Assignment Methods for Spatial Reuse TDMA

Jimmi Gronkvist Swedish Defence Research Establishment

Div. of Command & Control Warfare Tech. SE-58 1 11 Linkoping, Sweden

Abstract- Spatial reuse TDMA is an access scheme for multi-hop radio networks. The idea is to increase capac- ity by letting several radio terminals use the same time slot when possible. A time slot can be shared when the radio units are geographically separated such that small inter- ference is obtained. STDMA schedules can assign trans- mission rights to nodes or alternatively assign transmis- sion rights to links, i.e. transmitter/receiver pairs. Here we compare these two methods and determine which one is preferable. We show that only the connectivity of the network and the input traffic load of the network is needed in order to determine whether node or link assignment is preferable.

I. INTRODUCTION

We consider a radio network where a number of radio units are spread out in some terrain and where the mobility demands are moderate. If the received signal power is sufficient in re- lation to noise and interferences, it is assumed that any two radio units can communicate, i.e., establish a link. In a multi- hop network, the power consumption can be kept low and area coverage is achieved by letting messages be relayed over one or several intermediate nodes.

One problem in a radio network is the interferences caused by simultaneously transmitting nodes. These conflicts occur if the received signal is too weak in comparison with the in- terfering signals, An important issue is therefore to design access schemes that control the use of the channel. One used access scheme is time division multiple access (TDMA). For sparsely connected networks, however, this is usually ineffi- cient. To increase capacity one can instead use spatial reuse TDMA (STDMA), which is an extension of TDMA where the capacity is increased by letting several radio units share the same slot. An STDMA schedule describes the transmission rights for each time slot. In ad hoc networks with moder- ate mobility demands updates of an STDMA schedule can be managed [ 13.

Channel access STDMA for multi-hop packet radio net- works was formalized in [2]. A number of algorithms to gen- erate STDMA schedules have been proposed. Some of these algorithms assign transmission rights to nodes, i.e. a node can transmit to any of its neighbors. In other algorithms, transmis-

sion rights are assigned to links, in this case both transmitter and receiver node is determined in advance.

Example of node assignment algorithms can be found in [3, 41, and link assignment algorithms in [5]. Sometimes both versions of the algorithm have been proposed, node assigned [4] and link assigned [5] Furthermore, in [6] both link and node assignment algorithms are described. However, it has not been shown which approach that is preferable.

The main focus here is to determine for what situations link or node assignment is preferable. Our result suggests that the connectivity of the network and the input traffic are sufficient parameters for determining which approach that is preferable. Furthermore, link assignment achieves higher throughput than node assignment and the gain in throughput increases with the size of the network and decreases with increased connectivity.

We focus on point-to-point traffic. The question whether node or link assignment should be used is interesting essen- tially for point-to-point traffic. For broadcast or multicast traffic node assignment is intuatively preferable by common sense.

The node and link algorithms used are described in [7] and [8], respectively. Here we give a short description of these algorithms. The only differens between these algorithms is that they assign transmission rights to nodes and links, respec- tively. We use an interference based model of the network as suggested for STDMA scheduling in [9].

11. NETWORK MODEL

Here we describe the interference based model of a radio net- work that is used. The network is represented by a set of nodes V and the basic path-loss &(i , j ) between any two dis- tinct nodes (vi, wj), i # j . To simplify notation, we assume isotropic antennas.

For any two nodes, (vi, V j ) where vi is the transmitting node and vj # wi, we define the signal to noise ratio (SNR), rij, as follows

D

where Pi denotes the power of the transmitting node V i ,

Lb( i , j ) is the basic transmission path-loss between nodes vi and vj, and NR is the receiver noise. For convenience, we de- fine I'ii = 0 corresponding to the physical situation of a node not being able to transmit to itself.

0-7803-6534-8/00/$10.00 0 2000 IEEE

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We say that a pair of nodes (vi,vj) form a link, ( i , j ) , if the signal to noise ratio (SNR) is not less than a threshold, 70. That is, the set of links, IC, is defined:

K = {(i,j) : ru 2 To) . (2)

For a set of links, K C K, we define the set of transmitting nodes as follows:

VT(K) = {vi : (i,j) E K } . For any link, (2, j ) E K , we define the inteflerence:

Furthermore we define the signal to inteference ratio (SIR):

D.

(3)

We assume that any two radio units can communicate a. packet without error if the SIR is not less than a threshold, 71. A schedule S is defined as the sets vt, for t = 1,2, ..., T , where T is the period of the schedule. The sets vt contain the nodes or links assigned time slot t . A schedule is considered to be conjlict free if equation ( 3 ) holds for all scheduled trans- missions.

Furthermore, we assume that a node cannot transmit more than one packet in a time slot and that a node cannot receive and transmit simultaneously in a time slot.

The Signal-to-interference criteria (3) gives the following

( 5 ) condition

I I K ( i , j ) 2 71 v ( i l j ) E K. If the above two conditions, (4) and (5 ) , hold for a set of

links K E IC, we say that the links in K can transmit simulta- neously.

Similarly we will state two necessary conditions for the sit- uation when all the nodes in a set V are allowed to transmit packages simultaneously. Let the neighbors Q(v) to a node v in V be the set of all nodes that have a link from v to itself. The neighbors are the nodes that v possibly can transmit a packet to. Similarly, let Q(V) denote the union of all neighbors of all nodes in V .

Thefirst condition is that two neighbors can not transmit at the same time. Another way to say this is that the sets V and Q(V) must be disjoint:

V n Q(v) = 0. (6)

Let K ( V ) be the set of all links from the nodes in V to their neighbors in O(V) . Since all the links in K(V) must be possi- ble to use for transmission simultaneously, we state the second condition:

b q V ) ( i r j ) 2 71 for all (2, j ) E K ( V ) . (7)

If the above two conditions, (6) and (7), hold for a set of nodes V E V we say that the set of nodes can transmit simul- taneously.

IV. TRAFFIC CONTROLLED REUSE SCHEDULES 111. ASSIGNMENT METHODS

In a node assigned schedule, a node is allowed to transmit to any of its neighbors in its slot. If the schedule is to be conflict free this means that we have to guarantee that we will not have a conflict in any of the neighboring nodes. In link oriented assignment, the directed link is assigned a slot. A node can then only use this slot or transmission to a specific neighbor. In general this knowledge can be used to achieve a higher degree of spatial reuse. The effect is higher throughput.

In the following, we first describe the criterions needed for a set of links to be able to transmit simultaneously with suffi- ciently low interference level at the receiving nodes. Then, we do the same for a set of nodes.

We say that a link ( k , 1 ) is adjacent to link ( i , j ) E K iff {i, j } n {k, I} # 0. Furthermore we define Q(K) as the union of all adjacent links to the links in K . We assume that a node cannot transmit more than one packet in a time slot and that a node cannot receive and transmit simultaneously in a time slot. Alternatively, we say that a set of links K and the set of its adjacent links Q ( K ) must be disjoint:

K n Q ( K ) = 0. (4)

The relaying of traffic in multihop networks causes a consider- able variation of the traffic load on the links or nodes in a net- work. To achieve large capacities we have to use an efficient traffic controlled schedule to compensate for this problem.

In a traffic controlled schedule, links or nodes can use sev- eral slots, see [lo], according to the traffic load. We define hij as the number of slots allocated to link (i, j) within a frame in a schedule. The corresponding notation for node assignment is hi.

Note that the problem with varying traffic loads may be less severe in a node assigned schedule, since the variation of traf- fic is averaged over a node.

In our traffic model we assume point to point traffic, i.e., a packet entering the network has only one destination. Pack- ets enter the network at entry nodes according to a proba- bility function, p(v),v E V and packets exit the network at exit nodes. When a packet enters the network it has a desti- nation, i.e., an exit node from the network. The destination of a packet is modeled as a conditional probability function, q(ur(v), (tu, U) E V x V , i.e., given that a packet has entry node v, the probability that the packet's destination is w is q(z0lv). For simplicity we will assume a uniform traffic model,

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i.e., p ( v ) = 1 / N , and q(wlv) = 1 / ( N - 1 ) where N is the number of nodes, N = IV1.

Let X be the total traffic load of the network, i.e., the av- erage number of packets per time slot arriving to the network as a whole. Then, X/N(N - 1 ) is the total average of traffic load entering the network in node vi with destination node vj . However, as the network is not necessarily fully connected, some packets must be relayed by other nodes. In such a case, the traffic load on each link can be calculated first when the traffic has been routed.

Now, let TR denote the routing table, where the list entry TR(v, w) at U, w is a path from entry node v to exit node w. Let the number of paths in TR containing the directed link (i, j ) be equal to Aij .

Further, let X i j be the average traffic load on link ( i , j ) . Then X i j is given by

For node assignment we have, Xi as the average traffic load on. node vi. Where Xi is given by:

and where Aij and Ai is the relative traffic of a link and a node, respectively.

It is difficult to find a schedule S which achieves the highest maximum throughput of all schedules. Instead we will use the algorithm described in [7]. This algorithm compensates for the traffic variations, by assigning hi = Ai.

The schedule generated by the algorithm described in [7] is node assigned. To compare it with a link assignment schedule, we modify the algorithm described in [8], such that the guar- anteed number of time slots will be hij = Aij . The only differ- ence between these algorithms is then the assignment strategy.

The algorithms used in the comparison totally compensate the traffic variations. A comparison with algorithms without full compensation will differ from our results.This is due to the fact that traffic varies more over links than over nodes. Link assigned schedules are therefore more sensitive to insuf- ficient traffic compensation. However, efficient schedules that achieve high throughput must have a good traffic compensa- tion.

Furthermore, the algorithms use a fixed transmission power level rather than a varying power level optimized to achieve maximum throughput or minimal delay. Efficient power con- trol is more difficult to use in a node assigned schedule, since a node has to assume worst case power level at the other trans- mitting nodes.

A priority scheme is used to spread the time slots of nodes or links evenly over the frame. Because of this, the resulting schedule does not have increased average delay due to the traffic control.

The algorithm used is a greedy algorithm.

7

Figure 1: Average packet delay in a 30 node network. Dashed curve for node assignment and solid curve for link assignment.

In short the algorithm works as follows, choose the node with highest priority which has not yet been checked in the time slot. Assign it to the time slot if possible. If all nodes with time slots left have been checked, continue to the next slot. Proceed until all nodes have been scheduled their guaranteed time slots.

v. EVALUATION METHOD

We evaluate the average packet delay and the maximum throughput of a network. Packet delay is the time, in time slots, from the arrival of a packet at the buffer of the arrival node v k to the arrival of the packet to the destination node vl. This parameter has been estimated using simulations.

Maximum throughput is the largest network input traffic X giving bounded average packet delay.

The maximum throughput X i for a link assigned schedule S can be written as [ 113

where TL is the length of the link assigned schedule. The corresponding result for node assignment can be writ-

ten as [ 111 N ( N - 1)hi

X;V = min (i) TNAi ' (9)

where TN is the length of the node assigned schedule. In figure 1, average packet delay for different X is shown

for a network of 30 nodes. In this figure three areas of interest can be seen. At low traffic, the fraction of delay between node assignment and the delay of link assignment. At high traffic loads, the fraction between the throughput of node assignment

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Connectivity

Figure 2: The figure shows the relation between maximum through- put for link assignment and node assignment for networks of different connectivity. The relation is plotted for 400 networks of size 20 nodes.

and the throughput of link assignment. Finally the intersec- tion point between the curves, giving the X when the delay is equal. For traffic loads above this intersection link assignment is preferable and for traffic loads below this intersection node assignment is preferable.

The above equations, (8) and (9), have been used in sim- . ulated networks of sizes 5 , 10, 20, and 60 nodes. For each

of these network sizes, 400 networks have been generated with different connectivity, by varying the transmitting power. However, all nodes in a network use equal transmitting power, i.e., the transmitting power is a constant, P. We define con- nectivity as the fraction of the network that can be reached by a node, in one hop, in average, i.e., M / ( N ( N - l)), where M is the number of directed links in the network.

To generate realistic networks, a terrain-data based ground wave propagation model, Vogler's five knife-edge model, has been used for the calculation of the basic transmission path- loss, see [ 121 for more details.

In the simulations, these additional assumptions have been made:

Shortest route, i.e., packets sent between two nodes will al- ways use the path which requires the least number of transmis- sions. If several routes of the same length exists, all packets between two specific nodes will always use the same route.

When estimating packet delay, packets are generated by a pseudo Poisson process with intensity X and with a uniform traffic distribution. Since the schedule is collision free, all packets are assumed to be perfectly received, and no retrans- missions are considered.

1.8

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1.5-

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b.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Connectivity

Figure 3: The figure shows the average relation between maxi- mum throughput for link assignment and node assignment for networks of different connectivity. Relation between throughput for networks of different size 5,10,20, and 60 nodes. 5 nodes at the bottom and 60 nodes at the top.

VI. NODE OR LINK ASSIGNMENT?

The first parameter we study is the relation between maximum throughput XL of link assignment and maximum throughput X;V of node assignment, i.e.,

according to equation (8) and (9). As can be seen in figure 2, that this relation exhibits consid-

erable variations over the networks studied. One conclusion is that link assignment provides higher throughput. This is not so surprising since the degree of spatial reuse is higher for link assignment.

To determine how much better link assignment can be, we plot in figure 3, the ratio XE/X;V averaged over connectivity for networks of different sizes. As can be seen X;/X;V in- creases with the size of the network and decreases if connec- tivity is increased.

The second parameter studied is the relation between packet delay of link assignment and node assignment at low traffic loads. In figure 4 this parameter can be studied. The vari- ance is rather low and a linear relationship between delay and connectivity can be detected. This linear relationship can be explained as follows. Assume that the various slots assigned to a node are perfectly distributed over the schedule. The aver- age message delay through a node can then, at low traffic rates, be approximated as T~/2hi, where TN is the total number of . time slots in the schedule. This is an approximation since re- lay traffic does not arrive as a Poisson process. The average packet delay with node assignment can then be approximated

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QZ 8 - d

6 -

4 -

2-

I w*: t

Connectivity

Figure 4: The figure shows the relation between average packet delay for link assignment and average packet delay for node as- signment. The relation is plotted for 400 networks of size 20 nodes.

as TN N

2 N ( N - 1) ’ and for link assignment as T L M / ~ N ( N - 1). The relation between delays can then be written as M / N * TL/TN, which increases linearly with connectivity.

The last parameter studied is the input traffic load of the network which gives equal packet delay for node and link as- signment. This parameter is interesting since it determines for what traffic loads linWnode assignment is preferable. That is, for traffic loads higher than this parameter link assignment is preferable and at lower traffic loads node assignment is prefer- able.

As can be seen in figure 5 the variance over the simulated networks, of this parameter is less than for the other parame- ters, which means that the average value is highly interesting. This average is shown in figure 6.

Throughput and delay are of course not the only important parameters. It is interesting to note that link assignment can achieve lower power consumption since only the proper re- ceiver of a packet will try to receive it, the rest of the neighbors saving the power required to determine if they are the target of the packet.

A link assignment algorithm can also more efficiently take advantage of power control and directional antennas.

VII. CONCLUSIONS

The preferable assignment method can be determined with knowledge only of the connectivity of, and input traffic to, the

1

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0.7 -

0.6 -

4 0.5-

0.4 - 0.3 -

$1 012 0:3 0:4 0:5 016 017 0:8 019 Connectivity

Figure 5: The figure shows the input traffic level giving equal packet delay for different network connectivity. This is plotted for 400 networks of size 20 nodes.

network. For high traffic loads, link assignment is better than node

assignment due to higher reuse efficiency. However, for low traffic loads node assignment is preferable due to lower aver- age time between transmissions.

Furthermore, the intersection point, i.e. the traffic load XI when both methods achieve the same delay, increases with in- creasing connectivity of the network, according to figure 6. For traffic loads above this intersection, link assignment is preferable and for traffic loads below this intersection, node assignment is preferable.

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:::I , , , , , , , ,

8.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Connectivity

Figure 6: The traffic load which gives equal delay for 400 networks of sizes 10, 20, and 60 nodes. 10 nodes at the bottom and 60 nodes at the top.

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