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9 878 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 17, NO. 3, JUNE 2009 Fig. 7. Single hop delay. (a). (b). such as Voice-over-IP traffic can be derived similarly to the way we derive the scheduler for round-trip path TDMA delay [37]. A round-trip path in the topology graph corresponds to a cycle in the conflict graph. The cycle in the conflict graph can be obtained by finding the conflicts needed to visit the vertexes of the conflict graph listed in a path. For example, the round-trip path in Fig. 3(a) corresponds to the cycle, marked in Fig. 3(b). In the same topology, the round-trip path in Fig. 3(a) corresponds to the cycle, also marked in Fig. 3(b). We find the end-to-end scheduling delay for a path by finding the delay incurred while traversing the corresponding cycle in the conflict graph: also possible to calculate delay more precisely, by accounting for the delay from beginning of the first transmission on the path and to end of the last transmission on the path. This delay can be calculated using the unicast delay formula [37], which is very similar to (20). However, in this paper we restrict ourselves to the worst case round-trip delay. In the rest of the paper, we use vector notation to represent cycles to make formulae more compact. A cycle in the conflict graph is defined by the -dimensional vector, where if, if and if. For example, the cycle in Fig. 3, corresponds to the vector where, and all other entries in the vector are 0. In vector notation, scheduling delay on a round-trip path is (22) (19) where are the conflicts traversed in their direction and are the conflicts traversed in their opposite direction. Roundtrip scheduling delay depends on the transmission order only: (20) where starting times cancel out. For example, the summation over difference terms for cycle in Fig. 3(b) is (21) This cancellation in is a well known graph property [31, pp. 174]. The delay calculated by (20) is the worst case round-trip delay, since it measures the delay from the time the first link on the path transmits to the time the first link transmits again. It is where is a constant for the cycle. B. Min-Max TDMA Delay Scheduling We formulate a {0, 1}-integer program that finds a transmission order that minimizes the end-to-end scheduling delay on the set of round-trip paths, while ensuring that a feasible virtual schedule exists over slots. An objective function that finds such schedules is the min-max delay defined as. We combine the formula for the scheduling delay (20) with the polyhedron of feasible transmission orders to formulate a {0, 1}-integer program that finds a schedule with the min-max delay for a fixed : (23a) (23b) (23c) (23d) Constraints (23b) ensure that no path has a delay larger than, while is minimized; this achieves the goal of finding the

13 882 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 17, NO. 3, JUNE 2009 in the number of unique paths, Algorithm-TH uses more spatial re-use and comes closer to the performance of Algorithm-MM. Fig. 11. Fig. 12. Percentage of achieved bandwidth for the mesh topology. Maximum delay for the mesh topology. tree. The graph coloring approach has the maximum delay, approximately twice as large as the delay of Algorithm-MM. C. Example 3: 5 3 Mesh Topology With Random Sources In this section we examine performance of scheduling algorithms on a 5 3 grid. We place the base-station in the centre of the topology and pick sources at random. After picking the sources, we find paths to the base-station with the minimum spanning tree algorithm. Each source requests an end-to-end rate of 2.2Mbps. We repeat the selection of sources 50 times. We plot the results for averaged over all 50 scenarios in Fig. 11 and the results for averaged over all 50 scenarios in Fig. 12. We note again that despite spatial re-use, the graph coloring algorithm achieves at most as much bandwidth as the scheduling algorithm that does not use spatial reuse (Fig. 11). The delay of the graph coloring algorithm is better than the delay of the algorithm, however it is still significantly larger than the delay of Algorithm-MM or Algorithm-TH (Fig. 12). Finally, we note that the bandwidth performance of Algorithm-TH improves, compared to the performance of Algorithm-MM, as the number of sources increases. End-to-end bandwidths allocated with Algorithm-TH come within 5% of the end-to-end bandwidths allocated with Algorithm-MM. The reason for the improvement, is that as the number of sources increases, so does the number of unique paths. With the increase VII. CONCLUSION This paper introduces TDMA delay aware scheduling for multi-hop wireless networks. First, we have shown that the delay unaware TDMA scheduling problem can be solved in two parts. In the first part, a relative transmission order of the links (precedence) is found. In the second part, the relative transmission order is used together with the conflict graph as the input to a modified Bellman Ford algorithm, which can find a feasible schedule in polynomial time. For example, TDMA schedules can be found with the distributed Bellman Ford algorithm that does not require the direct knowledge of the full network topology. Second, we have shown that the precedence is related directly to scheduling delay in the network, and propose methods to find transmission orders with good TDMA delay properties. We devise a polynomial time algorithm that finds one-frame scheduling delay transmission orders on overlay tree topologies. Since the two most important and common examples of wireless multi-hop networks using overlay tree topologies are sensor networks and mesh networks, this algorithm is applicable in a wide array of scenarios. We have used simulations to compare the algorithms in this paper applied to mesh networks with other algorithms proposed for these networks. We have shown that our approach has better performance for two reasons. First, it allows spatial re-use so total bandwidth provided by the network is increased. Second, it limits the number of transmissions each link makes in the frame, so it increases the efficiency and end-to-end bandwidth. ACKNOWLEDGMENT The authors would like to thank the anonymous IEEE IN- FOCOM and IEEE/ACM TRANSACTIONS ON NETWORKING (ToN) reviewers who gave significant suggestions for improving the paper. The authors are especially grateful to the ToN reviewer who suggested the simple proof of correctness for the MINIMIZE-SCHEDULE-LENGTH algorithm, which is presented in the paper. REFERENCES [1] P. Djukic and S. Valaee, Link scheduling for minimum delay in spatial re-use TDMA, in Proc. IEEE INFOCOM, 2007, pp [2] IEEE Standard for Local and Metropolitan Area Networks Part 16: Air Interface for Fixed Broadband Wireless Access Systems, , [3] IEEE P802.11s/D1.01, Draft STANDARD for Information Technology Telecommunications and Information Exchange Between Systems Local and Metropolitan Area Networks- Specific Requirements- Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications Amendment: ESS Mesh Networking, s, [4] X. Lin, N. B. Shroff, and R. Srikant, A tutorial on cross-layer optimization in wireless networks, IEEE J. Sel. Areas Commun., vol. 24, no. 8, pp , Aug [5] K. Jain, J. Padhye, V. N. Padmanabhan, and L. Qiu, Impact of interference on multi-hop wireless network performance, Wireless Networks, vol. 11, pp , [6] S. Ramanathan, A unified framework and algorithm for channel assignment in wireless networks, Wireless Networks, vol. 5, no. 2, pp , Mar

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