Performance Comparison of 802.16 Centralized

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Performance Comparison of 802.16 Centralized Scheduling Algorithms Petar Djukic and Shahrokh Valaee

Abstract IEEE 802.16 mesh standard specifies uses Time Division Multiple Access (TDMA) technology to provide end-to-end Quality-of-Service (QoS). However, the performance in the mesh depends on TDMA scheduling algorithms. A new class of TDMA scheduling algorithms, based on the BellmanFord algorithm, was proposed recently. This class of TDMA scheduling algorithms takes into account many of the QoS issues not considered with previous scheduling approaches, such as overhead and TDMA scheduling delay. In this paper we apply the Bellman-Ford scheduling algorithms to 802.16 mesh networks and compare them with other 802.16 scheduling algorithms in terms of end-to-end delay and end-to-end throughput. We show that only the Bellnan-Ford scheduling algorithms have consistently low TDMA scheduling delay and high end-to-end throughput.

I. I NTRODUCTION Currently deployed mesh networks use 802.11 wireless devices for wireless mesh connectivity [1], [2]. However, 802.11 medium access control is not appropriate for commercial applications of mesh networks since the Distributed Coordination Function (DCF), used to coordinate 802.11 transmissions, cannot provide Quality of Service (QoS) [3]. To solve this problem, IEEE has ratified 802.16 [4] mesh standard, which uses Time Division Multiple Access (TDMA) to provide guaranteed link rates. In TDMA, link rates are allocated over frames with a fixed number of transmission opportunities. A schedule assigns transmission opportunities to links and during each transmission opportunity, a number of non-conflicting links can transmit simultaneously, 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 e-mail:{djukic,valaee}@comm.utoronto.ca This work was sponsored in part by the LG Electronics Corporation.

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taking advantage of spatial reuse. The rate of each link is given by the number of transmission opportunities it is assigned in the frame. The performance of 802.16 mesh networks is dependent on the ability of the management layers to schedule links. The 802.16 standard [4] proposes a simple scheduling algorithm to schedule links, however this algorithm does not allow any spatial re-use so it has small end-to-end throughput. Scheduling TDMA links to take advantage of spatial re-use has been a research topic for some time, however most of this research is not appropriate for the 802.16 mesh protocol. We identify three main reasons for the disconnect between the previous research in TDMA scheduling algorithms and requirements of the 802.16 mesh protocol. First, many of the previous approaches do not model all conflicts that exist in TDMA wireless networks [5]–[7]. In these works, the authors assume that only the links sharing a node in common conflict (primary conflicts). Second, TDMA scheduling algorithms in literature do not take transmission overhead into account, which introduces a huge overhead in 802.16 networks. In 802.16, every transmissions has an overhead of 28.8kbps at the lowest modulation and 259kbps at the highest modulation. Finally, these scheduling approaches ignore TDMA scheduling delay. TDMA scheduling delay occurs when packets arriving on an inbound link must wait for the subsequent frame to be transmitted on the outbound link. Since TDMA scheduling delay accumulates at every hop in the network, the end-to-end delay experienced on a path can be large. Recently [8], [9], a new class of TDMA scheduling algorithms was proposed to resolve all of these issues. This class of scheduling algorithms is based on the Bellman-Ford algorithm. In [8], [9], we recognized that the TDMA scheduling delay depends on the transmission order in the frame. This allows us to decompose the TDMA scheduling problem in two parts. First, the scheduling algorithm algorithm finds either an optimal or a heuristic transmission order. Finding the optimal transmission order requires an exhaustive search over all possible transmission orders, which is computationally hard. Nevertheless, we show in [8], [9] that heuristics perform well. In this paper, we show that the Bellman-Ford algorithms fit perfectly into the 802.16 scheduling framework and evaluate their performance in 802.16 mesh networks. There are many heuristic scheduling algorithms that have been proposed specifically for 802.16 mesh networks [10]–[12]. These algorithms take “secondary” conflicts into account and schedule links to take advantage of spatial re-use. One of the algorithms [12] also takes 802.16 overhead into account and schedules links once per frame. However, these algorithms still do not take June 21, 2007

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into account TDMA scheduling delay. This paper is the first to apply Bellman-Ford scheduling algorithms [8], [9] to 802.16 mesh networks. We also provide a realistic comparative analysis of centralized scheduling algorithms [8]–[12] in terms of end-to-end delay and end-to-end throughput. We take into account 802.16 mesh protocol overhead when evaluating end-to-end throughput, which is sometimes ignored [10], [11]. We also take into account TDMA scheduling delay, which is an important QoS parameter, but (other than our own work [8], [9]) is not considered in previous work. We show with simulations that the Bellman-Ford scheduling algorithms have the best performance in terms of end-to-end throughput and end-to-end TDMA scheduling delay. We have previously compared some of these algorithms in a limited fashion in [13]. In this work, we compare the performance of scheduling algorithms with further simulations and include new scheduling algorithms that have not been published prior to the publication of [13]. The rest of the paper is organized as follows: Section II summarizes 802.16 mesh standard and its two scheduling protocols. Section III defines the two QoS measures we use to evaluate scheduling algorithms: end-to-end throughput and end-to-end TDMA scheduling delay. Section IV summarizes 802.16 centralized scheduling algorithms available in literature [10]–[12] and explains how to implement the Bellman-Ford scheduling algorithms [8], [9] in 802.16 mesh networks. Section V simulates the algorithms and compares their performance in terms of endto-end throughput and end-to-end TDMA scheduling delay. II. 802.16 M ESH S TANDARD In this section, we describe the 802.16 mesh standard [4]. We start by describing the 802.16 Orthogonal Frequency Division Multiplexing (OFDM) physical layer. Then, we describe how TDMA is implemented in 802.16 mesh networks by combining OFDM symbols into 802.16 transmission opportunities and how transmission opportunities are assigned to links with 802.16 scheduling protocols. A. Physical Layer IEEE 802.16 mesh physical layer uses Orthogonal Frequency Division Multiplexing (OFDM) in the licensed and license-exempt 5GHz frequency bands. OFDM transforms blocks of bits into constant duration symbols carried on a set of frequency orthogonal carriers. The bandwidth of June 21, 2007

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the final OFDM signal is the frequency range occupied by all carriers. The raw data rate of 802.16 depends on the duration of each OFDM symbol, which depends on wireless bandwidth, and the number of bits carried in each OFDM symbol. In the license exempt 5GHz frequency band, 802.16 mesh networks are mandated to use OFDM with 20MHz total rate, resulting in 12.5µs OFDM symbol duration.1 The number of bits carried in a OFDM symbol depends on the modulation scheme used during its transmission. The lowest modulation is BPSK-1/2 with 96 bits per OFDM symbol, resulting in a raw data rate of approximately 7.7Mbps. The highest modulation is 64QAM-3/4 with 864 bits per OFDM symbol, resulting in the raw data rate of approximately 70Mbps. The full list of possible modulations is available in the standard [4] and our review of the standard [14]. B. 802.16 TDMA IEEE 802.16 groups symbols into equal duration frames. Framing has two purposes. First, the frame boundaries are used to synchronize mesh nodes. Second, the frame structure allows the division of control and data traffic into the control sub-frame and data sub-frames. OFDM symbols in both control and data sub-frames are grouped into transmission opportunities. In the control sub-frame, transmission opportunities consist of 7 OFDM symbols and there are a total of MSH-CTRL-LEN transmission opportunities in each control sub-frame. MSH-CTRL-LEN is a parmeter set by the network operator. In the data sub-frame, the size of transmission opportunities is determined by dividing the number of OFDM symbols in the data sub-frame with 256. 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 packets are 8 bits long. For example, if the frame size is 10ms, there are 800 OFDM symbols in the frame and if there are MSH-CTRL-LEN = 10 transmission opportunities in the control sub-frame, there are 730 OFDM symbols in the data sub-frame and each data sub-frame opportunity is DataTxOppSize = 3 OFDM symbols. Because the number of OFDM symbols in each transmission opportunity is always rounded up, there may be less than 256 transmission opportunities in each frame. For example, in the previous example, there are DataTxOppNum = 243 transmission opportunities in each data sub-frame. 1

The standard allows other symbol durations at 20MHz, however we chose this value because it allows integer number of symbols in TDMA frames. June 21, 2007

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The 802.16 standard mandates how transmission opportunities are assigned in the control subframe with specialized scheduling protocols and algorithms. The 802.16 standard mandates two scheduling protocols, which provide a mechanism to assign transmission opportunities to links in the data sub-frame and exchange packets in the control sub-frame. The standard leaves open how to decide the best assignment of transmission opportunities – scheduling algorithms. C. 802.16 Data Sub-frame Scheduling Protocols IEEE 802.16 specifies two different protocols for negotiation of schedules in the data subframe: centralized and decentralized scheduling protocols. The centralized scheduling protocol assigns transmission opportunities in the first MSH-CSCH-DATA-FRACTION×DataTxOppNum transmission opportunities of the data sub-frame, where MSH-CSCH-DATA-FRACTION is a percentage of data sub-frame transmission reserved for centralized scheduling, specified by the network operator. The decentralized scheduling protocol assigns transmission opportunities in the last (1 − MSH-CSCH-DATA-FRACTION) × DataTxOppNum transmission opportunities of the data sub-frame. In the centralized scheduling protocol, mesh nodes request end-to-end rates from the base station. The base-station uses the end-to-end rate requests and assigns link rates in the network. The assignment of link rates creates a routing tree in the mesh. After assigning link rates, the base station multicasts the assignment through the mesh. Mesh nodes use the link rate assignment to determine all starting times and link durations in the frame (a common, global, schedule). In the decentralized scheduling protocol, neighbouring mesh nodes negotiate local schedules. First, a node wishing to change its allocation of transmission opportunities sends a request for transmission opportunities to its neighbours. One, or more of the neighbours, correspond with a range of available transmission opportunities. The node chooses a subrange of these transmission opportunities and confirms that it will use them with a third message. The 802.16 standard does not specify any algorithms for the distributed scheduling protocol. In the rest of the paper we limit our investigation to scheduling algorithms for centralized scheduling protocols and assume that MSH-CSCH-DATA-FRACTION = 100%.

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III. E ND - TO -E ND Q O S IN 802.16 M ESH N ETWORKS We now define QoS metrics used to evaluate various scheduling algorithms in the next section. We start with an allocation scheme that determines the number of transmission opportunities required on each link to support requested end-to-end rates. Then, we show how to find end-toend throughput from a given schedule. Finally, show how to find end-to-end TDMA scheduling delay that is induced by the schedule. A. Link Durations from End-to-End Rates If two nodes are in the range of each other, they establish mutual links in the MAC layer, so the mesh can be represented with a connectivity graph G(V, E, ft ), where V = {v1 , . . . , vn } is the set of nodes, E = {e1 , . . . , em } are directional links between neighbouring nodes, and ft : E → V × V assigns links to pairs of nodes. The connectivity map ft enforces the fact that all links are directional, so for a link ek ∈ E, ft (ek ) = (vi , vj ) means that node vi uses link ek to transmit data to node vj . We model channel quality with link bit-rates. We define link bit-rate as the number of bits transmitted in an OFDM symbol. Bit-rates depend on the modulation, which depends on signalto-noise ratio for the link. The signal-to-noise ratio is divided into several discrete levels and each is associated with its maximum bit-rate. Link rates are represented with the map b : E → {96, . . . , 864}, where 96 is the number of bits carried in an OFDM symbol with the BPSK1/2 modulation and 864 is the number of bits carried in an OFDM symbol with 64QAM-3/4 modulation. We assume that a routing algorithm forms two spanning trees in the network. The first tree is associated with the uplink traffic and it consists of the links carrying the uplink traffic (to the base station). The second tree is associated with the downlink traffic and contains links carrying downlink traffic (to the mesh nodes). Each mesh node is connected to the base station with two, acyclic, directed paths. The first path is the uplink path directed from the mesh node to the base station and the second path is the downlink path directed from the base station to the mesh node. Paths are represented with ordered sets of links, so Pl = (ei , . . . , ej ), where ft (ei ) = (vk , vl ) and ft (ej ) = (vm , vn ), is a path connecting the nodes vk and vn . The uplink and the downlink paths are represented with separate sets: P up = {P2up , . . . , Pnup } and P down = {P2down , . . . , Pnup } where there are (n − 1) paths on the uplink, one for each node, and (n − 1) paths on the downlink, June 21, 2007

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also one for each node. We use the convention that v1 is the base station. There are no paths to the base station since the traffic from its wireless terminals does not traverse the wireless mesh. We represent the node uplink and downlink demands, in bits per second, with g up : V → R+ and g down : V → R+ , respectively.2 Since each node corresponds to an uplink and a downlink path, the uplink and downlink requests of each node are associated with the uplink and downlink paths for the node. The required rate on each link, r : E → R, can be found by adding up the traffic on each path traversing the link: rj =

n X

giup I(ej ∈ Piup ) +

i=2

n X

gidown I(ej ∈ Pidown ),

(1)

i=2

where I(·) is the indicator function, 1 when its argument is true and 0 when its argument is false. Since links are unidirectional, uplink and downlink trees are disjoint, hence links never carry both the uplink and the downlink traffic and one of the sums in (1) is always 0. Given the required link rate, the number of OFDM symbols used by each link in a frame is found from the modulation on the link and the frame duration. IEEE 802.16 mesh standard mandates that each transmission have one guard OFDM symbol before data is transmitted and one or two guard OFDM symbols after data is transmitted, so the number OFDM symbols link ej uses in each frame is: 

 rj Tf ∆j = DataTxOppSize + h , bj

(2)

where h is the number of guard OFDM symbols (h = 2 or h = 3), ⌈·⌉ is the ceiling function, bj is the number of bits in each transmission opportunity and Tf is the frame duration. However, the number of actual symbols may be different since 802.16 mandates that the number of OFDM symbols used in each transmission be a multiple of transmission opportunity size. In the sequel, we assume that ∆j is also rounded up to the nearest number of transmission opportunities, i.e. ∆j mod DataTxOppSize = 0. 2

Even though we are using the notation that rate demands real numbers, in reality the rates are discrete since they must be rounded to fit into 802.16 scheduling packets.

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B. Achievable End-to-End Rates Ideally, a centralized scheduling algorithm assigns the number of transmission opportunities to each link corresponding to the requested end-to-end rates. However, the scheduling algorithms we review in the following section may have to scale down the link durations in order to schedule all links, resulting in end-to-end throughput different from the requested rates. In some cases, the algorithms also have more than one transmission in a frame, which increases overhead and decreases end-to-end throughput supported by the schedule. We find the end-to-end throughput in two steps. First, we find the actual rate assigned to link ej , given the number of transmission ˆ j: opportunities assigned to the link, ∆ rˆj =

ˆ j × DataTxOppSize − nj × h ∆ × bj , DataTxOppSizeTf

(3)

where nj is the number of times link ej transmits in each frame. Second, we use the actual link rate, rˆj , to find the achievable end-to-end rates. We assume that the nodes use a mechanism such as Weighted Fair Queuing (WFQ) [15]. With WFQ, all end-toend connections can get an a proportional share of rate on the links they traverse. For example, if the end-to-end flows are weighted according to their requested rate, the share of uplink rate that node vi gets on link ej is rˆj /rj giup since by (1), rj is the total rate of all connections traversing the link. Similarly, the share of downlink rate that node vi gets on link ej is rˆj /rj gidown . The achievable end-to-end throughput is found by considering the minimum link rate the end-to-end connection gets on the path: gˆiup = minup ej ∈Pi

rˆj up g rj i

and gˆidown = min

ej ∈Pidown

rˆj down g . rj i

(4)

C. End-to-End TDMA Scheduling Delay TDMA scheduling delay occurs when packets arriving on an inbound link must wait for the subsequent frame to be transmitted on the outbound link [8], [9]. We show how TDMA scheduling delay occurs in Fig. 1. In this scenario, ei is an incoming link on a mesh node and link ej is an outgoing link on the same mesh node. The packets traverse link ei first and then link ej and are delayed at the mesh node, between the transmissions of ei and ej . We use the notation that sj is the first transmission opportunity link ej transmits in and fj is the last transmission opportunity ej transmits in. In Fig. 1a, fi < sj so packets forwarded on ej experience the delay June 21, 2007

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sj − si si

fi sj

sj − si + Tf Time

si + Tf

fj

sj

fj si

(a) Delay fi < sj

Fig. 1.

sj + Tf

fi

Time

(b) Delay fi > sj

TDMA Delay

of sj − si , from the time they were transmitted on ei to the time they are forwarded on ej . In Fig. 1b, fi > sj so when packets arrive at the mesh node on ei they have to wait for the next transmission of ej . In this case, packets forwarded on ej experience the delay of sj − si + Tf , from the time they were transmitted on ei to the time they are forwarded on ej . We note that the same delay can be observed if ei or ej transmit multiple times in a frame, since 802.16 fragments upper layer packets for multiple transmissions on ei and then the packets have to be fully de-fragmented at the common node, before they can be routed and forwarded on ej . On a path, the total TDMA scheduling delay is given by the sum of the waiting times at each node on the path and the time required to transmit the packet on the last link. On path Pl = (ei , . . . , ej ) the end-to-end delay is X

Dl = dj +

el ,ek ∈Pl :el

where we use the notation el

[sk − sl + Tf I(fl > sk )] ,

(5)

ek

ek to indicate ek follows el on the path and we add dj to account

for the time takes to transfer packets on the last hop. We ignore propagation delay since it is independent of scheduling. IV. 802.16 C ENTRALIZED S CHEDULING A LGORITHMS e1

e3

v1

v2 e2

e5

e8

e6 v3

v4 e7

e4

v5

Fig. 2.

A simple example of a mesh network

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In this section, we briefly review the operation of 802.16 centralized scheduling algorithm proposed in the 802.16 standard [4] and scheduling algorithms proposed in literature [10]–[12]. We also review the operation of the Bellman-Ford TDMA scheduling algorithms [8], [9] and show how these algorithms can be applied to 802.16 mesh networks. Due to space restrictions we do not get into the details of each of the algorithms. We compare the algorithms with simulations in the next section. Throughout this section, we use the topology in Fig. 2 to find 802.16 schedules. We compare the performance of the algorithms on this topology in the next section. In this topology, all nodes request 600kbps uplink and downlink rate from the base-station. We set the 802.16 frame duration to 10ms for a total of 800 Orthogonal Frequency Division Multiplexing (OFDM) symbols in each frame and we set the size of the control sub-frame to 70 OFDM symbols, leaving 730 OFDM symbols for the data sub-frame. We set MSH-CSCH-DATA-FRACTION = 100% so that the entire data sub-frame is used for centralized scheduling. With these parameters, each transmission opportunity consists of 3 OFDM symbols, for a total of 243 transmission opportunities in the data sub-frame. We set the overhead of 802.16 to 3 OFDM symbols, so for every transmission there is one transmission opportunity of overhead. For every transmission in the frame, this overhead is equivalent to 28.8kbps at the lowest modulation and 259kbps at the highest modulation. A. IEEE 802.16 Scheduling Algorithm IEEE 802.16 [4] uses an example to propose a simple algorithm to assign transmission opportunities. This algorithm assigns transmission opportunities during a breadth-first traversal of the routing tree. The first visited link, in the traversal of the tree, is assigned transmission opportunities at the beggining of the data sub-frame. The link traversed next is assigned the next set of transmission opportunities and so on, until all links are assigned transmission opportunities. If the total number of assigned transmission opportunities is larger than the number of transmission opportunities reserved for centralized scheduling, the algorithm scales down and rounds link durations, so they fit in the frame. Since this algorithm does not allow spatial reuse, it has low end-to-end rates. We show the schedule obtained with the 802.16 algorithm in Fig. 3a; as expected none of the link transmissions overlap, since there is no spatial re-use in the schedule.

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B. Graph Colouring Algorithms A graph colouring based 802.16 scheduling algorithm is proposed in [10]. This algorithm assigns transmission opportunities in rounds. In each round, the algorithm assigns one transmission opportunity to the link with the highest number of outstanding transmission opportunities and all other links that do not conflict with it. Those links then have one less outstanding transmission opportunity. Rounds are repeated until either there are no free transmission opportunities left or demands of all links are satisfied. The graph colouring algorithm introduces spatial reuse (Fig. 3b), but it allows the links to transmit multiple times in the frame. We see shortly that despite the spatial re-use, the graph algorithm performs worse than the 802.16 algorithm due the overhead of multiple transmissions. A similar algorithm, the Transmission-Tree scheduling (TTS) algorithm, is proposed in [11]. TTS algorithm also assigns transmission opportunities to non-interfeering links in iterations. However, the TTS algorithm leaves the decision of next assigned link open. For example, instead of choosing the link with the highest number of outstanding transmission opportunities, the algorithm may chose the next assigned link randomly. Since the TTS algorithm is very similar to the graph colouring algorithm, we only examine the performance of the graph colouring algorithm in the next section. C. Load-balancing Algorithm A load-balancing algorithm for 802.16 node scheduling was proposed in [12]. Node scheduling can be used for 802.16 link scheduling if the uplink and downlink traffic are scheduled in their own, non-overlaping, parts of the data sub-frame. For example in Fig. 2, the downlink schedule for nodes v2 , v3 , v4 and v5 corresponds to the schedule for links e1 , e3 , e6 and e4 . For the uplink, node schedule for v2 , v3 , v4 , v5 corresponds to the link schedule for e2 , e5 , e7 and e8 . We modify the node load balancing algorithm [12] to schedule links. The link scheduling version algorithm increases spatial re-use in the network, since it allows the uplink and downlink links to be scheduled simultaneously. The load-balancing algorithm works in iterations. At the begining of each iteration, the algorithm finds links’ satisfaction with the schedule from a previous iteration. The satisfaction is directly proportional to the ratio of the link rate achieved with the schedule and the rate assigned to the link by the base station. For link ej , the satisfaction is sj = rˆj /rj , where rˆj June 21, 2007

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is obtained from the previous schedule with (3) and rj is the required rate on the link. After finding link satisfactions, the algorithm schedules the links. First, the algorithm schedules the link with the smallest satisfaction at the beginning of the data sub-frame. Then, the algorithm schedules the rest of the links in the order of increasing satisfactions. Each link is scheduled as close to the beginning of the data sub-frame as possible, so that its transmission does not overlap with transmissions of its conflicting links. Scheduling links in this way takes advantage of spatial re-use in the network, however links scheduled towards the end of the frame may get a smaller number of transmission opportunities then what they were assigned by the base station (their transmissions are truncated). The links that were truncated get more transmission opportunities in the next iteration of the load-balancing algorithm. In the next iteration, the algorithm recalculates the satisfaction of the links. The links that were truncated in the previous schedule (previous iteration of the algorithm) have the lowest satisfaction, so in the next iteration of the algorithm they are scheduled towards the begining of the frame. Scheduling the links sooner in the frame ensures that their transmissions are longer, making their rate higher than in the previous frame. On the other hand, links that were not truncated in the previous iteration have the highest satisfaction index, so they are scheduled towards the end of the frame where they may be truncated. We show the schedule obtained in the first iteration of the load-balancing algorithm in Fig. 3c and the schedule obtained in the second iteration of the load-balancing algorithm in Fig. 3d. In the first iteration of the algorithm, the transmission of link e5 is truncated and link e8 is not even scheduled. The situation is improved in the next iteration of the algorithm, where link e8 is scheduled to transmit first and link e5 is scheduled to transmit second. They are allocated more transmission opportunities in the schedule, but at the expense of link e2 , which is scheduled last and truncated. We note that e7 is scheduled at the beginning of the frame in both iterations since in either iteration it does not interfere the links scheduled at the beginning of the frame. D. Bellman-Ford Scheduling Algorithm We propose the Bellman-Ford TDMA scheduling algorithm in [8], [9]. Bellman-Ford scheduling allows spatial re-use to increase the end-to-end rates in the mesh backbone, limits the number of link transmissions to one per frame to decrease overhead and also takes into account TDMA scheduling delay. We show in [8], [9] that TDMA scheduling delay is directly related to the June 21, 2007

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transmission order of link in the frame. This can also be seen from (5), where a frame duration Tf is added to the delay whenever a link transmits before a link traversed before it on the same path. We also show in [8], [9] that the transmission order is related to end-to-end throughput and provide a search method that finds transmission orders that are both throughput and delay optimal. Finding a throughput optimal schedule with the smallest TDMA scheduling delay is a computationally hard problem. However in [8], [9], we show how to decompose the problem into two parts: finding a good transmission order and then finding a throughput optimal schedule for this transmission order. Using the decomposition, we propose a polynomial time TDMA scheduling algorithm, with two steps [8], [9]. First, the algorithm finds a link transmission order with a TDMA scheduling delay of one frame for all paths in the network. The algorithm achieves this by traversing all paths and setting the transmission order of links further down the path after the transmission order of the links earlier in the path. In effect the algorithm is setting the Tf terms in (5) to 0. We prove in [8] that this transmission order limits the TDMA scheduling delay on all paths to one frame. Second, the algorithm finds a transmission schedule with the Bellman-Ford algorithm running on the conflict graph for the network. The conflict graph has links as vertices and conflicts between the links as arcs and links conflict if they cannot transmit at the same time. The link transmission times are calculated from the minimum distances in the conflict graph for the network. We show in [8], [9] that this schedule is throughput optimal for the transmission order. We note that since we prove the optimality of the Bellman-Ford algorithm in [8], [9], we expect this algorithm to perform better than other algorithms. The other algorithms are based on heuristics and no results are available about their optimality. The scheduling decomposition in [8], [9] fits perfectly into the 802.16 centralized scheduling protocol. First, the base station finds a transmission order either with the exhaustive search or with the one frame TDMA scheduling delay heuristic. The base station then sends the transmission order together with the assigned end-to-end rates. With the transmission order, the mesh nodes can find the schedule in polynomial time. We show in [8], [9] that due to the decomposition property of the Bellman-Ford scheduling there is no loss of optimality with this procedure. We show the schedule obtained with the optimum version of the Bellman-Ford algorithm in Fig. 3e and the schedule obtained with the one frame delay transmission order in Fig. 3d. June 21, 2007

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The optimum schedule is found with an exhaustive search for a transmission order with the highest end-to-end throughput, while in the schedule with one frame delay transmission only one order is considered. In this topology the two schedules are identical. We see later that one of the drawbacks of the one frame delay transmission order scheduling is that spatial re-use is not allowed between links on the same path. However, if there are more distinct paths in the network, there is more spatial re-use (as in the topology in Fig. 2) V. S IMULATION R ESULTS In this section, we compare the performance of scheduling algorithms [8]–[10], [12] from the previous section with simulations. We start with the simple topology in Fig. 4. We then simulate the performance of the algorithms on chain and grid topologies. In all simulations, we use the 802.16 parameters from the last section. A. Example 1: Small Topology We we now compare the performance of schedules in Fig. 3 terms of end-to-end throughout and TDMA scheduling delay in Fig. 4. We find the uplink and downlink throughput of the schedules with (4) and (3) and we find the end-to-end TDMA scheduling delay with (5). We do not take into account the overhead of upper layer protocols. Fig. 4a and Fig. 4b show the uplink and downlink throughput of each source. The graph colouring algorithm has the lowest end-to-end throughput (“Graph Colouring” in Fig. 4a and Fig. 4b). This may seem surprising since that algorithm uses spatial re-use to increase the capacity of the network, however this algorithm introduces a large amount of overhead, which decreases the end-to-end throughput. The 802.16 algorithm has the next higher end-to-end throughput, despite the fact that it is not using any spatial re-use (“IEEE 802.16” in Fig. 4a and Fig. 4b). The load-balancing provides different end-to-end throughput in each iteration. In the first iteration, the effect of link e8 having no transmission opportunities (Fig. 3c) manifests as source v5 has 0 uplink throughput (“Load Balancing, Integration 1” Fig. 4a). The situation is improved in the second iteration, where source v5 has some uplink throughput (“Load Balancing, Integration 2” in Fig. 4b). However, the uplink throughput of all sources is decreased in the second iteration because link e2 is truncated (Fig. 3d) and link e2 is the bottleneck for all uplink traffic flows

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(Fig. 2). The end-to-end throughput is the best for two Bellman-Ford algorithms (“BellmanFord (Optimum)” and “Bellman-Ford (MinDelay)”). This is expected since their performance is throughput optimal [8], [9]. We note that the only algorithms exhibiting “fairness” are the 802.16 algorithm and the two Bellman-Ford algorithms. For these algorithms, the end-to-end throughput is approximately the same for all nodes. On the other hand, the graph colouring algorithm is not fair. For example, the graph colouring algorithm allocates twice as much throughput to node v2 on the uplink than it does to node v5 . From iteration to iteration, the load-balancing algorithm can be even worse. For example, source v5 , which has 0 uplink throughput in the first iteration of the loadbalancing algorithm. However, the average of the load-balancing algorithm over 50 iterations (“Load Balancing, 50 Iterations” in Fig. 4a and Fig. 4b) shows that while the load-balancing is “unfair” of the short-term, the algorithm is more “fair” over the long term, since all nodes get about the same amount throughput over the 50 iterations of the algorithm. However, in this topology the algorithm favours uplink traffic over the downlink traffic. We compare the TDMA scheduling delay on the return paths in the network in Fig. 4c. 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 [16]. The lowest TDMA scheduling delay (10ms for all nodes) is obtained by the two Bellman-Ford algorithms. This is not surprising since both algorithms minimize the TDMA scheduling delay. The other algorithms have higher TDMA scheduling delay. The delay is the highest for source 4, which is furthest away from the base station. For example, in the load-balancing algorithm source 4 experiences the TDMA scheduling delay of 40ms, which is four times as much as the delay of the two Bellman-Ford algorithms. We note that if the frame size was increased to 20ms the TDMA scheduling delay would be 40ms since it is directly proportional to frame duration Tf . B. Example 2: Chains In this section, we examine performance of scheduling algorithms on chain topologies. In each scenario, we create a chain of mesh nodes where one of the end nodes is the base-station and all other nodes in the topology request uplink and downlink end-to-end rates of 3.0Mbps. We chose 3.0Mbps as the requested rate because, with our physical layer parameters, this is close to the maximum possible with just two nodes in the chain. June 21, 2007

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Fig. 5a and Fig. 5b show the performance of different algorithms in terms of uplink/downlink end-to-end throughput, as the size of the chain – the number of sources – increases (the naming is the same as in Fig. 4). We show the average end-to-end throughput among all nodes in the network. The optimum Bellman-Ford algorithm has the best performance in terms of end-toend throughput. The one frame TDMA scheduling delay Bellman-Ford algorithm the 802.16 scheduling algorithm have the same, lower achievable bandwidth. The reason for this is that when the end-to-end TDMA scheduling delay is limited to on frame, there is no spatial re-use among links on the same path. However, in general the transmission orders found with this transmission order do allow spatial re-use among links on the different paths (Fig. 4). Despite taking advantage of spatial re-use, the graph colouring algorithm has the worst performance due to multiple transmissions in the frame. Even with just one source, when it is possible to schedule all requested end-to-end bandwidths, the colouring algorithm only achieves 75% of the requested end-to-end bandwidths. The load-balancing algorithm performs worse than in the previous scenario. The reason for decreased performance is that the amount of bandwidth requested on the links exceeds the amount of bandwidth that can be scheduled in one frame. Only a few links transmit in each schedule, in effect cancelling out the benefits of spatial re-use. We show how this happens in Fig. 6, where we plot two iterations of the load-balancing schedule for the three node chain (Fig. 6a and Fig. 6b). The load-balancing schedule always oscillates between these two schedules. In both schedules link e2 is scheduled for about 100 OFDM symbols, adding up to less than 200 OFDM symbols over the two frames. On the other hand, link e1 is scheduled for over 500 slots over the two frames, giving it twice the bandwidth. So, nodes v2 and v3 have half as much uplink bandwidth than downlink bandwidth since link e2 is their bottleneck link (Fig. 6c). This observation also explains the variability in the end-to-end throughput of the load-balancing algorithm. Fig. 5c shows the maximum delay as the chain length increases. The one frame TDMA scheduling delay Bellman-Ford algorithm has a constant one frame TDMA delay, consistent with the claims in [8], [9]. The bandwidth optimum Bellman-Ford algorithm has higher delay than the one frame TDMA scheduling delay Bellman-Ford algorithm. This is expected, since the bandwidth optimum Bellman-Ford algorithm optimizes the schedule for bandwidth and delay at the same time. The 802.16 algorithm introduces an even larger delay because it ranks the links with the breadth-first search of the routing tree. In case of chains, the breadth first search June 21, 2007

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increases TDMA scheduling delay because it forces all the link on the uplink to transmit before the links on the downlink. The graph colouring approach has the largest delay, approximately twice as large as the delay of the optimum Bellman-Ford algorithm. We do not show the delay for the load-balancing algorithm since when some of the links do not transmit in a frame (Fig. 6), the formula for end-to-end scheduling delay does not apply anymore. C. Example 2: 5 × 3 Grid In this section we examine performance of scheduling algorithms on a 5 × 3 grid. We place the base-station in the center 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 3.0Mbps. We repeat the selection of sources 50 times. We plot the results for end-to-end throughput and end-to-end delay in Fig. 5. We do not run the optimum Bellman-Ford scheduling algorithm in our examples because it takes a very long time. We note again that despite spatial re-use, the graph colouring algorithm achieves at most as much bandwidth as the 802.16 scheduling algorithm that does not use spatial reuse. The delay of the graph colouring algorithm is better than the delay of the 802.16 algorithm, however it is still significantly larger than the delay of the Bellman-Ford algorithm. The load-balancing appears to work better than the Bellman-Ford algorithm, sometimes on the uplink. However, the short term behaviour of the algorithm is such that it often starves out links for one or more frames. This produces extremelly large delays in the network, making the algorithm unusable in 802.16 mesh networks. VI. C ONCLUSION We have compared 802.16 centralized scheduling algorithms [8]–[12] with a realistic comparative analysis in terms of end-to-end delay and end-to-end rates. We have taken into account 802.16 mesh protocol overhead when evaluating rates, and shown that allowing multiple transmissions in each frame has a detrimental impact on the performance of 802.16 mesh networks. We show that, despite taking advantage of spatial re-use, graph colouring algorithms, such as [12] have lower end-to-end throughput than the 802.16 algorithm which has no spatial re-use, due to the overhead of multiple transmissions. This paper is the first to take into TDMA scheduling delay. We show if TDMA scheduling delay is not included during scheduling [10], [12], resulting June 21, 2007

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schedules introduce huge end-to-end TDMA scheduling delays, which impacts upper network layers. The algorithms with by far the best performance [8], [9] take both overhead and TDMA scheduling delay into account. R EFERENCES [1] J. Camp, J. Robinson, C. Steger, and E. Knightly, “Measurement driven deployment of a two-tier urban mesh access network,” Rice University, Technical Report TREE0505, December 2005. [2] Nortel Networks, “Wireless mesh network - extending the reach of wireless LAN, securely and cost-effectively,” http://www.nortelnetworks.com/solutions/wlan/, November, 2003. [3] S. Xu and T. Saadawi, “Does the IEEE 802.11 MAC protocol work well in multihop wireless ad hoc networks,” IEEE Communications Magazine, vol. 39, no. 6, pp. 130–137, June 2001. [4] “IEEE standard for local and metropolitan area networks part 16: Air interface for fixed broadband wireless access systems,” 2004. [5] B. Hajek and G. Sasaki, “Link scheduling in polynomial time,” IEEE Transactions on Information Theory, vol. 34, no. 5, pp. 910–917, September 1988. [6] T. Salonidis and L. Tassiulas, “Distributed dynamic scheduling for end-to-end rate guarantees in wireless ad hoc networks,” in Proceedings of MobiHoc, 2005, pp. 145–156. [7] M. Kodialam and T. Nandagopal, “Characterizing achievable rates in multi-hop wireless mesh networks with orthogonal channels,” IEEE/ACM Transactions on Networking, vol. 13, no. 4, pp. 868–880, 2005. [8] P. Djukic and S. Valaee, “Link scheduling for minimum delay in spatial re-use TDMA,” in Proceedings of INFOCOM, 2007. [9] ——, “TDMA delay aware link scheduling for multi-hop wireless networks,” Submitted to IEEE Transactions on Networking, May 2007. [10] H.-Y. Wei, S. Ganguly, R. Izmailov, and Z. Haas, “Interference-aware IEEE 802.16 WiMax mesh networks,” in VTC Spring’05, 2005. [11] B. Han, W. Jia, and L. Lin, “Performance evaluation of scheduling in IEEE 802.16 based wireless mesh networks,” Computer Communications, vol. 30, pp. 782–792, 2007. [12] D. Kim and A. Ganz, “Fair and efficient multihop scheduling algorithm for IEEE 802.16 BWA systems,” in Broadnets, 2005, pp. 895–901. [13] P. Djukic and S. Valaee, “Scheduling algorithms for 802.16 mesh networks,” in Wimax/MobileFi: Advanced Research and Technology, Y. Xiao, Ed. Auerbach Publications, CRC Press, 2007. [14] ——, “802.16 mesh networking,” in Handbook of Wimax, S. Ahson and M. Ilyas, Eds. CRC Press, 2007. [15] S. Keshav, An Engineering Approach to Computer Networking.

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Stochastic Processes, 1997. [16] J. Padhye, V. Firoiu, D. F. Towsley, and J. F. Kurose, “Modeling TCP reno performance: A simple model and its empirical validation,” IEEE/ACM Transactions on Networking, vol. 8, no. 2, pp. 133–145, April 2000.

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