) factor provides a single-parameter means of expressing the achievable capacity region (and optimality) of one such scheme, greedy maximal scheduling (GMS). The exact LoP factor for an arbitrary network graph is generally difficult to obtain, but may be evaluated or bounded based on the network graph’s particular structure. In this paper, we provide rigorous characterizations of the LoP factor in large networks modeled as Erdős–Renyi (ER) and random geometric (RG) graphs under the primary interference model. We employ threshold functions to establish critical values for either the edge probability or communication radius to yield useful bounds on the range and expectation of the LoP factor as the network grows large. For sufficiently dense random graphs, we find that the LoP factor is between 1 2 and 2 3, while sufficiently sparse random graphs permit GMS optimality (the LoP factor is 1) with high probability. We then place LoP within a larger context of commonly studied random graph properties centered around connectedness. We observe that edge densities permitting connectivity generally admit cycle subgraphs that form the basis for the LoP factor upper bound of 2 3. We conclude with simulations to explore the regime of small networks, which suggest the probability that an ER or RG graph satisfies LoP and is connected decays quickly in network size.
As mentioned, checking LoP conditions can be computationally prohibitive, particularly under arbitrary interference models. Therefore, algorithms to easily estimate or bound ( ) and ( ) are of interest and immediate use in studying GMS stability. Joo @cite_9 provide a lower bound on ( ) by the inverse of the largest interference degree of a nested sequence of increasing subsets of links in (G ), and provide an algorithm for computing the bound. Li @cite_18 refine this algorithm to provide individual per-link bounds on ( ). Under the primary interference model, Joo @cite_30 show that ( (2 -1) ) is a lower bound for ( ). Leconte @cite_31 , Li @cite_18 , and Birand @cite_8 note that a lower bound for ( ) is derived from the ratio of the min- to max-cardinality maximal schedules.
@cite_30 @cite_18 @cite_8 @cite_9 @cite_31
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In this paper, we characterize the performance of an important class of scheduling schemes, called greedy maximal scheduling (GMS), for multi-hop wireless networks. While a lower bound on the throughput performance of GMS is relatively well-known in the simple node-exclusive interference model, it has not been thoroughly explored in the more general K-hop interference model. Moreover, empirical observations suggest that the known bounds are quite loose, and that the performance of GMS is often close to optimal. In this paper, we provide a number of new analytic results characterizing the performance limits of GMS. We first provide an equivalent characterization of the efficiency ratio of GMS through a topological property called the local-pooling factor of the network graph. We then develop an iterative procedure to estimate the local-pooling factor under a large class of network topologies and interference models. We use these results to study the worst-case efficiency ratio of GMS on two classes of network topologies. First, we show how these results can be applied to tree networks to prove that GMS achieves the full capacity region in tree networks under the K-hop interference model. Second, we show that the worst-case efficiency ratio of GMS in geometric network graphs is between 1 6 and 1 3. We consider the scheduling of arbitrary wireless links in the physical model of interference to minimize the time for satisfying all requests. We study here the combined problem of scheduling and power control, where we seek both an assignment of power settings and a partition of the links so that each set satisfies the signal-to-interference-plus-noise (SINR) constraints. We give an algorithm that attains an approximation ratio of O(log n ċ log log Δ), where n is the number of links and Δ is the ratio between the longest and the shortest link length. Under the natural assumption that lengths are represented in binary, this gives the first approximation ratio that is polylogarithmic in the size of the input. The algorithm has the desirable property of using an oblivious power assignment, where the power assigned to a sender depends only on the length of the link. We give evidence that this dependence on Δ is unavoidable, showing that any reasonably behaving oblivious power assignment results in a Ω(log log Δ)-approximation. These results hold also for the (weighted) capacity problem of finding a maximum (weighted) subset of links that can be scheduled in a single time slot. In addition, we obtain improved approximation for a bidirectional variant of the scheduling problem, give partial answers to questions about the utility of graphs for modeling physical interference, and generalize the setting from the standard 2-dimensional Euclidean plane to doubling metrics. Finally, we explore the utility of graph models in capturing wireless interference. In this paper, we characterize the performance of an important class of scheduling schemes, called greedy maximal scheduling (GMS), for multihop wireless networks. While a lower bound on the throughput performance of GMS has been well known, empirical observations suggest that it is quite loose and that the performance of GMS is often close to optimal. In this paper, we provide a number of new analytic results characterizing the performance limits of GMS. We first provide an equivalent characterization of the efficiency ratio of GMS through a topological property called the local-pooling factor of the network graph. We then develop an iterative procedure to estimate the local-pooling factor under a large class of network topologies and interference models. We use these results to study the worst-case efficiency ratio of GMS on two classes of network topologies. We show how these results can be applied to tree networks to prove that GMS achieves the full capacity region in tree networks under the K -hop interference model. Then, we show that the worst-case efficiency ratio of GMS in geometric unit-disk graphs is between 1 6 and 1 3. Selective families, a weaker variant of superimposed codes [KS64, F92, 197, CR96], have been recently used to design Deterministic Distributed Broadcast (DDB) protocols for unknown radio networks (a radio network is said to be unknown when the nodes know nothing about the network but their own label) [CGGPR00, CGOR00]. We first provide a general almost tight lower bound on the size of selective families. Then, by reverting the selective families - DDB protocols connection, we exploit our lower bound to construct a family of “hard” radio networks (i.e. directed graphs). These networks yield an O(n log D) lower bound on the completion time of DDB protocols that is superlinear (in the size n of the network) even for very small maximum eccentricity D of the network, while all the previous lower bounds (e.g. O(D log n) [CGGPR00]) are superlinear only when D is almost linear. On the other hand, the previous upper bounds are all superlinear in n independently of the eccentricity D and the maximum in-degree d of the network. We introduce a broadcast technique that exploits selective families in a new way. Then, by combining selective families of almost optimal size with our new broadcast technique, we obtain an O(Dd log3 n) upper bound that we prove to be almost optimal when d = O(n D). This exponentially improves over the best known upper bound [CGR00) when D, d = O(polylogn). Furthermore, by comparing our deterministic upper bound with the best known randomized one [BGI87] we obtain a new, rather surprising insight into the real gap between deterministic and randomized protocols. It turns out that this gap is exponential (as discovered in [BGI87]), but only when the network has large maximum in-degree (i.e. d = O(na), for some constant a > O). We then look at the multibroadcast problem on unknown radio networks. A similar connection to that between selective families and (single) broadcast also holds between superimposed codes and multibroadcast. We in fact combine a variant of our (single) broadcast technique with superimposed codes of almost optimal size available in literature [EFF85, HS87, I97, CHI99]. This yields a multibroadcast protocol having completion time O(Dd2 log3 n). Finally, in order to determine the limits of our multibroadcast technique, we generalize (and improve) the best known lower bound [CR96] on the size of superimposed codes.