Discrete models and algorithms for packet

scheduling in smart antennas

Edoardo Amaldi, Antonio Capone and Federico Malucelli

DEI, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy, E-mail:amaldi, capone, malucell@elet.polimi.it

Abstract

We investigate two combinatorial optimization problems arising when schedulingpackets in a smart antenna. To select a maximum number of users to be simultane-ously served in a given time slot, an arc-circular model is proposed and a polynomialalgorithm, which searches for a maximum weight path in an appropriate acyclicgraph, is presented. To partition the users so mimimize the number of time slotsneeded to transmit all the given packets, heuristics are developed.

Key words: packet scheduling, circular arc model, exact algorithm, heuristics

1 Introduction

The use of adaptive antennas arrays, known as “smart antennas”, has been re-cently considered for third generation mobile telecommunication systems (seee.g. [1,2]). The main advantages of smart antennas is that they are expected tobring higher capacity and enhanced signal quality as compared to traditionalantennas.

If the antennas elements of the array are close enough to each other (i.e., lessthan half the wavelength) it is possible to view a smart antenna as a set ofco-located directive antennas whose orientations can be adapted (via DSP)according to the user positions. It is as if a smart antenna simultaneouslytransmits to (or receive from) narrow ”beams” (approximately 12 degree wideangles) within the same cell. The interference between transmissions in differ-ent non intersecting beams can be considered as negligible. Since the spatialseparation of users is exploited to simultaneously receive signals transmittedon the same radio channel, this access scheme is usually referred to as SpaceDivision Multiple Access (SDMA) in the telecommunications literature. To

Preprint submitted to Elsevier Science 3 March 2003

allow multiple transmissions within the same antenna beam, SDMA is usu-ally combined with other access schemes like Code Division Multiple Access(CDMA), which is typical of third generation mobile telecommunication sys-tems. In this work we formulate and address two combinatorial optimizationproblems arising in packet scheduling with smart antennas.

2 Model, problems and results

For modelling purposes, the cell associated to any smart antenna can be con-sidered as a circular area and the users can be represented as points spatiallydistributed within this area. Due to the power control mechanism, which ad-justs the emission power so as to guarantee a received power equal to a giventarget value, all users can be assumed to lie at the same distance from theantenna. All points can thus be projected on the unit circumference C repre-senting the cell perimeter.

Let I = {1, . . . , n} denote the set of points on C and, for each i in I, let thenonnegative real weight wi be proportional to the priority of the ith user. Ifα > 0 denotes the length of the arcs on C corresponding to the beams, afeasible schedule of the smart antenna transmissions during a single time slotcan be formalized by a subset I ⊆ I of points, a set A of circular arcs of lengthα positioned around the unit circle C, and an assignment of the points in Ito the circular arc A satisfying the following requirements. Each point in Imust be assigned to exactly one circular arc in A among those containing thatpoint. Each circular arc in A must contain exactly one point in I. Notice thatpairs of circular arcs in A may have a nonempty intersection but each assignedpoint, i.e., each point in I, must be contained in exactly one circular arc in A.A subset of points I ⊆ I is said to be arc-independent if there exists a set A ofcircular arcs around the unit circle C with together with an assignment from Ito A satisfying the above conditions. It is worth emphasizing that the circulararcs are not part of the input but must be determined. From the smart antennapacket scheduling point of view, two main types of combinatorial optimizationproblems are of interest.

1) Maximum weighted arc-independent subset of points:

Given a set of points I = {1, . . . , n} on C with nonnegative weights {wi}i∈I anda real α > 0, find an arc-independent subset I ⊆ I of points with maximumtotal weight.

This clearly amounts to maximizing the total priority (number) of the usersthat can be served during a single time slot, since their transmissions do notinterfere with one another.

2

2) Minimum partition into arc-independent subsets of points:

Given a set of points I = {1, . . . , n} on C with nonnegative weights {wi}i∈I

and a real α > 0, find a partition of the points in I into a minimum numberof arc-independent subsets.

Although other objective functions could also be interesting, the one aboveaims at minimizing the number of time slots (of subsets of non interferingtransmissions) needed to transmit all the given packets.

In the basic versions of these two problems, a single user is considered pertransmission channel, but if a code division access scheme is adopted a channelcan be shared by multiple users. This gives rise to generalized versions of theproblems in which the assignment of points to the circular arcs can be many-to-one instead of one-to-one, namely at most k points can be assigned to eachcircular arc in A, with k ≥ 1. In this work we have investigated the bothcombinatorial optimization problems for k = 1 and k > 1.

We assume that the distance between all pairs of consecutive points on theunit circle C is strictly smaller than α. Otherwise, by cutting in the middleof all these pairs of consecutive points, one can reduce the problem on C intoone (or a collection) of independent a 1-dimensional subproblems that can besolved separately. The arc intervals are also assumed to be open on the leftand closed on the right.

A simple but important observation allows us to consider only a restrictedtype of possible placements for each circular arc. Given any arc-independentsubset of points I ⊆ I with a corresponding set of arcs A, without loss ofgenerality, we can focus attention only on the extremal arc arrangements, thatis arc arrangements obtained by shifting all arcs of A counterclockwise. Clearlyafter such a shift all arc intervals are in one of two confirgurations: either theright end point of the interval coincides with the selected point, or the left endpoint of the interval is just on the right of the immediately preceding selectedpoint. An extremal configuration is shown in figure 1, where the circumferencehas been stretched on a rectilinear line.

A2

A1

1

2

4

6

7

A4

A7

A6

5 8

Fig. 1. Example of an extremal interval arrangement: the set of selected points is{1, 2, 4, 6, 7}

3

By exploiting this fact, it is possible to reformulate the problem of findinga maximum weighted arc-independent subset of points, as that of finding amaximum weight path in an special acyclic graph. The graph, that has a layerLi for each point i ∈ I is constructed as follows. Each node of the graphrepresents a possible selection of a point together with the suitable positionof the interval associated to it. Considering a point i ∈ I we can distinguishtwo types of nodes: node i which represents the selection of point ii and arcAi having it right end point corresponding to the point, nodes (j, i) whichrepresent the selection of node i as immediate successor of point j and arc Ai

having its left end point just on the right of point j, for all points j not fartherthan α on the left of i. In figure 2 we exemplify the two types of nodes. Theweight of the point wi is associated to all nodes of layer Li.

i i

Ai

jj,i

iAj

Ai

Fig. 2. Example of nodes of layer Li

The arcs of the graph represent the compatibility of selecting pairs of pointsand the placement of their intervals. The number of nodes and arcs of thegraph is polynomial in the number of points, and being the graph acyclic,at least when we consider the problem on the line, the computation of theheaviest path, corresponding to the maximum priority set of arc independentpoints, can be carried out in polynomial time. The algorithm can be extendedto the circular case whitout loosing the polynomiality.

The problem of finding minimum partitions into arc-independent subsets ofpoints can be viewed as a particular cycle cover of the layered graph which canbe modeled as a flow problem with side constraints. It is still open whether itis NP-hard but so far we have devised heuristics.

References

[1] Perez-Neira, A.; Mestre, X. and Fonollosa, J.R., “Smart antennas in softwareradio base stations”, IEEE Communications Magazine , Vol. 39 (2) , Feb. 2001,166-173.

[2] Sheikh, K.; Gesbert, D.; Gore, D. and Paulraj, A., “Smart antennas forbroadband wireless access networks”, IEEE Communications Magazine, Vol.37 (11) , Nov. 1999, 100-105.

4