2005 Conference on Information Sciences and Systems, The Johns Hopkins University, March 16–18, 2005

Maximum throughput scheduling in time-varying-topology wireless ad-hoc networks

Anna Pantelidou, Anthony Ephremides, Andre TitsInstitute for Systems Research

Department of Electrical and Computer EngineeringUniversity of Maryland College Park, MD 20742

e-mail:{apantel, etony, andre }@umd.edu

Abstract — In wireless ad-hoc networks, signal interference andcollisions from simultaneous transmissions by neighboring nodessignificantly degrade throughput. Hence, when throughput is of theessence, it is necessary to devise scheduling policies for coordinatingthe wireless transmissions. In this paper, we focus on the problemof scheduling the transmissions of a finite number of nodes in a mo-bile ad-hoc wireless network. As a consequence of node mobility,the topology varies with time. This is modeled as an i.i.d topol-ogy process. Customers of different classes enter the network andmay be serviced differently, according to their class requirements.We propose a family of scheduling policies all of which maximizethe total stable throughput of the network, while taking into ac-count the different class requirements. Policies within this familyare parameterized by the preferential weights assigned to the re-spective customer classes. Finally, we report results obtained by aMonte Carlo simulation that confirm that mobility may consider-ably improve the network throughput under our proposed schedul-ing scheme.Keywords: wireless networks, scheduling, maximum throughput

I. I NTRODUCTION

One of the fundamental characteristics of wireless ad-hoc networksis the fact that the wireless medium is shared by multiple static or mobilenodes. As simultaneous transmissions by neighboring nodes interferewith each other and the transmitted signals get corrupted, the throughputof the network degrades significantly.

For this reason, medium access control (MAC) schemes coordinatethe access of the various wireless nodes to the medium. Two ma-jor medium access control schemes have been widely used. On theone hand, contention based schemes attempt to resolve collisions whenmultiple nodes compete for access to the medium. On the other hand,scheduling rules prevent collisions by restricting medium access to non-interfering nodes.

In dynamic environments, for example when the wireless nodes aremoving according to some mobility pattern, scheduling the wirelesstransmissions is a challenging issue. Mobility affects the link availabil-ity at each time instant and the set of topologies the network may evolveinto during the course of time. Since the above information is necessaryto the scheduler, the scheduling decision process gets perplexed.

In our work, we consider scheduling in mobile wireless ad-hoc net-works. Our objective is to find a centralized scheduling rule that maxi-mizes the total stable throughput of the network. Our work is inspiredby the work of [3] and [5] and generalizes their results. In [3], theauthors focus on scheduling in a finite node, static multi-hop wirelessad-hoc network. A centralized, optimal stable throughput stationaryscheduling policy is introduced and its corresponding stability regionis characterized. The authors show that non-stationary scheduling rules

do not achieve a better performance in terms of throughput, and con-clude that it suffices to restrict attention to the set of stationary policies.In [5], Tassiulas addresses the problem of scheduling in a finite node,time-varying-topology ad-hoc network. However, his model does notallow multi-hop forwarding of traffic.

In this paper, we generalize the results of [3] and [5] and identifya family of optimal centralized stationary scheduling policies that allmaximize the network throughput inmulti-hop time-varying-topologynetworks. Policies within this family are parameterized by the prefer-ential weight assigned to the respective customer classes.

This paper is organized as follows. In section II we introduce ourmodel. Then, in section III we present the proposed family of central-ized scheduling policies. In section IV we give our optimality results.Then, in section V we confirm our analytical results through a MonteCarlo simulation. Finally, we conclude the paper in section VI.

II. M ODEL

We consider a slotted time, time-varying-topology wireless, ad-hocnetwork, operating under a TDMA based medium access scheme. Thenetwork is comprised of a finite numberN of nodes, with mobile capa-bility, each equipped with one transceiver (transmitter/receiver pair) andan omni-directional antenna. Nodes share a common medium, hence itis essential to properly coordinate their transmissions to avoid signalinterference and collisions from simultaneously transmitting neighbor-ing nodes. Towards this end, a set ofprimary constraints is imposedon the set of simultaneously transmitting neighbors. These constraintsdictate that:

1. A node cannot transmit and receive simultaneously.

2. A node cannot transmit simultaneously to multiple nodes.

3. A node cannot receive simultaneously from multiple nodes.

Nodes exchange datagram packets of constant packet lengths, thatcan be transmitted in one time slot, in a unicast fashion. We considerJdistinct customer classes, each intended for a set ofexit nodesVj , j =1, . . . , J . The set of exit nodes is such that whenever a packet of someclass reaches an exit node for this class, the packet leaves the network.The different setsVj are allowed to overlap.

Each node is modeled as a set ofJ infinite buffer queues, each hold-ing separately the packets corresponding to different customer classes.We denote byXij(t) the non-negative integer queue size for classj atnodei at the endof time slot t. In addition,X(t) is a queue lengthmatrix defined asX(t) = (Xij(t), i = 1, 2, . . . N, j = 1, 2, . . . J) andfor eachj ∈ {1, 2, . . . J}Xj(t) is aN × 1 vector of all queue sizes ofclassj at timet, that is,Xj(t) = (Xij(t), i = 1, 2, . . . N). We denotebyX the space of all queue size vectors.

A link ` is said to exist between nodesi1 andi2, if nodei2 is withintransmission range of nodei1. We model each link as a server. Inparticular, a link that originates from nodes(`) and terminates at noded(`) is a server that serves a customer from queues(`) and upon service

completion, sends the served customer (packet) to the destination queued(`). All the servers are synchronized to start serving a customer at thebeginning of a time slot. The primary constraints on medium accessmake the servers interdependent, in the sense that not all of the serverscan be active at any time slot.

Node mobility causes some already established links to break, asparticipating nodes move far away from each other, and results in estab-lishing new connections between nodes that move closer to each other.Hence, the mobility pattern of the nodes affects the set of possiblenet-work topologies, as well as their respective probability distribution.Finiteness of the number of nodes implies finiteness of the setT oftopologies the network may evolve into, i.e.T = {T1, . . . , TNT }.We denote the topology at time slott by T (t) ∈ T . Each topologyTk is characterized by the set of linksL(k) that are present underTk.Let L be the set of all links that exist in at least one topology, i.e.L = ∪NT

k=1L(k). A uniform numbering of the links is used across alltopologies, i.e. if link ∈ {1, . . . , |L|} connects nodesi1 andi2 undertopologyk, then in every topology where nodei1 is connected toi2, thelink connecting them is numbered`.

Customers of each class may enter the network at any node, exceptfor the exit nodes of the corresponding class, and at each time instant.For each customer classj, the vector of arrivalsAj(t) = (Aij(t) : i =1, 2, . . . , N, i /∈ Vj) is a component-wise non-negative vector. Itsith

element,Aij(t), represents the number of customers of classj arrivingat queuei, during time slott. For all queuesi : i = 1, . . . , N and allcustomer classesj : j = 1, . . . , J , we defineaij = E[Aij(t)], that isassumed to be time-invariant, and we termmulti-class arrival vectorthe non-negative component-wise matrixa = (aij : i = 1, . . . , N, j =1, . . . , J).

To capture the dependence among link activations, due to themedium access constraints, following [3] and [5], we define a validac-tivation set for topologyTk to be a set of servers (a subset ofL(k)),that comply with the primary constraints and are allowed to be activatedsimultaneously. We also define the validactivation vector associatedto an activation set as an|L| element vector with its th component,1 ≤ ` ≤ |L|, set to1 if the `th server belongs to the activation setand set to0 otherwise. If link` is not present inTk ∈ T or mediumaccess constraints prevent it from being activated, then the`th elementof all the valid activation vectors for topologyTk must be set to0. Theconstraint setSk for topologyTk ∈ T is the set of all valid activationvectors associated with network topologyTk. Since the setSk is deter-mined by the primary constraints, it should be clear that for eachk, ifc ∈ Sk, all the vectors obtained by setting some of the active links ofcto inactive, belong inSk as well.

Next, we define the binary random variableE`j(t) ∈ {0, 1}, thatindicates whether server` is active during slott, serving a customer ofclassj. The random variableE(t) = {E`j(t) : ` = 1, . . . , |L|, j =1, . . . , J} is called a validmulti-class activation vector for topologyT (t), if the corresponding vectorsEj(t) = {E`j(t), ` = 1, . . . , |L|}are such that:

1. E`j(t) = 0, if Xs(`)j(t− 1) = 0.

2. E`j(t) = 0,∀` /∈ T (t).

3.∑J

j=1 Ej(t) ∈ Sk, whereT (t) = Tk.

We denote byE the collection of all valid multi-class activation vectorsthat correspond to all possible network configurations.

Finally, we define as the state of our system to be the process{X(t)}∞t=0 of the queue sizes at all network nodes and for all classesof customers at the end of the current time slot. Both the system state

X(t−1) as well as the current topologyT (t) are available to the sched-uler at the beginning of each time slott. Based on this information, thescheduler takes a scheduling decision in a centralized fashion. Accord-ingly, we define a deterministic,stationary scheduling policyπ, to bea mapX × T → E such that,π(x, T ) is a valid multi-class activationvector for the current topology, as this vector is defined previously.

A Queue size dynamicsClearly, the queue size at nodei for classj at the end of the next timeslot equals its value at the end of the current time slot, plus the externalarrivals for this class at this node during the next time slot, modified byinternal arrivals or departures for classj that involve nodei. Arrivalsand departures are determined by the next multi-class activation vec-tor and by whether or not service completes during the next time slot.Hence, the queue size process for classj in vector form is:

Xj(t + 1) = Xj(t) + RjM(t + 1)Ej(t + 1) + Aj(t + 1) (1)

In Equation 1,Rj is aN × |L| matrix that we dub the “combinedrouting matrix for classj”. It has one column for each link∈ L andits elements are:

rji` =

1 if d(`) = i, with i /∈ Vj

−1 if s(`) = i

0 otherwise

(2)

Matrix M(t) is a diagonal matrix, of dimensions|L| × |L|, that cap-tures the link “quality”. Its th diagonal element(M(t))` represents thebinary random variable that corresponds to the successful service com-pletion of a customer served by server` during time slott. If a customercompletes service and moves from queues(`) to queued(`) (or exitsthe system if the node where queued(`) resides belongs to the classof the exit nodes for the particular customer class), then(M(t))` = 1,otherwise(M(t))` = 0. The latter case may occur if theth link isnot present in the current topology at time slott or when although it ispresent, a customer did not receive full service and its service is beingdeferred.

Note that rule1 in the definition of a valid multi-class activationvector,E(t), guarantees that the elements ofX(t), evolving accordingto Equation 1, are non-negative at all times.

B AssumptionsThroughout this paper, we make the following assumptions.

Assumption 1 The topology process is a stationary and ergodic pro-cess, that is varying in an i.i.d. fashion. In particularpk = P [T (t) =Tk],∀k ∈ 1, . . . , NT does not depend ont.

Assumption 2 The process{M(t)}∞t=0 of link qualities is an i.i.d pro-cess. In addition, for allt, t′, M(t) andT (t′) are independent. Fur-thermore,E[M(t)|T (t) = Tk] is independent oft, ∀k ∈ {1, . . . , NT }.We will call this expected valueMk.

Assumption 3 The arrival process{Aij(t)}∞t=0 is independent ofthe topology process{T (t)}∞t=1 and of the service rates process{M(t)}∞t=0. Also, for a given nodei and customer classj the arrivalprocess{Aij(t)}∞t=0 is an i.i.d process. Finally,E[A2

ij(t)] < ∞.

III. A N OPTIMAL THROUGHPUT SCHEDULING RULESUPPORTING WEIGHTS

In this section, we introduce a general scheduling policy that treatsdifferently the customers of different classes. To achieve this class dis-tinction, a weightwj > 0 is assigned to each customer classj. Inaddition, we letw = (wj , j = 1, . . . , J) be the vector of of weights ofeach class.

Let us define for each link, the weighted differences between thesource and the destination queues of the link, for all customer classesj,weighed by the quality of the link and the weight of the correspondingclass. Specifically:

(Dwk (x))`j = wj (xs(`)j − xd(`)j)(mk)`, (3)

where(mk)` is the`th (diagonal) entry ofE[M(t)|T (t) = Tk].In addition, let the weight of server, (Dw

k (x))` be the maximumweighted difference in queue sizes, that is achieved for some customerclassj, i.e.,

(Dwk (x))` = max

j=1,...,J(Dw

k (x))`j . (4)

We also define the weight vector of each link` that is present undertopologyTk as:

Dwk (x) = {(Dw

k (x))` : ` = 1, . . . , |L|}

Then, the proposed scheduling policyπw0 is

(πwj0(x, Tk))` =

1, j = (jk(x))`, and(ck(x))` = 1 and xs(`)j ≥ 1,

0, otherwise,

whereck(x))` is given by

ck(x) = arg maxc∈Sk

{Dwk (x)T c} (5)

and

(jk(x))` = arg maxj=1,...,J

{(Dw

k (x))

`j

}. (6)

Ties in obtaining a solution for Equations 5 and 6 are resolved by se-lecting one of the solutions.

Thus, policiesπw0 preferentially activate links (Equation 5) and

customer classesj (Equations 4 and 6) for which the weighted differ-ence of queue sizes at the source and destination is largest; here theweights are the products of the probabilities(mk)` of successful trans-missions and of the assigned weights,wj . By selecting the links withthe higher differences in queue sizes this family of policies achieves aload balancing in the network queues: whenever a queue size increasesenough, one of its outgoing links will be activated, thus reducing thequeue sizes.

IV. STABILITY UNDER RANDOMLY VARYING TOPOLOGIES

A network is stable if the queue lengths reach a steady state anddo not tend to increase without bound. Under Assumptions 1 - 3, theprocess of the queue sizes described by Equation 1, is a Markov Chain.However, this Markov Chain is not guaranteed to be irreducible.

Let Zi, i = 1, 2, 3, . . . be the sets of its recurrent communicatingclasses andY be the set of all transient states. Following [3] we definestability of the network as follows:

Definition 1 The system isstable if for the queue size process{X(t)}∞t=0, whereX(0) = x ∈ Y :

P [τy < ∞] = 1, ∀y ∈ Y

and all statesz ∈ ∪∞i=1Zi are positive recurrent,

τy =

{ ∞, if X(t) ∈ Y, ∀t > 0min{t > 0 : X(t) /∈ Y }, otherwise

We also define a multi-class arrival vector to be stable if the system isguaranteed to be stable whenever it operates under this arrival vector.The modified version of Foster’s theorem ([3], [1]) that follows, givessufficient conditions for stability of our system.

Theorem 1 (Extended Foster Theorem) [3] Consider a MarkovChain{X(t)}∞t=0 with state spaceX . Suppose there exists a real val-ued, bounded from below, functionV : X → R, a numberε > 0 and afinite subsetX0 ofX such that:

{E[V (X(t + 1))− V (X(t))|X(t) = x] < −ε, if x /∈ X0

E[V (X(t + 1))|X(t) = x] < ∞, otherwise

Then,{X(t)}∞t=0 is stable in the sense of Definition 1.

A Optimality of πw0

Let us define the following sets of arrival rates. Initially, we define theset of arrival ratesA:

A =

{a ∈ RNJ

+ : ∃f jk ∈ R

|L|+ ,

aj = −NT∑

k=1

Rjf jkpk,

J∑j=1

f jk ∈ co(MkSk)

},

Next, we define a slightly smaller setA? as follows:

A? =

{a ∈ RNJ

+ : ∃f jk ∈ R

|L|+ , aj = −

NT∑

k=1

Rjf jkpk

δ

J∑j=1

f jk ∈ co(MkSk), for someδ > 1

}.

Let Cπ be the set of arrival rates for which the stationary schedulingpolicy π stabilizes the system. In addition, defineC to be the set ofarrival rates for which there exists some stationary policy that stabilizesthe system,i.e

C = ∪πCπ

Then, we have proved the following theorem.

Theorem 2 The following properties hold:

1. A is a convex polytope.

2. For everyw with wj > 0, ∀j, A?, A, C andCπ0 are relatedthrough the following set inequality:

relint(A) ⊆ A? ⊆ Cπw0⊆ C ⊆ A = Cπw

0(7)

The proof of Theorem 2 invokes Theorem 1 with functionV : X →R, such thatV (x) =

∑Jj=1

∑Ni=1 wj{xij}2. Theorem 2 implies that

the setA? contains the relative interior of polytopeA and differs fromit at most by points on its faces. In addition, it implies thatC andCπw

0

are “almost identical” and “almost identical” to the convex polytopeA.Specifically,C andCπw

0may differ fromA only by missing certain

points from its faces (withA viewed as a subset of the affine spaceA itgenerates). In particular,C andCπw

0have equal measure inA. In that

sense,πw0 is (near) optimal in the sense of maximum throughput.

V. SIMULATION RESULTS

In this section, we will consider a simple example of a time-varyingtopology network and characterize its stability region under the pro-posed scheduling scheme. We will observe that under a policyπw

0 al-though the total throughput under any of the individual topologies thenetwork takes is zero, the throughput of the time-varying network thatevolves through each one of them with non-zero probability is strictlypositive. Hence, mobility increases the maximum stable throughput thatthe network can sustain underπw

0 .We assume that the servers are perfect, implying thatM(t) is the

identity matrix for all t. The policy that is simulated isπ10 , in other

words for each customer classwj = 1, j = 1, . . . , J . Our graphs areobtained through a Monte Carlo simulation. The arrivals are Bernoulli.

We start by selecting an arrival rate vector randomly according to theuniform distribution, within the interval(0, 1]. We then run the simula-tion for 10, 000 time slots. We decide instability on a threshold basedcriterion, i.e. if any of the network queues exceeds a threshold undersome multi-class arrival vector, this rate vector is said to be unstable.If the rate vector is deemed unstable, we decrease some element of thisvector by a step =0.05 and start the simulation again. However, if itis a stable arrival rate vector, then we mark it by a dot (Figure 5) anda new arrival rate is again selected randomly according to the uniformdistribution in(0, 1].

A 4 nodes on a ringConsider a network comprised, of4 nodes that reside on a ring of radiusr. Assume two customer classes: customer class0, of ratea10, arrivingat node1, with exit node3 and customer class1, of ratea41, arriving atnode4, with exit node2. Further, assume that at each time instant thepositions of all the nodes are determined by the position of any one ofthem. Specifically, let a node be located at angleθ(t) ∈ [0, π/2], ∀t ≥0. The rest of the nodes will be placed at positionsπ−θ(t), π+θ(t) and2π − θ(t). Hence, each quadrant is allocated to exactly one node. Letthe nodes be numbered from1 to 4, as shown in Figure 1. Consider alsothat the power at the nodes is such, that a pair of nodes communicatesif their distance is at mostR. Assume that the radius of the ring is largeenough to guarantee the existence of some region where there exists noconnectivity among the nodes, in other wordsr > R√

2.

If θ(t) is small enough, then the network nodes that may communi-cate with each other will be the pair of nodes1 and2, as well as nodes3 and4. This is illustrated in Figure 2. This would put a constrainton the distance between these nodes to be less than the range of com-

munication, namely2rsin(θ(t)) ≤ R, i.e. θ(t) ≤ θ0∆= sin−1( R

2r).

Hence, as soon asθ(t) increases and becomes largerθ0, the distancebetween communicating nodes1 and2 and3 and4 becomes larger thanR, and hence they may not communicate any more. More specifically,the network will be completely disconnected for values ofθ(t), suchthat θ0 < θ(t) < π/2 − θ0. This is illustrated in Figure 3. Furtherincrease inθ(t), i.e. π/2 − θ0 < θ(t) < π/2 results in a different set

.

45ο

θ (t)

r

90

0ο

ο

180ο

270ο

4

32

1

Figure 1: A network of4 nodes, residing on a ring of radiusr.

2

1

3

4a10

a41

T0

V1 V0

p

Figure 2: TopologyT0 is present. The nodes that are able tocommunicate with each other are nodes1 and2 and nodes3 and4.

of communicating nodes, namely1 communicates with4 and2 with 3,as shown in Figure 4. So we have the following set of possible networktopologies:

• 0 ≤ θ(t) ≤ θ0: TopologyT0 is present (Figure 2).

• θ0 < θ(t) < π2 − θ0: TopologyT2 is present (Figure 3).

• θ0 ≤ θ(t) ≤ π/2: TopologyT1 is present.(Figure 4).

By considering a uniform distribution onθ(t), we observe thattopologiesT0 andT1 occur each with equal probabilityp = 4θ0

2πand

T2 occurs with probability1 − 2p. In each of the individual topolo-gies and since sources are completely isolated from their intended exitnodes the maximum throughput of the network is0, for both customerclasses. However, by considering a network that switches among thethree topologies, taking each one of them with a non zero probability,we observe that a strictly non-zero stability region is achieved. In par-ticular, consider the example whenp = 0.25. Then, the maximum

T2

2 3V V1 0

1 4 a41

a10

1−2p

Figure 3: TopologyT2 is present. All network nodes are discon-nected from each other.

32V V1 0

T1

1 4a10

a41

p

Figure 4: TopologyT1 is present. Nodes2 and3 communicateand nodes1 and4 communicate.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Stability region for a network of 4 nodes arranged on a ring

stab

le a

rriv

al r

ates

of c

usto

mer

cla

ss 1

(pk

ts/s

lot)

stable arrival rates of customer class 0 (pkts/slot)

Stable operating points from simulationLinear least squares fitStable throughput region

Figure 5: Stability region of the network consisting of4 nodeson a ring, when the occurrence probability of topologiesT0 andT1 is p = 1/4.

throughput scheduling policyπ10 as simulated by our simulation tool,

gives stable points that are marked with blue dots (Figure 5). Clearlyevery point that lies in the first quadrant and is bounded from abovethese points is stable as well. In Figure 5, we also drew the linearleast square fit for the set of stable points that is depicted by the linea10 + a41 = 0.5. The shaded region below the linea10 + a41 = 0.5 isthe total stability region of our system. Our simulation results coincidewhat we would guess intuitively about the stability region of this sim-ple time-varying topology network. For example, consider the customerclass0. Although the links from node1 to 2 and from node2 to 3 areavailable0.25 fraction of time each, we can make the arrival ratea10 tobe stable for values strictly larger than0.25 by letting node1 forwardits traffic through both nodes2 and4.

VI. CONCLUSIONS

In this paper, we introduced a family of centralized stationaryscheduling policies applied on time-varying-topology wireless ad-hocnetwork. We characterized the common set of stable arrival rates that allof these policies achieve. In addition, we proved optimality of all poli-cies in this family in terms of maximizing the total stable throughput ofthe network. We did that by showing that the set of rates that are stableunder any member of this family is essentially identical to the union ofthe sets of stable arrival rates achieved by all stationary policies. Fi-nally, we confirmed through simulations that mobility at the nodes mayindeed increase the total stable throughput of the network. This is due to

the fact that relaying through intermediate nodes can be used to createpaths from source nodes to their corresponding exit nodes.

REFERENCES

[1] Pierre Bremaud. Markov Chains: Gibbs Fields, Monte Carlo Simulationand Queues. Springer, 1999.

[2] Anna Pantelidou. Scheduling transmissions in wireless ad-hoc networkswith time-varying topologies. Master of science, Department of Electricaland Computer Engineering, University of Maryland, College Park, Decem-ber 2004.

[3] L. Tassiulas and A. Ephremides. Stability properties of constrained queue-ing systems and scheduling policies for maximum throughput in multihopradio networks.IEEE Transactions on Automatic Control, 37(12), Decem-ber 1992.

[4] Leandros Tassiulas.Dynamic link Activation Scheduling in Multihop RadioNetworks with Fixed or Changing Connectivity. PhD thesis, University ofMaryland-College Park, 1991.

[5] Leandros Tassiulas. Scheduling and performance limits of networks withconstantly changing topology.IEEE Transactions on Information Theory,43(3), May 1997.