WMNs are being considered within several wireless tech- nologies, including IEEE 802.11 WLAN [3], IEEE 802.16. This work has been partially supported by ...
Scheduling Optimization in Wireless MESH Networks with Power Control and Rate Adaptation Antonio Capone and Giuliana Carello Dipartimento di Elettronica e Informazione, Politecnico di Milano, Italy {capone,carello}@elet.polimi.it
Abstract— Wireless MESH networks are a new networking paradigm that allow to extend the coverage of traditional wireless access networks with multi-hop connections through fixed wireless mesh routers. Wireless MESH networks partially replace wired backbone networks, and it is reasonable to carefully plan radio resource assignment in order to provide quality guarantees to traffic flows. Differently from ad hoc networks, energy consumption is usually not a problem with wireless MESH routers, routes are quite stable and bandwidth requirements of traffic flows can be considered almost constants. In this paper we study the scheduling optimization problem in wireless MESH networks assuming a time division multiple access (TDMA) scheme, a dynamic power control able to vary emitted power slot-by-slot, and a rate adaptation mechanism that sets transmission rates according to the signal-to-interference-andnoise ratio (SINR). Traffic quality requirements are expressed in terms of minimum bandwidth and modelled with constraints defining the number of information units (packets) that must be transmitted on each link per frame. We propose an alternative problem formulation where decision variables represent compatible sets of links active in the same slot. Approaches to solve both lower and upper bound for the problem are proposed: since compatible set variables are exponentially many, we use column generation to compute a lower bound for the problem. Heuristic approaches to compute feasible integer solutions are proposed and tested. Index Terms— Wireless MESH Networks, Scheduling, Optimization, Power Control, Rate adaptation, Cross-layer.
I. I NTRODUCTION Wireless Mesh Networks (WMNs) have emerged recently as a new network architecture able to extend the coverage and increase the capacity of wireless access networks [1], [2]. WMNs are a promising solution to provide both indoor and outdoor broadband wireless connectivity in several environments without the need for costly wired network infrastructures. The network nodes in WMNs, named mesh routers, provide access to mobile users, like access points in Wireless Local Area Networks (WLAN) or base stations in cellular systems, and they relay information hop by hop, like routers, using the wireless medium. Mesh routers are fixed and usually do not have energy constraints. Therefore, WMNs are characterized by infrequent topology changes mainly due to node failures. WMNs are being considered within several wireless technologies, including IEEE 802.11 WLAN [3], IEEE 802.16 This work has been partially supported by European Network of Excellence EURO-NGI.
Wireless Metropolitan Area Networks (WMAN) [4] and next generation cellular systems [5]. In all cases, WMNs partially replace the wired backbone network and should be able to provide similar services and quality guarantees. For cellular systems in particular, but also for WMAN, the backbone network is usually devised to provide an almost static resource assignment to traffic flows between base stations and network gateways. This approach allows to simplify the radio resource management at the interface between the network and the mobile users and to provide quality of service guarantees. Therefore, in these scenarios traffic engineering methodologies able to provide bandwidth guarantees to traffic flows and to optimize transmission resources utilization in WMNs appears to be a key element. Advanced multiple access schemes based on time division, power control mechanisms, and adaptive modulation and coding techniques are the most appropriate tools for defining radio resource management algorithms able to reserve the required rate to traffic flows and to achieve high network efficiency. These tools are already available for IEEE 802.16 networks and are commonly considered for next generation cellular systems [6]. In a wireless environment, network topology depends on the position of nodes and propagation conditions. We can assume that a link between two nodes (i, j) exists if transmitting a signal at maximum power in i, the Signal-to-Noise Ratio (SNR) in j is sufficiently high to correctly decode the signal. To achieve high network efficiency, parallel transmissions on more than one link must be considered by the scheduling scheme. However, parallel transmissions generate interference at receiving nodes that may affect the correct decoding. Therefore, some constraints on resource reuse must be considered in order to guarantee correct network operation. A simple model that has been proposed for reuse constraints is based on a conflict graph [7], [8]. In the conflict graph there is a link (l, g) if the received signal level in g is above a carrier sense threshold when a signal is transmitted by l. If i is transmitting to j, no other node h can transmit if link (h, j) is in the conflict graph. Moreover, assuming that nodes cannot transmit and receive at the same time, no other node k can transmit to i on link (k, i). Within this context, scheduling problem has some similarities with the graph coloring problem and several solutions have been proposed both for point-topoint and broadcast/multicast transmissions [7], [10], [11], [12], [13], [14], [15]. In order to take into account topology changes even more general schemes can be considered [16],
[17]. Obviously, the model based on conflict graph does not consider that the effect of several interfering transmissions is cumulative. Therefore, the carrier sense threshold must be set to a quite low value and the resulting scheduling policies are not very efficient [18]. A more accurate model takes into account the cumulative effect of interference evaluating the Signal-to-Interference and Noise Ratio (SINR) at receivers [8]. With this model a set of parallel transmission on a set of links is admissible if the SINR at all receivers is above a given threshold [9]. Since SINR values depends on the power emitted by all transmitting nodes, power control mechanisms can improve resource reuse adjusting powers in order to set the SINR at receivers just above the threshold [19]. In this case a set of parallel transmissions is feasible if it is possible to find a power assignment that satisfies power limitations and provides the required SINR. Several scheduling approaches that jointly assign transmission time slot and emitted power have been proposed considering different objective and constraints [20], [21], [22], [23]. A further improvement can be achieved considering also transmission rate control within a cross-layer approach. Modern adaptive modulation and coding schemes allow to adapt transmission rate to the actual propagation and interference conditions. According to the value of the SINR the best transmission rate that provides an error rate sufficiently low can be selected. Since in the considered scenario SINR values are determined by the set of parallel transmissions and the transmission powers, it appears quite reasonable to consider transmission scheduling, power control and rate control at the same time [24], [25], [26], [27], [28]. Even if optimization approaches to different types of scheduling problems have been considered within the general framework of ad hoc and multi-hop wireless networks, WMNs represent a practical scenario where the usually high computation times required by these schemes can be tolerated within an almost static radio resource planning approach. In this paper we study the scheduling optimization problem in wireless MESH networks assuming a time division multiple access (TDMA) scheme, a dynamic power control able to vary emitted power slot-by-slot, and a rate adaptation mechanism that sets transmission rates according to the SINR. Traffic quality constraints are expressed in terms of minimum required bandwidth. Since the time frame defined by the TDMA scheme is fixed, the bandwidth requirement can be translated into the number of information units (packets) that must be transmitted on each link per frame. Moreover, according to a discrete set of possible transmission rates, the number of packets that can be transmitted per time slot depends on the SINR at receivers. In order to get more insights on the characteristics of the problem and the effect of different control mechanisms, we consider three different versions of the problem with increasing complexity. In the first one we assume fixed power and rate, in the second one variable power and fixed rate, and finally in the third one variable power and rate. Given a number of available slots, our goal is to provide an assignment of time
slots to links such that bandwidth constraints are satisfied and the number of available time slots is not exceeded. To solve such problem, it is possible to look for the minimum number of needed time slots: if it is smaller than the number of available slots, a feasible assignment exists. The solution approach we propose is based on an alternative problems formulation where decision variables represent compatible sets of links active in the same time slot. Since variables are exponentially many, we use column generation to solve the continuous relaxation of problem which provide a lower bound of the optimal solution. In several cases the solution provided by the column generation is even integer, however computation times increase remarkably with problem size. To provide good solutions in reasonable time we propose some heuristics. The paper is organized as follows. In Section II we revise related work and carefully define system model. In Section III we present our problem formulations, while in Section IV we introduce the proposed solution approach. In Section V we show first some results on small size instances that allow to understand the basic characteristics of the proposed schemes and then more detailed numerical results on random generated instances. Finally, in Section VI we provide concluding remarks and an outlook of current research activities. II. S CHEDULING APPROACHES According to the well known classification proposed in [8] two possible interference models can be considered for multihop wireless network, namely the Protocol Interference Model and the Physical Model. The Protocol Interference Model describes interference constraints according to a conflict graph but does not allow to take into account the cumulative effect of interference, while the Physical Model directly considers SINR constraints at receivers. The network can be described with a directed graph G(V, E), where vertices are wireless routers and edges physical links. The SINR level at receiver j when a signal is transmitted by i is given by: SIN Rj =
ηj +
pij Gij
(l,m)=(i,j)
plm Glj
(1)
where pij is the power emitted by i on link (i, j), Gij is the gain of the radio channel between i and j, and ηj is the thermal noise at receiver j. When a single modulation rate is considered, SINR constraints can be modelled just requiring that SIN Rj ≥ γ for all j ∈ V . Several different scheduling schemes have been proposed according to this interference model. Usually, additional constraints preventing nodes to transmit and receive at the same time are added assuming that nodes are equipped with a single radio element and that a single channel is available. In [20] a scheduling and power control strategy is proposed. The general objective is to maximize network throughput. However, the solution approach proposed is based on the decomposition of the problem into two sub-problems. The first sub-problem has the goal of finding the maximum subset of parallel transmissions where transmitting nodes are separated
by at least a given distance. This sub-problem is equivalent to scheduling problems defined according to a conflict graph. The second sub-problem sets powers according to a scheme which is equivalent to the power control mechanism in cellular systems. The power controlled scheduling algorithm proposed in [21] is also based on a two steps approach, however the set of links obtained in the first phase is usually unfeasible and it is pruned in the second phase through a greedy algorithm that directly considers SINR constraints. In [22] also the endto-end routing is included in the optimization problem. The routing problem is solved through the shortest path approach defining a link metric based on the number of interference conflicts caused by each link. The scheduling and power control problems are then solved with a greedy integrated approach. The impact of power control on the performance of multi-hop wireless networks is deeply investigated in [23], considering the joint routing and scheduling problem. However, in the proposed scheme power levels can be different from node to node but they are assumed constant over time. Rate control is another element that can be considered for the optimization of radio resource utilization. It requires a cross-layer approach that is able to directly control transmission rate through the use of a set of physical layer settings based on the estimation of propagation and interference conditions. When multiple modulation rates are considered, SINR constraints can be modelled requiring that SIN Rj ≥ γw for all j ∈ V , where γw is the minimum required SINR when rate w is adopted. In [24] the joint routing, scheduling, and power control problem is considered taking into account also long term minimum rate requirements of traffic flows. A two steps approach is considered where the first phase focuses on the scheduling and power control only. The data rate on the radio link is assumed to be a continuous linear function of the SINR and the optimization objective is total power minimization. A similar problem is considered in [27] where, however, rate requirements are guaranteed within the frame period and not only in the long term. The solution approach is based on greedy heuristics. A mixed integer linear programming (MILP) formulation for the joint scheduling, power control, and rate control problem is proposed in [26]. The goal here is throughput maximization and the solution approach is based on a greedy algorithm. In [28] the joint routing, scheduling and power control problem is studied considering a nonlinear optimization problem where achievable transmission rate is given by the Shannon capacity. In [25] an opportunistic scheduling scheme is proposed considering time-varying channel conditions. Emitted powers and data rates are dynamically adapted according to scheduling decisions and channel state. Finally, in [29] a general framework for the evaluation of flowlevel performance in multi-hop wireless networks is presented focusing on elastic flows and balanced-share concept. In this paper we study the joint scheduling, power control and rate control problem. For each link of the network we considers traffic constraints defining the number of information units (packets) that must be transmitted per frame. Considering a discrete set of possible transmission rates, the number of packets transmitted per time slot is set according to SINR at
receivers. Differently from previously proposed models, there is no a priori model of the relation between the data rate and the SINR. We just consider the value of the rate, expressed as numbers of packets transmitted per slot, and the corresponding threshold γw for the SINR. The lowest rate allows to transmit one packet per slot, while higher rates allow to transfer more packets. The objective of the optimization problem is to find the scheduling settings that minimize the number of used time slots. Obviously, this approach allows to find a feasible assignment if the minimum number of slots is smaller than the number of available slots. Moreover, differently from other approaches based on power minimization or throughput maximization, it provides a compact scheduling that may be used for easing and expediting admission control procedures that need to estimate spare bandwidth. The solution approach is based on problem formulations where decision variables represent compatible sets of links active in the same time slot. Since variables are exponentially many, we use column generation [9]. III. S CHEDULING PROBLEMS FORMULATIONS In the following two different formulations of the scheduling problem are presented. The first formulation directly considers time slot and rate assignment variables as well as power variables. It allows to test the performance of state-of-theart ILP (Integer Linear Programming) solvers with different sizes instances. The second formulation considers as decision variables the set of compatible links that can be activated at the same time. Since there are exponentially many variables, this formulation is specially devised for a solution approach based on column generation. A. Linear number of variables model First consider the problem in which the power of each link is equal to a fixed value P . Each link requires to send Rij packets. Denote with N the set of devices belonging to the network and with L the set of links. The minimum value of SINR required is denoted by γ. Let Gij be the gain of the channel between devices i and j. The problem can be modelled with two sets of binary variables, zijk and xk – where k takes value from 1 to K, representing an upper bound on the number of available slots – such that zijk =
1 if device i transmits to device j in time slot k 0 otherwise
xk =
1 if in slot k at least one link is active 0 otherwise
The objective function minimizing the number of needed slots is min
K k=1
and the constraints are
xk ,
(2)
zijk ≤ xk ∀(i, j) ∈ L, k = 1, . . . , K (3) zijk + zjik ≤ 1 ∀i ∈ N , k = 1, . . . , K
(i,j)∈L
(j,i)∈L K
(4) zijk ≥ Rij
k=1
∀(i, j) ∈ L
(5)
P Gij ≥ γzijk η +
P Ghj zhmk
(h,m)∈L: h=i
(6)
zijk ∈ {0, 1}
∀(i, j) ∈ L, k = 1, . . . , K
(7)
xk ∈ {0, 1}
∀(i, j) ∈ L, k = 1, . . . , K
(8)
The objective function (2) minimizes the number of used time slots. Constraint (3) forces variable xk to one if at least one link is active in time slot k. Equation (4) implies that each device is active in at most one link in each slot, while constraint (5) guarantees that the traffic requirement of each link is fulfill, provided that a link sends a packet in each slot in which is active. Equation (6) guarantees that traffic quality requirements are satisfied in each time slot, η representing the noise. The model can be extended to deal with the problems in which power control and rate control are considered. To model the problem in which the power of each link is to be optimized, a nonnegative variable pijk , representing the amount of power of link (i, j) in time slot k, must be introduced. The objective function (2) and constraints (3)-(5), together with binary constraints (7) and (8), hold also for this case, while constraint (6) is to be changed. The new constraint is:
pijk Gij ≥ γzijk η +
phmk Ghj
(h,m)∈L: h=i
1 if i transmits to j in slot k with rate w 0 otherwise,
must replace variables zijk in all the constraints. Constraint (5) is changed into equation K
∀(i, j) ∈ L
(12)
and in constraint (6) γw must replace γ. Although the models of the three different problems are non linear, since in the constraints forcing the signal to noise ratio variables are multiplied, they can be linearized – as it will be showed in Section IV – and then solved by commercial solvers, like CPLEX [30]. Nevertheless, only small size instances can be solved with this kind of models: in fact, solving a problem with five devices and one packet to be sent for each link requires more than four hours of computing. A different formulation is proposed in next section. B. Column generation model To model the problem in a different way, we define a configuration as a compatible set of links, i.e. a set of links that can be all active together without violating the signal to noise and interference ratio requirement. Denote with S the set of all possible compatible configurations. Besides, let denote with Sij the set of configuration in which link (i, j) is active. To formulate the problem, an integer variable λs is defined for each configuration s ∈ S representing the number of slots to which s is assigned. As only one configuration can be assigned to one time slot, the number of configurations assigned to time slots is equal to the number of used slots. The problem of minimizing the number of needed slots can be modelled as follows: min
(9)
∀(i, j) ∈ L, k = 1, . . . , K Besides, two constraints, setting the bounds of the power variables and connecting power and transmission variables, must be added: pijk ≤ Pmax zijk
∀(i, j) ∈ L, k = 1, . . . , K
(10)
γη zijk Gij
∀(i, j) ∈ L, k = 1, . . . , K
(11)
pijk ≥
Tw zijkw ≥ Rij
k=1 w∈W
∀(i, j) ∈ L, k = 1, . . . , K
zijkw =
If both power and rate are to be optimized, a set W of rates must be introduced as well as a parameter Tw representing the number of packets per slot which can be sent with rate w. Furthermore, the required SINR value depends, in this case, on the rate and is denoted by γw . To model such problem, binary variables zijkw , related to the link (i, j), to the time slot k and to the rate w, such that
λs
(13)
s∈S
λs ≥ Rij
∀(i, j) ∈ L
(14)
∀s ∈ S
(15)
s∈Sij
λs ∈ Z+
Objective function (13) imposes the minimization of the number of needed time slots, while constraint (14) guarantees that each link is active in at least one slot for each packet to be sent. To complete the mathematical description of the problem, we must provide equations describing compatible configurations. Consider a binary variable uij equal to one if link (i, j) is active in a given configuration and zero otherwise. A feasible configuration s must satisfy two kinds of constraints: (16) implies that each device is involved in only one link while (17) guarantees the required traffic quality. (i,j)∈L
uij +
(j,i)∈L
uji ≤ 1
∀i ∈ N
(16)
P Gij ≥ γ
P Ghj uhm uij
(h,m)∈L, i=h
(17)
∀(i, j) ∈ L. uij ∈ {0, 1}
∀(i, j) ∈ L
(18)
As the model proposed in Section III-A, the above model turns out to be non linear due to constraint (17). The model can be modified to deal with the other considered problems. If the power control is considered a nonnegative variable pij is introduced for each link and constraint (17) should be changed into
pij Gij ≥ γ
phm Ghj uij
∀(i, j) ∈ L
(19)
(h,m)∈L, i=h
and constraints pij ≤ Pmax uij pij ≥
γη uij Gij
∀(i, j) ∈ L
(20)
∀(i, j) ∈ L,
(21)
connecting u and p variables, must be added, where uij ∈ {0, 1} pij ≥ 0
∀(i, j) ∈ L
(22)
∀(i, j) ∈ L
(23)
Finally, for the problem with power and rate control, the rate must be decided for each link belonging to a configuration: a binary variable uijw is defined for each link (i, j) and each possible rate w ∈ W such that uijw = 1 if link (i, j) is active in the considered configuration with rate w. Different SINR thresholds for different rates are denoted by γw . The equations describing a compatible configuration become:
uijw +
(i,j)∈L w∈W
ujiw ≤ 1
∀i ∈ N
(24)
(j,i)∈L w∈W
pij Gij ≥ γw
phm Ghj uijw
(h,m)∈L, i=h
(25)
∀(i, j) ∈ L, w ∈ W pij ≤ Pmax uijw pij ≥
γη uijw Gij
uijw ∈ {0, 1} pij ≥ 0
∀(i, j) ∈ L, ∀w ∈ W
(26)
∀(i, j) ∈ L, w ∈ W
(27)
∀(i, j) ∈ L, w ∈ W ∀(i, j) ∈ L
(28) (29)
As the number of sent packets changes with the rate, a constant vijs is defined for each link (i, j) and for each configuration s, representing the number of packets sent by the link (i, j) in configuration s, depending on the rate chosen for link (i, j) in the considered configuration, and constraint (14) changes accordingly:
vijs λs ≥ Rij
∀(i, j) ∈ L.
(30)
s∈Sij
IV. S OLUTION APPROACH Consider the continuous relaxation of the problem (13)-(15), i.e. the problem described by (13)-(14) in which λs variables are continuous, and denote such problem with P. As P is the continuous relaxation of problem (13)-(15), the optimal solution of P provides a lower bound of the minimum number of needed slots. Since the number of variables in problem P is huge, it is not possible to enumerate them all. To solve P and provide a lower bound on the number of slots, we apply a column generation approach. In the column generation only a subset of variables is considered. The problem P involving a subset of variables, called master problem (MP), is solved: the solution is optimal for the MP but it may not be optimal for the original problem P as only a subset of variables is considered. Then, we need a procedure, called pricing, to check if the solution found is optimal also for P or to find out the variables to be added to the set to improve the solution. The pricing procedure is based on the properties of the dual problem of P. We recall that, given a solution of the problem P, if the dual variables related to such solution are feasible for the dual problem then the given solution is optimal for P. Besides, each variable (constraint) of P is associated to a constraint (variable) of the dual problem. Given a primal variable, if the associated dual constraint is violated, the considered variable has a negative reduced cost and therefore can produce an improvement in the objective function if it is added to the set of the considered variables. Thus the aim of the pricing procedure is to verify whether the dual variables associated to the primal solution found are feasible for the dual problem and, if they are not, to build a feasible configuration such that the related dual constraint is violated. The variable associated to such configuration must then be added to the considered set and MP is to be solved again. The continuous relaxation optimum is reached when no configuration can be found such that the related dual constraint is violated. Consider first the problem with fixed power P . Denoting with σij the dual variable related to constraint (14), the dual constraint associated to a given configuration s is
σij ≤ 1
(31)
(i,j): s∈Sij
To solve the pricing problem we must look for a configuration s satisfying (16) and (17) such that
σij uij
(32)
(i,j)∈L
is maximum. If the maximum is greater than one, the variable related to such configuration must be added to the set. The pricing problem is then formulated as an integer linear problem (ILP):
max
σij uij
(33)
(i,j)∈L
yij =
σij uijw
(34)
(i,j)∈L w∈W
subject to (24), (25), (26), (27), (28) and (29). In both cases the pricing problem is formulated as a mixed integer problem (MIP). The pricing problem as formulated above turns out to be non-linear for all the proposed problems. The formulation can be linearized and the ILP (or MIP in case of power and rate control) pricing problem can be solved to optimality with commercial solver as CPLEX. We proposed two ways to linearize constraint (17) – or (19) or (25), respectively –. In a similar way constraint (6) can be linearized as well. The first linearization can be applied to all the proposed problems, with fixed power, with power control and with power and rate control. The linear formulation of the traffic quality constraint is:
P Gij + M (1 − uij ) ≥ γ η +
P Ghj uhm
(h,m)∈L, i=h
(35)
∀(i, j) ∈ L, where M is a constant such that:
M ≥ γ η +
P Ghj uhm
∀(i, j) ∈ L.
(36)
(h,m)∈L: i=h
Different values of M satisfying equation (36) have been proposed and tested: according to computational results the value of M has been set as follows:
|N | , M = γ η + Gmax P 2 where Gmax = max Gij . i∈N , j∈N
A second linearization can be applied only to the problem in which power and rate control are not considered. A continuous nonnegative variable yij is defined for each link (i, j) such that
0
(h,m)∈L: i=h
P Ghj uhm
if uij = 1 otherwise
Using such variable, constraint (17) can be rewritten as P Gij uij ≥ γ yij
∀(i, j) ∈ L
(37)
To force the correct value of variables yij , the following constraints must be added:
subject to (16), (17) and (18). In case of power control the pricing problem is the problem of maximizing (32) subject to (16), (19), (20), (21), (22) and (23), while, if both power and rate control are envisaged the pricing problem becomes max
η+
τij uij +
P Ghj uhm + η − yij ≤ τij
(h,m)∈L: i=h
(38) ∀(i, j) ∈ L,
where τij = η +
P Ghj .
(39)
h∈N : i=h
Besides solving to optimality the pricing problem with CPLEX, to reduce the computational effort it is possible to solve the problem heuristically applying greedy approaches, that build the new configuration variable by adding the links one by one. We apply two different kinds of greedy algorithms. The links are sorted according to different criteria, such as non decreasing values of Gij , non decreasing values of σij or non increasing values of σij Gij . Then, the first kind of greedy considers the links one by one and adds the considered link if the constraints are not violated, while the second one, after adding a link, removes all the links not considered yet that would cause infeasibility. If the greedy approach fails in finding an improving compatible configuration, the exact solution is computed by solving ILP or MIP formulation to check whether the optimum of P has been reached or not. Preliminary computational tests have shown that in most of the cases the greedy approaches do not manage to find the optimal pricing solution and the exact approach is to be run. Thus the total computational time does not improve. The initial set of configurations L0 must provide a feasible solution for problem P. It is computed applying greedy approaches similar to those described above, with the aim of providing a significant variety of possible configurations. The column generation approach provides a lower bound, as it solves the continuous relaxation P. Besides, we compute an integer feasible solution by solving the problem (13)–(15) over the final set of configuration selected by the column generation. Furthermore, a feasible solution can be computed without involving the column generation procedure, by solving the problem P over the initial set L0 . In both cases the obtained solution is a heuristic one, as we consider only a subset of variables, heuristically chosen. As shown, the whole approach – column generation and heuristics for the integer solution – can be applied also to the different versions of the problem.
V. N UMERICAL RESULTS The different versions of the algorithm (considering exact or heuristic pricing solution and different values of M for the linearization) have been tested on several randomly generated instances. We have first considered a preliminary analysis of the numerical results in order to assess which version of the algorithm provides the best performance. Obtained results showed that the best tradeoff between computational effort and quality of the results is obtained by solving exactly the pricing problem applying the linearization based on equations (35) and (36). These results are not presented here for the sake of conciseness. The algorithm has been implemented using modeler AMPL and solver CPLEX. Computational tests have been run on a Intel Pentium 4 at 3 GHz and with 1 GB RAM, running under Unix. In Section V-A the behavior of the proposed algorithms is analyzed and the influence of fixed power, power control and power and rate control strategies is shown by solving two small size instances, while in Section V-B computational results on instances with up to 30 nodes are reported. A. Small instances We consider two small instances, one with random distribution of the devices, the other with regular distribution. The first one has eight devices. In Figure 1 the position of devices together with the traffic requirements are shown – the arc labels representing the number of packets to be sent. The devices coordinates have been randomly chosen in the range 0 - 70m, the noise value is set to 10−6 mW while SINR threshold is 2.5.
TABLE I F IXED POWER SCHEDULING - RANDOM DISTRIBUTION Configuration 1 2 3 4 5 6 7 8 9 10
Active links (4,3)(7,1)(8,5) (4,3)(8,5) (2,3)(5,8) (6,8) (2,4) (3,5) (6,1) (7,4) (8,1) (3,7)
# slot 4 0 1 8 1 2 4 1 1 1
For the considered problem, the column generation does not add any configurations, thus the initial solution, which is integer, represents the optimal solution of the problem, requiring 23 time slots. The required computational time is lesser then one sec: the column generation based model provides clearly a better performance. Consider now the power control version of the problem. The initial set of configurations, and therefore the initial solution of the master problem, is the same as for the previous problem, but the column generation procedure adds new configurations, proving that the optimal solution for the fixed power problem is not optimal for the power control problem. The set of added configurations together with the final solution is shown in Table II. TABLE II P OWER CONTROL SCHEDULING ( RANDOM DISTRIBUTION ). Configuration 11 12 13 14 15 16 17 18
Active link (5,8)(3,7) (2,3)(8,1) (6,8)(2,4) (6,8)(7,4) (4,3)(8,1) (6,8)(4,3) (8,5)(6,1) (7,1)(6,8)(3,5)
# slot 1 1 1 1 0 4 4 2
Power control allows to exploit better the transmission and the number of needed slot decreases to 14. In case of both power and rate control, we considered the rates and SINR thresholds described in Table III. TABLE III N UMBER OF PACKETS AND SINR THRESHOLDS
Fig. 1.
First small instance. Devices positions and traffic demands.
As explained above, even for this small size instance solving to optimality the models described in Section III-A would require more than four hours, thus the problem has been solved applying the column generation based approach. First let us consider the fixed power strategy. In Table I the initial set of configurations is shown together with the number of slot to which each configuration is assigned in the solution computed over the initial set.
SINR threshold 2.5 5 10 20
# packets 1 2 4 8
In that case only eight slots are needed in the final solution: the solution is shown in Table IV, in which for each active link the chosen rate is also given. In the second instance, twelve devices are located on a square grid. In Figure 2 the position of devices together with the traffic requirements are shown – the arc labels representing the number of packets to be sent. The parameters are kept.
TABLE IV P OWER AND RATE CONTROL SCHEDULING . Configuration 8 10 11 12 13 14 15 16
Active link (7,4,1) (3,7,1) (2,4,4)(6,8,4) (8,1,4) (6,1,4) (4,3,4)(5,8,4) (7,1,4)(2,3,4)(8,5,4) (3,5,4)
# slot 1 1 1 1 1 1 1 1
Concerning the problem with fixed power, the continuous relaxation on the initial configurations set turns out to be the optimal solution of the problem P and, as all the variables are integer, the optimal integer solution as well. In this case, in fact, yje value of the integer solution found coincides with the column generation lower bound. It requires 13 slot and is given in Table V.
TABLE VI P OWER CONTROL SCHEDULING ( GRID DISTRIBUTION ). Configuration 3 8 9
10
11
Active link (2,3) (5,6) (1,2) (3,4) (3,4) (5,6) (10,11) (3,4) (6,7) (9,10) (11,12) (3,4) (5,6) (7,8)
Power 6.4e−02 1.3e−01 2.4e−01 7.5e−02 6.6e−02 1.4e−01 8.1e−02 4.1e−01 1.2e+00 2.5e+00 5.8e−01 6.3e−01 2.5e+00 8.7e−01
# slot 1 4 1
1
2
TABLE VII P OWER AND RATE CONTROL ( GRID DISTRIBUTION ). Configuration 1 10 11 15 19
Active link (1,2,1)(6,7,1)(10,11,1) (3,4,2)(5,6,4) (2,3,4)(9,10,4) (7,8,2)(1,2,4) (3,4,4)(11,12,1)(5,6,2)
# slot 1 1 1 1 1
B. Extended results
Fig. 2.
Second small instance. Devices positions and traffic demands.
TABLE V F IXED POWER SCHEDULING ( GRID DISTRIBUTION ). Configuration 1 2 3 4 5 6 7
Active link (1,2)(6,7)(10,11) (1,2)(5,6)(9,10) (2,3)(5,6) (1,2)(5,6)(11,12) (1,2)(5,6) (3,4)(7,8) (3,4)
# slot 1 2 1 1 0 8 0
If power control is allowed, the initial solution of the master problem is not the optimal one as four configurations are added by the column generation approach. The final optimal solution is shown if Table VI, together with the power of each link. The number of needed slots is decreased down to 9. Finally consider the case of power and rate control, keeping parameters described in Table III. In that case, the number of needed slots is 5, as shown in Table VII. The column generation adds 13 configurations to the initial set and 4 of them belong to the final solution.
The proposed approach has been tested on a set of instances with 5, 10, 20 and 30 nodes. The instances have been generated by randomly locating the devices over a 100m × 100m square area and by randomly setting the entries of the traffic matrix. The maximum number of packets have been set to 15 for each link, η has been set to 10−6 mW, Pmax = 30mW and SIR≥10. For each network dimension eight instances have been generated for each kind of problem. The computational times have been limited: for fixed power instances the limit has been set to two hour, while for instances with power and rate control the time limit has been set to four hours. In fact solving the pricing problem to optimality becomes difficult for bigger problems and therefore the whole approach becomes slow. In Table VIII results on instances with fixed power are given, Table IX is devoted to the results for the case in which power control is considered, while Table X gives results for the instances in which both power and rate are optimized. The tables show the behavior of the algorithm according to different instances dimensions. For each dimension, reported in the first column, columns two to six are devoted to the results provided by the initial set of configurations, while the last columns report results provided by the final configurations set. Denote with IM P R the optimal solution of the continuous master problem on the initial set of configuration L0 , with IH the heuristic integer solution computed by solving the integer master problem over the set L0 , with LB the lower bound, i.e. the optimal solution of P computed by column generation, while F H denotes the heuristic computed by solving the integer master problem over the final set of configurations. In columns two and three the average and maximum error of
TABLE VIII R ESULTS FOR FIXED POWER CASE .
Size
error av 0 0.21 0.21 0.35
5 10 20 30
[%] max 0 1.69 0.60 0.57
Initial set T [s] av max 0.00 0.01 0.01 0.02 0.66 0.85 6.94 8.12
imp [%] av 0 0.21 0.21 0.35
error [%] av max 0 0 0 0 0 0 0 0
Final set T Tot. [s] av max 0.01 0.02 0.23 0.56 51.44 142.57 4675.65 7566.64
TABLE IX R ESULTS FOR POWER CONTROL CASE .
Size 5 10 20
error av 7.75 6.73 1.52
[%] max 15.91 11.01 2.85
Initial set T [s] av max 0.01 0.01 0.02 0.02 0.69 1.01
imp [%] av 7.75 6.70 1.52
Final error [%] av max 0 0 0.02 0.1 0.01 0.04
set T Tot. [s] av max 0.06 0.1 10.8 20.4 -
TABLE X R ESULTS FOR POWER AND RATE CONTROL CASE .
Size 5 10
error av 684 664
[%] max 767 739
Initial set T [s] av max 0.01 0.01 0.10 0.11
the initial integer solution with respect to the lower bound IH−LB are given and the related computational times, LB average and maximum, are reported in columns four and five. The sixth column gives the average error of the initial master problem solution with respect to the final lower bound, IM P R−LB , i.e. the improvement in the solution of the conLB tinuous problem P obtained by column generation procedure. Columns seven and eight give the average and maximum error of the final heuristic solution with respect to the final lower FR , while the last two columns report the bound M FM−M FR total computational time, average and maximum. Table VIII shows that, in case of fixed power optimization, the final heuristic solution is always the optimal one; in fact, it reaches the same number of needed slots provided by the continuous relaxation. Furthermore, even the initial heuristic solution provides good results as its average error with respect to the bound is always below 1% and it is 1.69% in the worst case. The column generation procedure improves only slightly upon the initial continuous solution: the initial set L0 is very close to the final variables set. The computational times are negligible for small size instances but increase with instances sizes. Table IX shows the behavior of the algorithm in case of power control. As the problem becomes more difficult, the computational effort increases and the time limit, set to four hour, is exceeded even for instances with 20 devices. For instances with five and ten nodes the final heuristic provides goods results, as its maximum error is at most 0.1%. For five nodes instances the final heuristic solution is the optimal one. Instead, the initial set does not provide good results, the maximum error rising up to about 16%. Thus, the
imp [%] av 684 664
Final set error [%] T Tot. [s] av max av max 48 64 0.89 2.05 46 59 4798 10827
column generation procedure must add a significant number of variables and the number of pricing iteration increases. Due to the increased number of pricing iterations and to the fact that the pricing problem is more difficult to solve than the corresponding fixed power pricing, the total computational time increases and requires more than four hours even for 20 devices instances. For such instances the procedure has been stopped before reaching the continuous relaxation optimum. Thus the comparison is done with the last computed solution of the master problem. Nevertheless, the algorithm provides heuristic solutions that can be implemented. Finally, Table X is devoted to the results related to power and rate control instances. As the problem becomes more and more difficult, the computational time increases and results for 20 nodes instances are omitted as the procedure is stopped too early to provide significant results. Results for five and ten nodes instances show that the performance of the algorithms get worse both concerning the initial and the final configurations set. Nevertheless, the column generation procedure provides a significant improvement with respect to the initial set solution. Further, the whole approach allows to compute feasible integer solution in reasonable CPU time even for 20 or 30 nodes instances, by solving the integer problem P over the variables set L0 . VI. C ONCLUSION In this paper we study the joint scheduling, power control and rate control problem in wireless MESH networks where traffic engineering methodologies able to provide bandwidth guarantees to traffic flows and to optimize transmission resources utilization are of paramount importance.
For each link of the network we considers traffic constraints defining the number of packets that must be transmitted per frame. The number of packets transmitted per time slot is selected according to a discrete set of possible transmission rates, based on the SINR at receivers. Data rates and corresponding SINR values are an input of the problem and no a priori model of the relation between the data rate and the SINR is considered. The goal of the optimization problem is to find the scheduling settings that minimize the number of used time slots. This approach provides a compact time-slot scheduling that allows to simplify admission control procedures that need to estimate spare bandwidth when a new traffic flow must be added to the network. We consider two scheduling problem formulations, the first based a linear number of decision variables, and the second with an exponential number of variables that define feasible sets of links over which parallel transmissions can safely occur. Since the number of variables is huge even with small instances we propose a solution approach based on column generation: starting from an initial set of variables the continuous relaxation is solved over the initial set to the optimum and then improving additional variables are added, which are generated by solving the pricing problem. The described approach has been tested on randomly generated instances. Numerical results shows that the column generation approach can solve small size instances while for bigger problems solving the pricing problem to optimality becomes difficult and therefore the whole approach becomes slow. However column generation provides good bounds and often the column generation solution is an integer one. The heuristic solution obtained solving the problem over the initial set L0 provides good solutions in reasonable CPU time. We are currently extending the models presented in this paper to consider also routing optimization. Moreover, we are also modelling multi-radio devices that require to include frequency assignment in the optimization process. R EFERENCES [1] I.F. Akyildiz, X. Wang, and W. Wang. Wireless mesh networks: a survey. Computer Networks, vol. 47, no. 4, pages 445-487, March 2005. [2] I.F. Akyildiz and X. Wang. A survey on wireless mesh networks. IEEE Communications Magazine, vol. 43, no. 9, pages 23-30, Sept. 2005. [3] R. Bruno, M. Conti, and E. Gregori. Mesh networks: Commodity multihop ad hoc networks. IEEE Communications Magazine, vol. 43, no. 3, pages 123-131, March 2005. [4] C. Eklund, R.B. Marks, K.L. Stanwood, S. Wang. IEEE standard 802.16: a technical overview of the WirelessMAN air interface for broadband wireless access. IEEE Communications Magazine, vol 40, no. 6, pages 98–107, June 2002. [5] R. Pabst, B. H. Walke, D. C. Schultz, P. Herhold, H. Yanikomeroglu, S. Mukherjeee, H. Viswanathan, M. Lott, W. Zirwas, M. Dohler, H. Aghvami, D. D. Falconer, and G. P. Fettweis. Relay-Based Deployment Concepts for Wireless and Mobile Broadband Radio. IEEE Communications Magazine, vol. 42, no. 9, pages 80–89, Sept. 2004. [6] H. Abramowicz et al. The Wireless World Initiative: A Framework for Research on Systems Beyond 3G, In Proc. of IST Mobile and Wireless Communications Summit, Lyon, France, June 2004.
[7] S. Ramanathan and E. L. Lloyd. Scheduling algorithms for multihop radio networks. IEEE/ACM Trans. on Networking, vol. 1, no. 2, pages 166-177, April 1993. [8] P. Gupta and P. R. Kumar. The capacity of wireless networks. IEEE Trans. on Information Theory, vol. 46, no. 2, pages 388-404, March 2000. [9] P. Bjrklund, P. Vrbrand, Di Yuan. A column generation method for spatial TDMA scheduling in ad hoc networks. Ad hoc networks, vol. 2, no. 4, pages 405–418, October 2004. [10] I. Cidon and M. Sidi. Distributed assignment algorithms for multihop packet radio networks. IEEE Trans. Comput., vol. 38, no. 10, pages 1353-1361, October 1989. [11] A. Ephremides and T. Truong. Scheduling broadcasts in multihop radio networks. IEEE Trans. on Commun., vol. 38, no. 4, pages 456-460, April 1990. [12] C. Zhu and M. S. Corson. A five-phase reservation protocol (FPRP) for mobile ad hoc networks. Wireless Networks, vol. 7, no. 4, pages 371-384, July 2001. [13] J. L. Hammond and H. B. Russell. Properties of a transmission assignment algorithm for multiple-hop packet radio networks. IEEE Trans. on Wireless Communications, vol. 3, no. 4, pages 1048-1052, July 2004. [14] S. Ramanathan. A unified framework and algorithm for channel assignment in wireless networks. Wireless Networks, vol. 5, pages 81-94, 1999. [15] M. Alicherry, R. Bhatia, Li Li. Joint Channel Assignment and Routing for Throughput Optimization in Multiradio Wireless Mesh Networks. In Proc. of ACM Mobicom 2005. [16] I. Chlamtac and A. Farag‘o. Making transmission schedules immune to topology changes in multihop packet radio networks. IEEE/ACM Trans. on Networking, vol. 2, no. 1, pages 23-29, February 1994. [17] J.-H. Ju and V. O. K. Li. An optimal topology-transparent scheduling method in multihop packet radio networks. IEEE/ACM Trans. on Networking, vol. 6, no. 3, pages 298-306, June 1998. [18] A. Behzad and I. Rubin. On the Performance of Graph-based Scheduling Algorithms for Packet Radio Networks. In Proc. of IEEE GLOBECOM 2003. [19] N. D. Bambos, S. C. Chen, and G. J. Pottie. Radio link admission algorithms for wireless networks with power control and active link quality protection. In Proc. of IEEE INFOCOM, 1995. [20] T. ElBatt and A. Ephremides. Joint scheduling and power control for wireless ad hoc networks. IEEE Trans. on Wireless Communications, vol. 3, no. 1, pages 74-85, January 2004. [21] A. Behzad and I. Rubin. Multiple access protocol for power-controlled wireless access nets. IEEE Trans. on Mobile Computing, vol. 3, no. 4, pages 307-316, October-December 2004. [22] Y. Li and A. Ephremides. Joint scheduling, power control and routing algorithm for ad-hoc wireless networks. In Proceedings of the 38th Annual Hawaii International Conference on System Sciences, 2005. [23] A. Behzad and I. Rubin. Impact of Power Control on the Performance of Ad Hoc Wireless Networks. In Proc. of IEEE INFOCOM 2005. [24] R. L. Cruz and A. V. Santhanam. Optimal routing, link scheduling and power control in multi-hop wireless networks. In Proc. of IEEE INFOCOM, 2003. [25] J. W. Lee, R. R. Mazumdar, and N. B. Shroff. Opportunistic resource scheduling for wireless ad-hoc networks. In Proc. of BroadWISE04, October 2004. [26] G. Kulkarni, V. Raghunathan, and M. Srivastava. Joint end-to-end scheduling, power control and rate control in multi-hop wireless networks. In Proc. of IEEE GLOBECOM, 2004. [27] U. C. Kozat, I. Koutsopoulos, and L. Tassiulas. A framework for crosslayer design of energy-efficient communication with QoS provisioning in multi-hop wireless networks. In Proc. of IEEE INFOCOM, 2004. [28] R. Bhatia, M. Kodialam. On power efficient communication over multihop wireless networks: joint routing, scheduling, and power control. In Proc. of IEEE INFOCOM, 2004. [29] A. Penttinen, J. Virtamo and R. Jntti, Performance analysis in multi-hop radio networks with balanced fair resource sharing, Telecommunications Systems, vol. 31, no. 4, pp. 315-336, 2006 [30] ILOG CPLEX 8.0 user’s manual, 2002. [31] R. Fourer, D. M. Gay, and B. W. Kernighan, AMPL A modeling language for mathematical programming, 1993.
