NEW APPLICATIONS of wireless multi-hop networks,

Transcription

1 1 TDMA Delay Aware Lin Scheduling for Multi-hop Wireless Networs Petar Djuic and Shahroh Valaee Abstract Time division multiple access (TDMA) based medium access control (MAC) protocols can provide QoS with guaranteed access to wireless channel. However, in multi-hop wireless networs, these protocols may introduce delay when pacets are forwarded from an inbound lin to an outbound lin on a node. Delay occurs if the outbound lin is scheduled to transmit before the inbound lin. The total round trip delay can be quite large since it accumulates at every hop in the path. This paper presents a method that finds schedules with minimum round trip scheduling delay. We show that the scheduling delay can be interpreted as a cost collected over a cycle on the conflict graph. We use this observation to formulate a min-max program for the delay across a set of multiple paths. The min-max delay program is NPcomplete since the transmission order of lins is a vector of binary integer variables. We design heuristics to select appropriate transmission orders. Once the transmission orders are nown, a modified Bellman-Ford algorithm finds the schedules. The simulation results confirm that the proposed algorithm can find effective min-max delay schedules. Index Terms TDMA Scheduling, Networ Flows, Bellman- Ford I. INTRODUCTION NEW APPLICATIONS of wireless multi-hop networs, such as commercial mesh networs, require guaranteed Quality-of-Service (QoS) in the MAC layer. This has prompted development of new multi-hop MAC protocols based on Time Division Multiple Access (TDMA), such as s and [2] [4]. The new protocols provide guaranteed lin bandwidth with scheduled access to wireless channel. The lin bandwidth is allocated over frames with a fixed number of slots. A schedule assigns slots to lins and during each slot, a number of non-conflicting lins can transmit together taing advantage of spatial reuse. The bandwidth of each lin is given by the number of slots it is assigned in the frame. Previously, TDMA scheduling algorithms were designed to minimize the number of slots required to schedule all lins [5] [10]. However, this type of scheduling ignores TDMA scheduling delay. TDMA scheduling delay occurs when pacets arriving on an inbound lin must wait for the subsequent frame to be transmitted on the outbound lin. Since TDMA scheduling delay accumulates at every hop in the networ, the end-to-end delay experienced on a path can be large. In this paper, we address the following important problem: Given an assignment of lin bandwidths, what is the minimum The authors are with The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, 10 King s College Road, Toronto, ON, M5S 3G4, Canada {djuic,valaee}@comm.utoronto.ca A preliminary version of this wor was presented at INFOCOM 2007 [1]. This wor was sponsored in part by the LG Electronics Corporation. length TDMA schedule that also minimizes end-to-end TDMA scheduling delay? We call this problem the TDMA delay aware lin scheduling problem. We solve the TDMA delay aware lin scheduling problem in two parts. First, we develop a new class of TDMA algorithms that find conflict-free TDMA schedules with minimum utilization. A TDMA schedule is conflict-free if all lins whose pacets collide in simultaneous transmissions transmit at separate times. We start by stating the necessary and sufficient conditions that a TDMA schedule is conflict-free as a set of linear inequalities. The conflict-free linear inequalities correspond to pairwise conflicts in the networ and each inequality is defined by the transmission duration of the lins in the conflict, the activation times of the lins in the conflict and the transmission order of the lins. We use the inequalities in a linear search algorithm that finds the minimum utilization TDMA schedules. We show that for fixed transmission orders, the inequalities can be solved in polynomial time with the Bellman-Ford algorithm, so schedules can be found polynomial time. Second, we show the transmission order defines the end-toend TDMA scheduling delay. This allows us to formulate a {0, 1}-integer linear program that finds a min-max delay for a subset of paths in the networ. The number of binary variables in the linear program is equal to the number of conflicts in the networ and corresponds to the lin transmission order. Since the {0, 1}-integer linear program is hard to solve in on-line situations, we propose a heuristics for tree networs that finds transmission orders whose TDMA delay is fixed to one frame. We examine the applicability of the scheduling algorithms to mesh networs with numerical simulations. We compare the performance of the heuristic to the performance of the full scheduling algorithm and show that the heuristic performs well. We also compare the performance of our algorithm to another algorithm proposed for networs [11], which is based on graph colouring. The graph colouring approach allows multiple transmissions in the frame, thus introducing a large amount of overhead. Indeed, schedules produced with the graph colouring approach have so much overhead that they have a significantly lower end-to-end bandwidth than either of our scheduling approaches. A. Related Wor Previously, TDMA scheduling algorithms were designed to use some type of graph colouring to find schedules with minimum frame length [5] [10]. In [5], the authors provide a polynomial time algorithm that finds a minimum length

2 TDMA schedule that can carry a given bandwidth allocation. The authors assume that the only conflicts in the networ are the primary conflicts between lins sharing a neighbour, so they are able to define conflict-free scheduling conditions in terms of edge colouring polyhedron for the networ. In [6], the authors also use the assumption that secondary conflicts can be eliminated in the networ and find a TDMA with edge colouring on a multi-edged version of the topology graph. With an additional operation, the secondary conflicts can be removed from schedules found with edge colouring [7], however this approach requires all lins to have the same bandwidth allocation, which is not practical. The edge colouring approach can be expanded to tae secondary conflicts into account if the colouring is restricted so that lins that are two or more hops away also have different colours [8]. This approach approximates secondary conflicts by excluding lins from having the same colour in an extended neighbourhood, rather than just the conflicting lins. The precise secondary conflicts can be specified with the conflict graph for the networ [9]. With conflict graphs, schedules can be found with vertex colouring. However, the approach in [9] only allows for allocation of a single slot in each frame. The conflict graph for the wireless networ is also used in [10] where authors use the independent set polytope of the conflict graph to state the existence of a schedule with arbitrary lin durations. 1 Since there are an exponential number of independent sets in the graph, [10] proposes a heuristic which iteratively adds independent sets to the feasibility conditions, thus coming closer to the optimal solution. We also use the conflict graph, but we do not require the set of all independent sets in the conflict graph to state the feasibility conditions. In comparison to [10], our feasibility conditions only have a linear number of constraints. B. Contributions In this wor, we introduce the TDMA delay aware lin scheduling problem. Our formulation significantly extends the recent related wors [5] [10]. From a theoretical point of view, we provide a set of linear constraints that comprise necessary and sufficient conditions for the existence of a TDMA transmission schedule. Although some results have been reported before [6], [8], [10], the advantage of our approach is threefold. First, unlie [6] where scheduling constraints capture the necessary conditions only, our linear constraints comprise both necessary and sufficient conditions for the existence of a schedule. Second, there are only a few linear constraints one for every conflict in the networ so we do not have to resort to an approximation as in [10]. Third, the linear constraints capture the exact TDMA conflicts in the networ maing our scheme more applicable to and s networs than [5], [6], and more efficient than [8]. We limit the number of transmissions each lin is allowed in the frame. The limitation on the number of transmissions is 1 While not explicitly using graph colouring, the independent set approach is closely related to graph colouring due to the relationship between the independent sets and graph colouring [12]. Control (T c = N ct ss) Fig. 1. TDMA Framing Active (ρn dt ss) Data (T d = N dt ss) required in current TDMA based protocols [2] and it significantly reduces overhead. We also separate the complexity of finding optimal schedules from scheduling and show that the scheduling problem on its own is a networ flow problem on the conflict graph. This has profound consequences on future scheduling approaches. For example, practical TDMA schedules, taing both the transmission overhead and the required lin durations, can be found with the distributed Bellman-Ford algorithm that does not require the direct nowledge of the full networ topology [13]. II. NETWORK AND TRANSMISSION MODEL We assume that the time is divided into slots of fixed duration, which are then grouped into frames. The duration of each slot is T s seconds and there are a total of N f slots in each frame maing the frame duration T f = N f T s seconds. The first N c slots (T c = N c T s seconds) of the frame are called the control sub-frame and are reserved for control traffic, while the last N d slots (T d = N d T s ) seconds, of the frame are called the data sub-frame and are reserved for data traffic (Fig. 1). Wireless nodes exchange messages in the control sub-frame to negotiate a common transmission schedule in the data subframe. The schedule repeats in every frame until the load (traffic demands) change. The percentage of active slots, which carry traffic in the data sub-frame, is called utilization ρ. We model the networ with a topology graph connecting the nodes in the wireless range of each other. We assume that if two nodes are in the range of each other, they establish lins in the MAC layer, so the TDMA networ can be represented with a connectivity graph G(V,E,f t ), where V = {v 1,...,v n } is the set of nodes, E = {e 1,...,e m } is the set of directional lins and f t : E V V assigns lins to pairs of nodes. The connectivity map f t enforces the fact that all lins are directional, so for a lin e E, f t (e ) = (v i,v j ) means that the traffic on the lin is transmitted from v i to v j. We note that this model allows non-symmetrical lins in the MAC layer. Since all transmissions are over the wireless channel, we associate bitrates with lins to model channel quality. We define the lin bitrate as the number of bits transmitted on the lin in one slot. Lin bitrate depends on the modulation and coding, which are chosen based on the signal-to-noise ratio for the lin. The signal-to-noise ratio is divided into several discrete levels and each is associated with its maximum bitrate. Lin bitrates are represented with the mapping b : E {B 1,B 2,...,B max }, where B 1 is the number of bits carried in a slot with the minimum modulation and coding and B max is the number of bits carried in a slot with the maximum modulation and coding. In the sequel, we use the notation b j b(e j ) (we use the same notation for all maps). 2

3 A. Assigned and Achievable Connection Rates We assume that a routing protocol establishes routes between nodes. The routes are represented with ordered sets of lins (end-to-end paths in the topology graph). For example, P l = (e i,...,e j ), where f t (e i ) = (v,v l ) and f t (e j ) = (v m,v n ), is a path connecting the nodes v and v n. The set of all paths is represented with P. In mesh networs, each mesh node is connected to the base station with two paths [2]. The first path is the uplin path directed from the mesh node to the base station and the second path is the downlin path directed from the base station to the mesh node. The uplin and the downlin paths may be represented with disjoint sets: P up = {P up up 2,...,Pn } and P down = {P2 down,...,pn up } where there are (n 1) paths on the uplin, one for each node, and (n 1) paths on the downlin, also one for each node. We use the convention that v 1 is the networ sin (base-station), so all pacets enter and leave the networ through v 1. Since the traffic from the basestation does not traverse the mesh, there are no paths to or from the base station. The two sets of paths combine to form P = P up P down. Paths are associated with end-to-end connections, which introduce load into the networ. We associate the load of g : P R + bits per second with each connection (path), where R + is the set of non-negative numbers. 2 Given the connection loads, we find lin rates, r : E R +, by adding up the traffic of the connections traversing each lin: r j = g l I(e j P l ), e j E (1) P l P where I( ) is the indicator function, which is 1 when its argument is true and 0 when its argument is false. In mesh networs, the uplin and downlin connection loads can be represented with separate maps g up : P up R + and g down : P down R +, so each e j E requires a lin rate of r j = g P up l Pup up l I(e j P up l ) + gl down Pl down P down I(e j Pl down ). (2) We note that one of the sums in (2) is always zero, since lins are unidirectional and uplin and downlin paths are disjoint. Ideally, the number of slots each lin uses in the frame matches the lin rate: j = rj b j T f + h, e j E, (3) where is the ceiling function, r j is the lin rate, b j is the number of bits e j transmits in each slot, T f is the frame duration in seconds and h is the overhead of each transmission. The duration of lin s e j transmission is found by multiplying the number of slots by the duration of each slot d j = j T s, e j E. However, a TDMA schedule may result in lin rates different from the required lin rates. Lin rates may be different 2 Even though we are using the notation that end-to-end rates are real numbers, in reality the rates are discrete values since these values must be rounded to fit protocol pacets that negotiate bandwidths, e.g scheduling pacets. because the scheduler may decrease some of the lin durations in order to schedule the lins, or because the scheduler allows lins to transmit multiple times in a frame. In either case, the actual rate on lin e j is obtained with: ˆr j = ( ˆdj T s n j h ) b ( ) j bj = ˆ j n j h, (4) T f T f where ˆd j is the actual time allocated to the lin with the schedule, ˆ j = ˆd j /T s the resulting number of slots allocated to the lin, and n j is the number of times the lin transmits in the frame. For example, in mesh networs operating at the 20MHz bandwidth slot duration is T s = 12.5µs and if the frame duration is T f = 10ms there are N f = 800 slots in the frame. Each lin transmission has an overhead of 3 slots. At the lowest modulation and coding of BPSK-1/2 each slot carries 96 bits so if a lin needs 115.2bps it should be allocated 12 data slots and 3 overhead slots, adding up to 15 slots (187.5µs) in each frame. However, if the lin transmits twice in the frame, it should be allocated 18 slots, increasing the overhead. In the worst case, the lin may transmit five times in the frame in which case it is allocated 30 slots, maing the overhead 100%. We find achievable connection rates from the assigned connection rates and the actual lin rates. We assume that the nodes use a mechanism such as Weighted Fair Queuing (WFQ) [14] to share the bandwidth among multiple connections traversing a lin. With WFQ, connections get a share of bandwidth on the lins they traverse proportional to their weight on the lin. For example, if connections traversing a lin are weighted according to their end-to-end rates, the share of the bandwidth the connection associated with path P P gets on lin e j is g ˆr j /r j, since r j is the sum of all end-to-end rates traversing the lin (1). The achievable connection rate is found by considering the minimum bandwidth the connection gets on all the lins on the path: ĝ l = min e j P l B. Wireless Interference Model ˆr j r j g l, P l P. (5) We eep trac of conflicts between lins with conflict graphs. Conflict graphs can be defined with a triplet G c (E,C,f c ), where E is the set of lins, C = {c 1,...c r } is the set of TDMA conflicts, one for each of the r conflicting pairs of lins, and f c : C {{e i,e j }, for all e i,e j E} associates the conflicts with pairs of lins. We use the notation {, } for unordered sets and (, ) for ordered sets, so f c defines an undirected graph. The graph is undirected since conflicts are symmetrical. In this paper, we use a conflict graph with an arbitrary assignment of directions to the arcs, G c (E,C, f c ), where f c : C E E. The arbitrary orientation of arcs does not cause any loss of generality, however it simplifies the derivation of the results in this paper. We use a six node topology (Fig. 2a) to demonstrate how the arcs in the conflict graph are created. The vertices in the conflict graph are the 10 lins from the connectivity graph. 3

4 e 3 v 3 e 5 e 8 v 5 NcTS ρndts (1 ρ)ndts v 1 e 1 v 2 e 2 e 4 e 7 v 4 e 9 (z 1)Tf ztf (z + 1)Tf Time e 10 e 6 i πi N πi = σi + zin πi + N Slots (ei) v 6 (a) Topology Graph j πj N πj = σj + zjn πj + N Slots (ej) (z 1)N zn (z + 1)N (z + 2)N c26 c30 e1 c12 e3 c15 e5 c20 e8 Fig. 3. Time Axis and Virtual Slots Axis c11 e2 Fig. 2. c1 c6 c29 c28 c13 c27 c14 e4 e10 c18 c3 c2 c16 c7 c8 c31 c17 θ1 c22 c21 c5 (b) Conflict Graph Topology and Conflict Graphs for a Simple Topology The lins that conflict with each other (e.g. e 1 and e 6 ) are connected with an arc in the conflict graph (c 27, Fig. 2b). On the other hand, the lins that do not conflict with each other (e.g. e 2 and e 6 ) are not connected with an arc in the conflict graph. The orientation of lins corresponds to lin identifiers: f c (c ) = (e i,e j ) i < j, c C. (6) c19 e7 e6 c23 III. TDMA SCHEDULING In this section, we discuss the TDMA scheduling problem with no restrictions on TDMA delay. We discuss the problem of delay constrained TDMA scheduling in the next section. In both cases, we find a conflict-free TDMA schedule that minimizes utilization ρ. A. Generalized Search For Minimum ρ We represent TDMA schedules with a pair of maps S(s,d), where s = [s 1,...,s m ] T is the vector of activation times, corresponding to how long (in seconds) after the beginning of the data sub-frame lins start their transmissions, and d = [d 1,...,d m ] T is the vector of lin durations (in seconds), corresponding to the amount of time each lin transmits when active. Since the transmissions are restricted to the data subframe (Fig. 1), start times and lin durations are both restricted to be non-negative and less than the duration of the data subframe T d, s R m [0,T d ] and d Rm [0,T d ]. We find schedules that minimizes ρ in two steps. First, we map TDMA schedules onto an axis that measures time in terms of slots the virtual slots axis (Fig. 3). Each active period in the data sub-frame is mapped into N slots on the virtual slots axis (control sub-frames do not exist on the virtual axis). While N d is the maximum possible length of a schedule, N c9 c4 c10 c24 c33 c32 θ2 e9 c25 Algorithm 1 Find Minimum ρ 1: N 0 2: while S v (σ, ) over N slots do 3: N N + 1 4: end while 5: α N d /max{n,n d } // this is the minimum N 6: e j E : ˆ i α i 7: Find schedule S v (σ, ˆ ) for N d slots 8: e j E : s i T s σ i is the number of slots required to schedule all lins. In the case that, N N d the requested end-to-end flows can be scheduled. Otherwise, N > N d and the end-to-end rates need to be scaled down in order to find the schedule. On the virtual slots axis, a TDMA schedule S(s, d) corresponds to a virtual TDMA schedule S v (σ, ), where σ = [σ 1,...,σ m ] T is the vector corresponding to the indices of slots in the virtual frame in which lins begin transmitting and = [ 1,..., m ] T is the vector corresponding to the number of consecutive virtual slots each lin uses while active. Each of the m lin activation times, s i, is mapped onto a virtual activation time σ i on the virtual slots axis and each of the m lin durations d i is projected onto i slots on the virtual slots axis. We note that σ is restricted to be non-negative and less than N, σ Z m [0,N). Second, we use an iterative procedure to find the schedule with a minimum ρ (Algorithm 1). The algorithm first searches for the minimum number of slots required to schedule all lins (steps 1 4). At each iteration, the algorithm tries to find a virtual schedule S v (σ, ) that fits into N slots. If the schedule that fits into N slots cannot be found, N is incremented. The scheduling algorithms used in step 2 of the algorithm, are the main focus of this paper and are covered in the rest of this section and in the next section. Later, we show that N has a certain monoticity property that ensures that N at step 5 of the algorithm is minimum and also allows us to use the binary search instead of the linear search in steps 2 4 of the algorithm. After finding the minimum N, the algorithm scales the lin durations (step 6) with α = N d max{n,n d }, (7) resulting in a scaled down virtual schedule S v (σ, ˆ ), where ˆ = [ˆ 1,..., ˆ m ] T. Finally, the algorithm uses the scaled down lin durations, ˆ, to find a schedule that fits into N d slots and produces an actual schedule by multiplying each virtual start time with the 4

5 slot duration T s (steps 7 and 8). Since N is the minimum number of slots required to schedule all lins, α is maximum, ensuring that frame utilization ρ of the final schedule is minimum. Later, we show that the algorithm is guaranteed to find a virtual schedule that fits into N d slots since it uses the floor function when the lin durations are scaled down. There are two reasons to use the scaling approach. First, the scaling approach is used in the standard [2]. Second, the scaling eeps the relative proportions of connection rates the same since all connection rates are scaled by α. By substituting ˆ j = α j and (3) into (4) and (5), ignoring rounding, we get ( ĝ l = min ˆ j n j h e j P l [ α = min e j P l ) bj ( rj b j T f + h = αg l (1 α)h g l b j min, T f e j P l r j g l = min (α j h) b j g l T f r j e j P l T f r ) ] j bj g l h T f r j where we used the fact that n j = 1. So, with no overhead (h = 0) using maximum α for the scaling maximizes the concurrent throughput (the overhead simply introduces an error in the optimum). B. Conflict-Free Scheduling We now develop TDMA scheduling algorithms that can be used in step 2 of Algorithm 1 to find schedules without considering the TDMA delay. We start by deriving the necessary and sufficient conditions that mae a virtual schedule S v (σ, ) conflict-free for a fixed N. A schedule is conflict-free if the transmissions of conflicting lins do not overlap. We note that since the virtual TDMA schedule is scaled down to obtain the TDMA schedule, the resulting TDMA schedule is also conflict-free. 3 We observe that the repetitive nature of TDMA schedules means that a virtual activation time σ i for lin e i actually represents a series of activation times on the virtual slots axis. The series of virtual activation times can be derived from σ i by adding multiples of N slots, Π i = {σ i + z i N,z i Z} (Fig. 3). Conversely, σ i can be found from any activation time π i Π i with the modulo operator: (8) σ i = π i (mod N). (9) We establish the conflict-free conditions by considering a pair of lins e i and e j that conflict with c : f c (c ) = (e i,e j ). Tae any activation time π i Π i for lin e i and choose the next activation time for a conflicting lin e j, π j = min{π Π j : π π i } (Fig. 3). In this case, e j should not transmit before e i finishes its transmission π j π i + i π j π i i, (10) and e j should stop transmitting before e i transmits again π j + j π i + N π j π i N j. (11) 3 If the virtual schedule transmissions do not overlap neither do the TDMA schedule transmissions after the scaling. The equations can be combined to arrive at the following conflict-free condition: i π j π i N j. (12) In the above example, we assume that e i transmits first in each frame. If we change the order of transmissions we have: j π i π j N i. (13) We can combine the two conflict-free conditions further since their ranges of π j π i are mutually exclusive: i π j π i + o N N j, (14) where o = 0 if π j π i > 0 and o = 1 if π j π i < 0. The extra variable o specifies a relative order of transmissions, which prompts us to refer to it as the transmission order in the rest of the paper. A subset of activation times π i Π i, e i E, can be interpreted as a potential on the conflict graph, if we tae π = [π 1,...,π m ] to be a function on the vertices of the conflict graph, π : E Z. In this case, the conflict-free conditions for a schedule can be stated in terms of potentials in the conflict graph: Proposition 1 (Conflict-Free Schedules): A set of slot demands has a conflict-free virtual schedule S v (σ, ) over N slots if and only if there exists a potential π : E Z, such that for every arc c C with its corresponding pair of conflicting lins f c (c ) = (e i,e j ) the conflict-free condition (14) is satisfied. Proof: Suppose we have a conflict free schedule S v (σ, ). This means that for every pair of conflicting lins e i and e j, starting times σ i and σ j satisfy (14) and so present a valid potential. On the other hand, if there is a potential π, for which (14) is true, then for every lin e j, π j Π i, by construction of (14). We can use the modulo property to find the starting times σ. The importance of potentials becomes clear from the only if part of the proposition, which gives a way to find feasible conflict-free schedules for N slots with the following {0, 1}- integer linear program: Find π, o (15a) s.t l C T π on N1 u (15b) π Z m,o {0,1} r, (15c) where o = [o 1,...,o m ] T, π = [π 1,...,π m ] T, 1 is a column vector of m 1 s, C is the m r incidence matrix of the conflict graph defined as 1, if f c (c ) = (e i,e j ) C i = 1, if f c (c ) = (e j,e i ) (16) 0, otherwise, l = [l 1,...,l r ] T and u = [u 1,...,u r ] T where for each c C with corresponding vertices f c (c ) = (e i,e j ), u = j, l = i. It can be shown that this search problem is NPcomplete by reduction to Graph K-Colourability [15]. Even though the general scheduling problem is NPcomplete, if the transmission order o is fixed, the scheduling 5

6 problem becomes that of finding a solution to a set of difference inequalities. In the next section, we show that the transmission order is directly related to TDMA delay and propose exact and heuristic methods to find transmission orders with minimum delay. For now, we assume that the transmission order is nown prior to solving the difference inequalities (15). The inequalities can be solved efficiently using the Bellman- Ford shortest path algorithm [16] or the Dijstra shortest path algorithm [17]. We have proposed centralized [1] and decentralized versions [13] of TDMA scheduling algorithms for fixed transmission orders based on the Bellman-Ford shortest path algorithm. Here we briefly outline the centralized version of the TDMA scheduling algorithm; full description of the algorithm can be found in [1]. The algorithm starts by adding an extra vertex s to the conflict graph and connecting the new vertex to each of the original vertices in the graph with an arc of cost 0. The rest of the arcs in the graph get their costs from the upper and lower bounds in (14). The upper bound is treated as the cost of traversing an arc in the positive direction, while the lower bound is treated as the cost of traversing the arc in the negative direction. The feasible potential π is the minimum distance from s to every node in the graph, which is found with the Bellman-Ford algorithm. The final step in the algorithm is to use the modulo operator to find start times σ from π with the modulo operation. The search for a feasible o with the {0,1}-integer search (15) and the Bellman-Ford scheduling algorithm lead us to two different refinements of steps 2 and 7 in Algorithm 1. First, if the transmission order o is unnown prior to the start of the algorithm, step 2 is the {0, 1}-integer search (15), which produces a feasible transmission order, o, while step 7 is the Bellman-Ford scheduler. Second, if o is nown prior to the start of the algorithm, both step 2 and step 7 of the algorithm are the Bellman-Ford scheduler. In the rest of this section, we assume that Algorithm 1 uses one of these two refinements. C. Validity of the Iterative Search for Minimum ρ We now establish that a linear search is sufficient to find the minimum ρ. To establish this result, we use a wellnown networ flow theorem [17], applied to the conflict-free scheduling conditions: Theorem 1 (Feasible Differential Theorem): For a fixed N and a fixed transmission order o, there is a conflict-free schedule if and only if for every cycle θ in the conflict graph Gc (E,C, f c ): ) (N o N u ( ) l o N 0, (17) c {θ} + c {θ} where a cycle is an ordered set of conflicts θ C r, {θ} + is the set of arcs in cycle θ traversed in the positive direction and {θ} is the set of arcs in cycle θ traversed in the negative direction. Proposition 2 (Monoticity of N): If there exists a virtual schedule S v (σ, ) for N slots, then there exists a schedule for N + 1 slots. Proof: We re-arrange (17) to get: (1 o ) + o 1 u + l. N c {θ} + c {θ} c {θ} + c {θ} (18) We note that the right side of the expression is positive and decreases with N, proving the proposition. Corollary 1 (Optimality of Algorithm 1): If Algorithm 1 uses optimization (15) in step 2, ρ is minimum. If Algorithm 1 is given a transmission order and it uses the Bellman-Ford algorithm to find the schedule in step 2, ρ is minimum for that transmission order. For a fixed transmission order, the Bellman-Ford scheduling algorithm finds a feasible schedule in O(mr) steps, where m is the number of lins and r is the number of conflicts [16]. Since N has the monoticity property, we can use the binary search in steps 2 4 of Algorithm 1 instead of the linear search. So, with a nown transmission order, the minimum utilization schedule can be found in at most O(mr lnn max ) steps, where N max = m i=1 i is the number of slots required to schedule all lins without any spatial re-use. The monoticity of N can also be used to show that scaling down of the schedule does not change the feasibility of transmission orders. Corollary 2 (Scaling Preserves Feasibility): If there exists a virtual schedule S v (σ, ) for N slots, then there exists schedule S v (ˆσ, ˆ ) for N d slots, where e i E : ˆ i = α i, (19) ˆ = [ˆ 1,..., ˆ m ] T, ˆσ = [ˆσ 1,..., ˆσ m ] T and α is given by (7). Proof: If N d N, α = 1 and by the monoticity of N, the proposition is true. If N d < N, we observe that the right side of (18) becomes 1 N u c {θ} + + l c {θ} = α N d u 1 N d c {θ} + + αu c {θ} + + l c {θ} αl, c {θ} (20) where the equality holds because of the definition of α and the inequality holds because of rounding down. So, the scaling does not introduce any negative cycles and we can find ˆσ with the Bellman-Ford algorithm. D. Equivalent Transmission Orders In schedules found with Algorithm 1, lins transmit once in the virtual frame. However, when the starting times are transfered bac to the frame, the lin s transmission may be split over two frames. Lin e i will be scheduled twice for transmission in the frame if σ i + i > N (Fig. 4). The first transmission starts at time T c +s i with the duration of T d s i seconds and the second transmission starts at the beginning of the next data sub-frame with the duration of s i + d i T d 6

7 i ei ej ei ej v vl vm v vl vm Fig. 4. zn σ i (z + 1)N slots (e i) T c T d s i Ts c i + d i T d zt f (z + 1)T f (z + 2)T f time Split transmissions 0 si sj Tf si + Tf di Tc dj sj si + tp 0 sj si Tf sj + Tf di Tc dj sj si + Tf + tp sj + Tf si + Tf seconds. So, this scheme limits the number of transmissions by any lin to at most two in a frame. The number of split transmissions can be minimized in at most N steps. The algorithm that minimizes the number of split transmissions shifts the feasible potential π in iterations. In each iteration a new potential π = [π 1,...,π m ] T is obtained from the potential in the previous iteration by increasing every π i by 1. Then, the algorithm converts the shifted potential the virtual schedule S v (σ, ) with the modulo operation and counts the number of lins with split transmissions. The algorithm pics the potential whose virtual schedule has the fewest lins with split transmissions as the potential to be converted to the final virtual schedule. Each of the shifted potentials π has the same transmission order, o, as the original feasible potential π. However, each virtual schedule σ obtained from a shifted potential π has its own transmission order o (σ), which may be different from o. The two orders may be different since the potential on each vertex in the conflict graph π i is related to the lin s virtual activation time with π i = σ i + z i N and for an arc c, with the endpoints f c (c ) = (e i,e j ), z i may be different from z j, while the feasibility conditions (14) are still true for the virtual activation times σ i and σ j. If we substitute π i = σ i + z i N into (14), we get: where i o (σ) N σ j σ i N j o (σ) N, (21) o (σ) = z j z i + o, c C : f c (c ) = (e i,e j ). (22) So, the change in the relative transmission order may happen if o = 0 and z j z i = 1, or if o = 1 and z j z i = 1. IV. TDMA DELAY AWARE LINK SCHEDULING In this section, we propose exact and heuristic methods to find transmission orders with minimum TDMA delay. First, we show that the TDMA delay is directly related to the transmission order in the frame. Then, we formulate a {0, 1}- integer program that finds the transmission order for which the maximum TDMA delay of all paths is minimized. This optimization can be used in step 2 of Algorithm 1 to find a minimum TDMA delay schedule for a fixed N. With the refinement, Algorithm 1 finds the minimum delay schedule among all schedules with maximum utilization. Finally, we propose a polynomial time heuristics that finds transmission orders whose TDMA schedules have a TDMA delay of one frame size. The heuristic can be used to find transmission orders and then Algorithm 1 can be used with Fig. 5. time (a) s j > s i Single Hop Delay time (b) s j < s i the Bellman-Ford scheduler in step 2 to find TDMA schedules with a fixed maximum TDMA delay, but sub-optimal utilization, in polynomial time. We compare the utilization of the heuristics with the exact method in the next section. A. TDMA Delay We start by observing what happens to data pacets when they traverse a TDMA networ. Consider a path P l = {...,e i,e j,...} where e i {v l } + and e j {v l } (the fragment of the path containing e i and e j is shown in Fig. 5). We use the notation that e i {v l } + are outgoing lins of v l and e j {v l } are the incoming lins of v l. In every frame, lin e i sends ˆr i T f bits to node v l, which forwards them on lin e j. We assume that the end-to-end rate for the connection is limited by (5), so data arriving on e i is always transmitted by the next transmission of e j. We can expand this observation to conclude that with end-to-end rate control at every router in the path data only wait for the difference in time until the transmission of the next lin in the path. Since the end-to-end delay is completely defined by the TDMA schedule, we call it end-to-end TDMA delay. 4 First, we find the expression for TDMA delay for a single-hop since TDMA delay occurs when a pacet arrives to a node and has to wait for the next outgoing lin to transmit. Then, we add up TDMA delay at each hop on the path to get the expression for the end-to-end TDMA delay. We show how single-hop TDMA delay occurs in Fig. 5 for two different schedules S a (s,d) (Fig. 5a) and S b (s,d) (Fig. 5b). We align the time axis to the beginning of the data sub-frame to simplify exposition. In schedule S a (s,d), s j > s i, so the first bit of a pacet sent from v to v m experiences the delay of s j s i +t p, where t p is the propagation delay. In schedule S b (s,d), s i > s j, so when the pacet arrives at v l it has to wait for e j to transmit in the next frame. In this case, the TDMA delay from v to v m is s j s i + T f + t p, where T f is frame duration. In the sequel we drop the propagation delay t p since it is independent of scheduling. The single-hop TDMA delay is directly related to the transmission order for the schedule. For example, the TDMA schedule S a (s,d) corresponds to a virtual schedule S v (σ, ) 4 If end-to-end rate limiting is not used, TDMA delay is the delay experienced by the first bit of a pacet traversing the networ with empty queues. 7

8 for which o (σ) = 0 and f c (c ) = (e i,e j ). However if the virtual schedule S v (σ, ) corresponds to TDMA schedule S b (s,d), the transmission order is o (σ) = 1 and f c (c ) = (e i,e j ). We conclude that if a conflict c is traversed in the positive direction, i.e. pacet is transmitted from e i to e j, TDMA delay is: t = s j s i + o (σ) T f. (23) On the other hand, if f c (c ) = (e j,e i ), i.e. the conflict is traversed in the opposite direction, TDMA delay is: t = s j s i + (1 o (σ) )T f. (24) In the rest of the paper, we concentrate on TDMA delay on return paths. The return path delay is important for applications that use TCP as the transport protocol since the throughput of TCP is inversely proportional to the return path delay [18]. Schedulers that minimize TDMA delay of unicast end-to-end connections such as Voice-over-IP traffic can be derived similarly to the way we derive the scheduler for return path TDMA delay [19]. The set of return paths P R can be obtained by fusing unicast paths. For example, in mesh networs the set of return paths can be obtained by fusing the downlin and uplin paths: P R = {P l : P l = P up l P down l,l = 2...n} (25) A return 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 vertices of the conflict graph G c (V,E, f c ) listed in P l. For example, the return path P 1 = (e 1,e 3,e 6,e 10,e 4,e 2 ) in Fig. 2a corresponds to the cycle θ 1 = {c 12,c 10,c 22,c 18,c 13,c 11 }, mared in Fig. 2b. In the same topology, the return path P 2 = (e 1,e 3,e 5,e 8,e 9,e 7,e 4,e 2 ) in Fig. 2a corresponds to the cycle θ 2 = {c 12,c 15,c 20,c 25,c 24,c 17,c 13,c 11 }, also mared in Fig. 2b. The two cycles share arcs c 11,c 12,c 13 in the conflict graph. We find the end-to-end TDMA delay for a path P l by finding the delay incurred while traversing the corresponding cycle θ l in the conflict graph: D l = c {θ l } + ( τ + o (σ) T f ) c {θ l } ( τ + o (σ) T f T f ), (26) where τ s j s i is tension on c C : f c (c ) = (e i,e j ), {θ P } + is the set of conflicts traversed in their direction and {θ P } is the set of conflicts traversed in their opposite direction. TDMA delay depends on the transmission order only: D l = c {θ l } + o (σ) T f + c {θ l } ( 1 o (σ) ) T f, (27) where all tension terms cancel out. For example, the summation over the tension terms for the cycle θ 1 in Fig. 2b is: τ 12 +τ 10 + τ 22 τ 18 τ 13 τ 11 = (π 3 π 1 ) + (π 6 π 3 ) +... (π 2 π 1 ) = 0. (28) The cancellation of tensions in (26) is a well nown graph property [17]. In the rest of the paper, we use vector notation to represent cycles to mae formulas more compact. A cycle θ l in the conflict graph is defined by the vector θ l = [θ (l) 1,...,θ(l) r ] T, where θ (l) = 1 if c {θ l } +, θ (l) = 1 if c {θ l } and θ (l) = 0 if c / θ l. For example, the cycle θ 1 in Fig. 2, corresponds to the vector θ 1 {0,1} 33 where θ 1 (10) = θ 1 (12) = θ 1 (22) = 1, θ 1 (11) = θ 1 (13) = θ 1 (18) = 1 and all other entries in the vector are 0. In vector notation, TDMA delay on a return path is: ] D l = [θ T l o (σ) + K l T f (29) where o (σ) = [o (σ) 1,...,o(σ) r ] T and K l = c {θ P } 1 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 TDMA delay on the set of return paths P R, while ensuring that a feasible virtual schedule exists over N slots. An objective function that finds such schedules is the min-max delay defined as min max Pl P R D l. We combine the formula for the virtual delay (27) 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 N: min t (30a) π,o,t ] s.t [θ T l o + K l T f t, P l P R (30b) l C T π No N1 u π Z m,o {0,1} r,t 0, (30c) (30d) Constraints (30b) ensure that no path has a delay larger than t, while t is minimized; this achieves the goal of finding the min-max optimum. The other constraints (30c) define the polyhedron of conflict-free schedules for a fixed N. We use the max-min optimization (30) in step 2 of Algorithm 1 (step 7 is the Bellman-Ford algorithm). In the sequel, we refer to this refinement of Algorithm 1 as Algorithm-MM. We note that Corollary 1 still holds for Algorithm-MM since a feasible solution to (30) is also a feasible solution to (15). Algorithm-MM therefore finds the min-max delay optimum schedule among all schedules that maximize ρ. If optimization (30) has a solution, it provides a transmission order o as well as a potential π corresponding to a virtual schedule. Algorithm-MM uses the Bellman-Ford scheduler to find the final schedule (step 7 of Algorithm 1). Since the last step in the Bellman-Ford scheduler is to use the modulo operator to find starting times σ, the final transmission order o (σ) may be different from o. We now show that despite the difference the delay produced by the two transmission orders is the same. Proposition 3 (Modulo Operator Does Not Change Delay): Suppose that transmission order o is used by the Bellman- Ford scheduler to find a virtual schedule S v (σ, ) with 8

9 transmission order o (σ). The TDMA delay induced by o (σ) is the same as TDMA delay induced by o. Proof: For a conflict c C : f c (e i,e j ), o (σ) and o are related with o (σ) = z j z i +o by (22), where z j,z i Z. If we define tension between the z variables as τ z = z j z i for c C : f c (e i,e j ) and the vector of tensions τ z = [τ1 z,...,τm] z T, it follows that θ T l o = θ T l τ z + θ T l o (σ) = θ T l o (σ) (31) since the sum of tensions around a cycle adds up to zero θ T l τ z = 0 [17]. This leads us to the conclusion that ] ] [θ T l o + K l T f = [θ T l o (σ) + K l T f = D l, (32) meaning that the two transmission orders have the same TDMA delay. C. A Frame Transmission Order for Tree Topologies The min-max formulation (30) is hard to solve because it requires a search for o over the space of all {0,1} r vectors, for example this search can be performed with the standard branch-and-bound technique [20]. In this section, we tae advantage of the fact that many multi-hop wireless networs have tree topologies and propose an algorithm that finds transmission orders on tree topologies that have a one frame TDMA delay T f. Since the two most important and common examples of wireless multi-hop networs with tree topologies are sensor networs and mesh networs, the heuristic is applicable in a wide array of scenarios. This heuristic can be used to find transmission orders before Algorithm 1 is used with the Bellman-Ford scheduler in steps 2 and 7. In the sequel, we refer to this refinement of Algorithm 1 as Algorithm-TH. The algorithm defines a raning function R : E Z, which indicates the preferred order of transmissions and then uses the raning function to find the transmission order o. Initially, the algorithm sets the ran of all the nodes to zero. The algorithm then examines each return path P l P R, lin-by-lin, and assigns a ran to each lin as a function of the distance from the root of the routing tree. We assume that the base-station is v 1 V. For lins in a return path P l = {e i,...,e,e l,...e j }, where e i {v 1 } + and e j {v 1 }, the ran is assigned as follows: R = max{r,r l + 1}, e,e l P l : e l e, (33) where we use the notation that e l e means that e follows e l on the path. We note that the distance of the lin is defined as its placement on the return path and not its topological distance from the root of the tree. For the example in Fig. 2a, we have R 1 = 0,R 3 = 1,R 5 = R 6 = 2,R 8 = R 10 = 3,R 9 = 4,R 7 = 5,R 4 = 6 and R 2 = 7. Given the raning, the transmission order o is assigned with the following rule: c C, with f c (c ) = (e i,e j ), { 0, if R j R i o = (34) 1, otherwise Proposition 4: If the ordering o, derived from the raning R : E Z has a feasible schedule, then for P l P R : D l = T f. (35) Proof: Consider a return path P l = {e i,...e j }, where e i {v 1 } + and e j {v 1 }, and its corresponding cycle θ l in the conflict graph. Assume that c l is the last conflict in θ l, connecting e j and e i in the conflict graph. Since R j > R i, c l = 1 if c l {θ l } + and c l = 0 if c l {θ l }. On the other hand for all other c {θ l }\{c l }, o = 0 if c {θ l } + \{c l } and o = 1 if c {θ l } \ {c l }. The delay can be found as: D l = o T f + (1 o )T f c {θ l } + c {θ l } = o l T f + (1 o )T f (36) c {θ l } \{c l } = o l T f + {θl } Tf {θl } \ {c l } Tf = T f, where is the cardinality of a set. We note that the raning function does not allow spatial re-use between lins on the same path since it orders all lins on the path to transmit in a sequence. However, the raning function still allows spatial re-use for lins on different paths. V. SIMULATION RESULTS In this section we compare the performance of scheduling algorithms from the previous section when they are applied to mesh networs [2]. We do not get into the details of the protocol in this paper; more information about the protocol and scheduling algorithms are available in the standard [2], our summary of the protocol [21], and our survey of scheduling algorithms for mesh networs [22]. In all our examples, we set the frame duration to 10ms giving a total of 800 Orthogonal Frequency Division Multiplexing (OFDM) symbols in each frame. We set the size of the control sub-frame to 70 OFDM symbols, leaving 730 symbols for the data sub-frame. The mesh protocol mandates the assignment of data sub-frame symbols to lins in terms of transmission opportunities. The size of the transmission opportunity is the smallest integer higher than the result of dividing the number of data sub-frame slots with With the 10ms frame size and 70 control OFDM symbols, each transmission opportunity consists of 3 OFDM symbols, for a total of 243 transmission opportunities in the data sub-frame. In order to mae our scheduling approaches consistent with mesh protocol, in the sequel, we equate transmission opportunities to TDMA slots. Frame utilization ρ has a nice practical interpretation in mesh networs. The first ρ N d slots of the data sub-frame are allocated with centralized scheduling and the last (1 ρ) N d slots of the data sub-frame are assigned with decentralized scheduling. In centralized scheduling endto-end bandwidth is assigned by the base station, while in decentralized scheduling, nodes negotiate slot assignments with their neighbours with no guarantees on end-to-end rates. 5 The standard restricts the number of transmission opportunities in the data sub-frame to at most 256 because the duration fields in the scheduling control pacets are 8 bits long. 9

10 In the sequel, we fix the percentage of the frame used for centralized scheduling to ρ = 80%, leaving 194 centralized scheduling transmission opportunities in every frame. In terms of mesh networs, the search for minimum N can be interpreted in two ways. First, if the number of centralized scheduling transmission opportunities is unnown, the search for minimum N maximizes the percentage of the frame used for best-effort traffic (decentralized portion). Second, if the number of centralized scheduling transmission opportunities is nown, minimum N (maximum scaling factor α) maximizes the concurrent throughput of connections scheduled with the centralized scheduling algorithm (8). Our TDMA scheduling approach fits perfectly into centralized scheduling. First, the base-station uses Algorithm- MM or Algorithm-TH to find the transmission order (lin ranings) and the scaling factor α that maximizes the concurrent end-to-end throughput of mesh nodes. Then, the basestation multicasts the raning and the scaled down lin bandwidths. Second, the nodes use the raning and the lin bandwidths they receive from the base-station and their nowledge of the networ topology to calculate the networ wide TDMA schedule with the Bellman-Ford algorithm. 6 We use two performance indices to measure the performance of the algorithms. The first performance index shows the ratio between the requested end-to-end rates and the endto-end rates achieved with a particular schedule. For this measure, we calculate the average percentage of achieved bandwidth over the active sources in the networ: ᾱ = 1 S n i S ĝ up i g up + 1 i S n i S ĝi down gi down, (37) where S is the set of of active sources, is the cardinality of a set and ĝ up i and ĝi down are obtained by first finding the actual rate on each lin resulting from the schedule with (4) and then finding the end-to-end rate with (5). The second performance index is the maximum return path delay: A. Example 1: Small Topology D max = max P l P D l. (38) We examine schedules applicable to the topology shown Fig. 2a. All nodes except the base station (v 1 ) act as sources with fixed uplin and downlin bandwidth demands of 900bps. Since the topology is a tree, finding all paths and allocating rates to lins with (2) is straightforward. We use Algorithm-MM (Fig. 6a), Algorithm-TH (Fig. 6b), the centralized scheduling algorithm (Fig. 6c), and the graph colouring algorithm [11] (Fig. 6d). Fig. 6b shows the schedule obtained with the centralized scheduling algorithm proposed in the standard [2]. The standard proposes this algorithm as an example of how to allocate transmission opportunities, however the standard does not mandate its use. The first step in the algorithm 6 We assume that the networ wide topology is nown to all nodes. If this assumption is not true the nodes can use the distributed version of the scheduling algorithm [13], which relies on local topology information only to find networ wide TDMA schedules. e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e1 e2 e3 e4 e5 e6 e7 e8 e9 e Control Sub-frame Centralized Scheduling Decentralized Scheduling (a) Algorithm-MM Control Sub-frame Centralized Scheduling Decentralized Scheduling (b) Algorithm-TH Control Sub-frame Centralized Scheduling Decentralized Scheduling (c) IEEE Control Sub-frame Centralized Scheduling Decentralized Scheduling (d) Graph Colouring Fig Schedules for the Small Topology (Fig. 2) is to find a lin raning during a breadth-first traversal of the routing tree. With the breadth first search the raning on the topology becomes R 1 = R 2 = 0,R 3 = R 4 = 1,R 5 = R 6 = R 7 = R 10 = 2 and R 8 = R 9 = 3. Second, the algorithm assigns transmission opportunities to lins in the order of their ran. The lin with the lowest ran is assigned transmission opportunities at the beginning of the data sub-frame. The lin with the next highest ran is assigned the subsequent transmission opportunities and so on, until all lins are scheduled. If the total number of assigned transmission opportunities is larger than the number of transmission opportunities reserved for centralized scheduling, the algorithm scales down the lin transmissions to fit in the frame. Fig. 6d shows the schedule obtained with a vertex colouring scheduling algorithm [11]. This algorithm assigns transmission opportunities in rounds. In each round, the algorithm assigns 10