SCHEDULING the wireless links and controlling their

Transcription

1 3738 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 7, JULY 2014 Minimum Length Scheduing With Packet Traffic Demands in Wireess Ad Hoc Networks Yacin Sadi, Member, IEEE, and Sinem Coeri Ergen, Member, IEEE Abstract Traditiona approach to the minimum ength scheduing probem ignores the packet eve detais of transmission protocos, meaning that a packet transmission can be divided into severa data chunks each of which is transmitted at a different rate due to the difference in the set of concurrenty transmitting nodes. This soution requires incuding packet headers for each data chunk resuting in both an increase in the system overhead and underutiization of the time sots. In this paper, we extend the previous works on minimum ength scheduing by considering the transmission of the packets of arbitrary sizes in the time sots of arbitrary engths. Given the packet traffic demands on the inks, we formuate the joint optimization of the power contro, rate adaptation and scheduing for minimizing the schedue ength of a wireess ad hoc network and demonstrate the hardness of this probem. Upon soving the power contro and rate adaptation probem separatey, we formuate the scheduing probem as an integer programming (IP) probem where the number of variabes is exponentia in the number of the inks. In order to sove this arge-scae IP probem fast and efficienty, we propose Branch and Price Method and Coumn Generation Method based heuristic agorithms. Index Terms Wireess ad hoc networks, packet traffic demand, power contro, rate adaptation, scheduing. I. INTRODUCTION SCHEDULING the wireess inks and controing their transmission power and rate in Spatia-reuse Time Division Mutipe Access (STDMA) wireess networks has been investigated to minimize the detrimenta effects of the interference for various objectives incuding maximizing tota throughput [1], [2], maximizing minimum throughput [3], [4], minimizing tota transmit power [5], [6] and minimizing schedue ength [7] [22]. Among these objectives, minimizing schedue ength given the traffic demands of the inks is increasingy used with the proiferation of the time critica appications of wireess networks such as rea-time surveiance, networked contro [23]. Moreover, minimum ength schedue determines the traffic-carrying capabiity of the wireess network since minimization of the tota time required for a set of inks to transmit their data packets is effectivey maximizing the tota Manuscript received May 7, 2013; revised December 22, 2013 and March 15, 2014; accepted Apri 6, Date of pubication Apri 17, 2014; date of current version Juy 8, This work was supported in part by Marie Curie Reintegration under Grant IVWSN, PIRG06-GA and by The Scientific and Technoogica Research Counci of Turkey under Grant 113E233. The associate editor coordinating the review of this paper and approving it for pubication was G. Xing. The authors are with the Department of Eectrica and Eectronics Engineering, Koc University, Istanbu, Turkey (e-mai: ysadi@ku.edu.tr; sergen@ku.edu.tr). Coor versions of one or more of the figures in this paper are avaiabe onine at Digita Object Identifier /TWC data transmission rate whie providing fairness among the inks, which are aocated enough time to transmit their packets. Earier work on minimum ength scheduing adopts Protoco Interference Mode, which describes the interference constraints among the active inks according to a confict graph, where nodes within a certain distance can communicate as ong as the receiver is separated by at east a fixed distance from any other active transmitter. Since this interference mode does not aow controing the transmission power and rate of the inks due to the predetermined confict graph, the time is partitioned into sots of fixed ength aowing the transmission of a fixed ength packet. The agorithms deveoped for minimum ength scheduing using Protoco Interference Mode aim to satisfy either the uniform traffic demand of the inks activating each ink for at east one time sot during the schedue [7], [8] or nonuniform traffic demand considering many-to-one transmission activating each ink for mutipe time sots during the schedue [9], [10]. Athough the scheduing agorithms deveoped for Protoco Interference mode take into account the packetized transmission aocating one packet transmission to each time sot, this mode is inadequate in modeing today s radios where mutipe power and rate eves are supported. Besides, this interference mode does not take into account the cumuative effects of the interference due to simutaneous transmissions thus requiring the overcompensation of the interference by using a arge distance of separation between the active transmitters and receivers. Recenty, Physica Interference Mode that considers the cumuative effect of the interference in the form of Signa-to- Interference-pus-Noise Ratio (SINR) and aows incuding the transmission power and rate of the inks as a variabe in minimizing the ength of the schedue has gained wider acceptance. Despite the fexibiity introduced by this interference mode to incude variabe transmission power and rate, some of the agorithms deveoped for this optimization probem restricts the partition of the time into sots of fixed ength. The ength of the time sot corresponds to the transmission time of either a fixed ength packet at fixed transmission rate [12], [13] or an integer number of packets the vaue of which is determined as a function of the rate chosen from a finite set of possibe transmission rates [14], [15]. These agorithms generate suboptima soutions since the fixed ength time sots may be underutiized by the inks. The agorithms that partition the time into sots of variabe ength on the other hand ignore the packet eve detais of the transmission protocos by aocating any amount of data to the time sots with the ony goa of satisfying the tota traffic demand of each node [16] [22]. This means that in the optima soution, a packet transmission can be IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

2 SADI AND ERGEN: MINIMUM LENGTH SCHEDULING WITH PACKET TRAFFIC DEMANDS 3739 Fig. 1. Soution to the traditiona formuation of minimizing the ength of the schedue for the scenario where each of 4 inks has one packet of 100 bits to transmit; the inks are of 1 m ength and uniformy distributed over an area of 15 m 15 m; the inks can ony transmit at discrete rates 1, 2, 3 Mbps if the SINR vaue at the receiver of the ink exceeds the SINR eve 10, 20, 30 db respectivey [22]. Time axis vaues are T 0=0μs, T 1= μs, T 2= μs, T 3= μs andt 4 = 100 μs. divided into severa data chunks each of which is transmitted at different rate due to the difference in the set of concurrenty transmitting nodes. Fig. 1 shows the soution to the probem of minimizing the ength of the schedue formuated in [22] for a scenario where each ink has one packet to transmit. Since the traffic demand in this probem is specified as the tota number of bits ignoring packet eve detais, in the optima soution, a the packet transmissions are transmitted in severa data chunks requiring interrupting the packet transmission. For instance, packet transmission of ink 1 starts at T 0 but is interrupted at T 1 and restarts at T 3. Moreover, the data rate of the inks is different for each data chunk due to the different set of nodes concurrenty transmitting in each time interva. For instance, the data rate of ink 1 is different in T 0 T 1 and T 3 T 4 intervas since ink 2 is transmitting concurrenty with ink 1 in T 0 T 1 interva whereas ink 4 is transmitting concurrenty in T 3 T 4 interva. The data chunks of the same packet must therefore be transmitted as individua packets by incuding a packet header at the beginning in the impementation. This additiona packet header increases the system overhead. Moreover, since the data rates of the inks aocated to the same time sot are different, the transmission time of these headers are different for each ink resuting in the underutiization of the time sots. The goa of this paper is to determine the optima power contro, rate adaptation and scheduing with the objective of minimizing the schedue ength given the packet traffic demands of the inks and the constraints of packet transmissions in a wireess ad hoc network. We first formuate the joint optimization of the power contro, rate adaptation and scheduing for minimizing the schedue ength of a wireess ad hoc network and demonstrate the hardness of this probem. We then show that the power contro and rate adaptation probem can be separated from the scheduing probem and introduce a nove probem for the optimization of power contro and rate adaptation where the time required for the concurrent transmission of a set of inks each having an integer number of packets is minimized. This probem determines the ength of the time sot corresponding to this ink set and the corresponding number of packets to be transmitted during the sot. Soving this power contro and rate adaptation probem for a possibe subsets of the inks in the network transmitting any possibe number of packets then aows formuating the probem of minimizing the schedue ength as an IP probem. The resuting IP probem is arge-scae with the number of variabes exponentia in the number of the inks. We therefore incorporate eegant Branch and Price Method and Coumn Generation method based heuristic agorithms to sove this IP probem. Coumn generation method has been previousy used for the minimum ength scheduing probem in different contexts: fixed transmission rate and variabe power [16] and variabe transmission rate and variabe power [19] [22] but without considering packetized transmission; variabe transmission rate and variabe power but considering ony a imited set of packet transmission scenarios for fixed ength time sots [14], [15]. In this paper, we extend these previous works on minimum ength scheduing probem by considering the transmission of the packets of arbitrary sizes in the time sots of arbitrary engths in the optimization of power contro, rate adaptation and scheduing. The rest of the paper is organized as foows. Section II describes the system mode and the assumptions used throughout the paper. In Section III, the joint optimization of power contro, rate adaptation and scheduing with the objective of minimizing the schedue ength is formuated considering the constraints of packet transmissions for constant, continuous and discrete rate transmission modes and the soution strategy based on the decomposition of the power contro and rate adaptation, and scheduing probems is described. Section IV presents the optima power contro and rate adaptation probem with the goa of determining the minimum time-sot ength for the concurrent packet transmissions of a subset of the inks in the network and propose both optima and heuristic agorithms for each transmission mode. Section V formuates the optima scheduing probem as an IP probem where the number of variabes is exponentia in the number of the inks whereas Sections VI and VII introduce Branch and Price and Coumn Generation Method based heuristic agorithms respectivey to sove this IP probem fast and efficienty. Simuations and

3 3740 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 7, JULY 2014 performance evauation are presented in Section VIII. Finay, concuding remarks and future work are given in Section IX. II. SYSTEM MODEL AND ASSUMPTIONS The system mode and assumptions are detaied as foows: 1) The wireess ad hoc network consists of a set of L directed inks with positive traffic demand. 2) A centra controer executes the agorithm for the joint optimization of power contro, rate adaptation and scheduing based on the avaiabe information on the network topoogy, traffic demands and channe characteristics of the inks. This centraized framework is appropriate for the communication of the routers in wireess mesh networks (WMNs) that have emerged recenty to partiay repace the costy wired network infrastructures [24]. The changes in WMN topoogy is quite infrequent since the ocations of the WMN routers are fixed. Moreover, the bandwidth requests of the WMN inks are typicay sowy varying over time and can be guaranteed with an amost static resource aocation. Besides, the centraized optima soution provides an upper bound on the performance of any distributed agorithm since distributed agorithms rey on the oca topoogy information in contrast to the centraized agorithms using the knowedge of the compete topoogy. Such an upper bound aows evauating the performance of the distributed agorithms. Introducing distributed agorithms within the proposed framework is out of scope of this paper and subject to future work. 3) Link has traffic demand of G packets of ength R bits. Athough the mathematica formuations derived in the paper aow having packets of different engths for the same ink, the formuations are easier to express with this assumption. The ink traffic demands can be either given for singe-hop networks or cacuated by using end-toend traffic demands in a muti-hop network with predetermined routing. The proposed framework can be extended to incude routing in the optimization probem by either a two phase approach in which scheduing and routing are soved iterativey [25], [26] or a joint approach in which routing is incuded via fow-baance equations that map the end-to-end fow demands to the ink traffic demands [3], [4], [14]. To avoid compexity in the first step of the study and better focus on the packetized scheduing, the incusion of the routing in the optimization probem is out of scope of this paper and subject to future work. 4) We consider Time Division Mutipe Access (TDMA) as MAC protoco since TDMA provides deay guarantee to the networks with predetermined topoogy and data generation patterns [27]. The time is partitioned into frames, which are further partitioned into sots of possiby variabe engths. Two possibe interpretations can be considered for studying the probem of minimum ength scheduing given traffic demands as detaied in [22]: In the first scenario, the traffic demand, i.e., G packets for ink, is defined as the number of packets to be transmitted in every frame. On the other hand, in the second scenario, the traffic demand is defined as the ong-run average data transfer rate of G s packets per second. This can then be accompished by dividing the time axis into frames of arbitrary positive constant duration T requiring the satisfaction of G = G s T packet transmissions for every ink during each frame. Minimizing the frame ength given these traffic demands is then usefu for both scenarios because it permits a arger number of frames per unit time. Extending this work for the dynamic adaptation of the schedue, power and rate of the inks to the arriva of packets at any instant in time is beyond the scope of this paper and subject to future work. 5) A node cannot receive and transmit simutaneousy. Aso, a node cannot receive from or transmit to more than one node simutaneousy. 6) The transmit power can take any vaue beow a maximum eve p max. Athough the radios are restricted to support a finite number of transmit power eves in practice, the assumption of continuous power is commony used in most of the previous work such as [12], [16], [20] [22] due to the simpification of the resuting probem and high approximation accuracy with the arge number of discrete power eves that can be supported by the existing radios. 7) Let g and g k be the channe gain of ink and the channe gain from the transmitter of ink k to the receiver of ink respectivey. We assume that the fading is sow such that the channe gain between every transmitter and receiver is fixed during the schedue. This is a common assumption used in the prior formuations of the minimum ength scheduing probem in wireess ad hoc networks [12], [16], [17], [20] [22]. Extending this work for fast fading wireess networks by either minimizing the mean vaue of the schedue ength as in [19] or providing an optima poicy by empoying the principes of dynamic programming whie incorporating a ot of overhead for coecting the channe information at the granuarity of the time sot duration as in [18] is beyond the scope of this paper and subject to future work. 8) First transmission mode used in the formuations is constant rate transmission mode in which each active ink assigned to a time sot n transmits at constant rate r if the SINR of ink is above a fixed threshod as p (n) g N 0 + k p(n) k g β (1) k where p (n) is the transmit power of ink in time sot n, N 0 is the background noise and β is the SINR threshod to be kept by ink [12], [16]. 9) Second transmission mode used in the formuations is continuous rate transmission mode in which Shannon s channe capacity formuation for an Additive White Gaussian Noise (AWGN) wireess channe is used in the cacuation of the maximum achievabe rate as a function of SINR as x (n) W og ( 1+ 1 W p (n) g N 0 + k p(n) k g k ) (2)

4 SADI AND ERGEN: MINIMUM LENGTH SCHEDULING WITH PACKET TRAFFIC DEMANDS 3741 where x (n) is the transmission rate of ink in time sot n, W is the channe bandwidth [19] [22], [28], [29]. 10) Third transmission mode used in the formuations is discrete rate transmission mode in which a finite set of transmit rates r =(r 1,r 2,...,r M ) and a finite set of SINR eves γ =(γ 1,γ 2,...,γ M ) are determined such that ink can transmit at rate r c in a time sot n if the SINR achieved at the ink γ (n) = p (n) g N 0 + k p(n) k g k is greater than or equa to γ c from the set γ [17], [18]. Here r 1 =0and γ 1 = db so that if γ (n) is beow the threshod γ 2, the transmission rate is 0. This is in genera more reaistic mode than the continuous rate mode since a wireess transmitter can ony work with a imited number of rate eves as suggested by the practica reaization of mutipe data rates in [30]. III. PACKETIZED MINIMUM LENGTH SCHEDULE PROBLEM A. Probem Formuation The joint optimization of power contro, rate adaptation and scheduing with the objective of minimizing the schedue ength given the packet traffic demands and packetized transmission constraints is mathematicay formuated for constant rate, continuous rate and discrete rate transmission modes. 1) Constant Rate Transmission Mode: The formuation of the optimization probem for constant rate transmission mode is as foows: minimize (3) N t (n) (4) n=1 subject to x (n) t (n) α (n) R, [1,L],n [1,N] (5) N α (n) G, [1,L] (6) p (n) p (n) g n=1 N 0 + k p(n) k g k x (n) p max, [1,L],n [1,N] (7) =1 { p (n) β 1 { } p (n) >0, [1,L],n [1,N] >0 (8) } r, [1,L],n [1,N] (9) a k +1 { } p (n) >0 +1 } {p 2,,k [1,L],n [1,N] (n) k >0 (10) variabes t (n) 0, p (n) 0, x (n) 0, α (n) [0,G ], [1,L], n [1,N] (11) where N is the number of time sots, a k is a constant that takes the vaue 1 if inks and k share a common node and 0 otherwise. The variabes of the optimization probem are t (n), the ength of the n-th time sot; p (n), the transmit power of ink in time sot n; x (n), the transmission rate of ink in time sot n and α (n), the number of packets sent by ink in time sot n. The goa of the optimization probem is to minimize the ength of the schedue. Equation (5) represents the packetized transmission requirement where each node transmits an integer number of packets within each time sot. Equation (6) gives the constraint on the tota number of packets generated at each node. Note that in the case where the packetized transmission within each time sot is not taken into account, (5) and (6) are repaced by one equation ony representing the tota number of bits that needs to be transmitted by each node as N n=1 x(n) t (n) G R. Equation (7) states the upper bound for the transmit power of the inks. Equation (8) represents the condition that the SINR of ink is above a fixed threshod if it is active. Equation (9) represents the transmission at fixed rate r for the active inks. Equation (10) states that any node in the network cannot transmit to and receive from more than one node simutaneousy. This optimization probem is a Mixed Integer Non-Linear Programming probem thus difficut to sove for the goba optimum [31]. We prove the hardness of the probem next. Theorem 1: The packetized minimum ength schedue probem for constant rate transmission mode is NP-hard. Proof: The packetized minimum ength schedue probem as formuated by (4) (11) for constant rate transmission mode is equivaent to the integrated ink scheduing and power contro probem in [12], which is shown to be NP-hard, for the instance in which the packet ength of a the inks in the network is the same. 2) Continuous Rate Transmission Mode: The formuation of the optimization probem for continuous rate transmission mode is very simiar to that formuated for constant rate transmission mode except that (8) and (9) are repaced by the foowing constraint: x (n) W og ( ) 1+ 1 W p (n) g N 0 + k p(n) k g, k [1,L],n [1,N] (12) This probem is a non-convex optimization probem for which there is no known poynomia time agorithm [31]. 3) Discrete Rate Transmission Mode: The formuation of the optimization probem for discrete rate transmission mode is aso very simiar to that formuated for constant rate transmission mode except that (8) and (9) are repaced by the foowing constraints: p (n) M i=1 x (n) = g z (n) i γ i N 0 + k p (n) k g k 0, [1,L], n [1,N],i [1,M] (13) z (n) i =1, [1,L],n [1,N] (14) M i=1 z (n) i r i, [1,L],n [1,N] (15)

5 3742 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 7, JULY 2014 and the incusion of the variabe z (n) i {0, 1}, where [1,L], n [1,N], i [1,M], that takes vaue 1 if ink transmits at rate r i in time sot n and 0 otherwise. Equation (13) states that the assigned rate r i to ink satisfies the corresponding SINR requirement as γ (n) γ i. Equations (14) and (15) state that each ink is assigned to exacty one rate. Coroary 1: The packetized minimum ength schedue probem for discrete rate transmission mode is NP-hard. Proof: Discrete rate transmission mode is equivaent to constant rate transmission mode for the instance in which M =2. Since the packetized minimum ength schedue probem for constant rate transmission mode is NP-hard based on Theorem 1, the packetized minimum ength schedue probem for discrete rate transmission mode is NP-hard. B. Soution Strategy The exponentia compexity of the packetized minimum ength schedue probem for different transmission modes necessitates the use of heuristic agorithms. The soution strategy proposed in this paper is based on the decomposition of the power contro and rate adaptation, and scheduing probems as described beow: The constraints of the joint power contro, rate adaptation and scheduing probem provided in equations (5) (15) incude the variabes for the transmit power and rate of the inks and the sot ength corresponding to ony one time sot at a time except the constraint on the tota number of packets generated at each node given in (6). Therefore, given the number of packets to be transmitted by each ink, the ength of the time sot shoud be minimized independent of the other sots in order to attain minimum tota schedue ength. A nove power contro and rate adaptation probem formuated as the minimization of the time sot ength required for the concurrent transmission of a set of simutaneousy active inks each transmitting an integer number of packets is formuated and soved for constant rate, continuous rate and discrete rate transmission modes. The optima scheduing contains the time sots each of which corresponds to a subset of the inks with each ink in the subset sending a certain number of packets. Therefore, given the minimum time sot engths corresponding to a the subset of the inks concurrenty transmitting any possibe number of packets, cacuated by using the formuation of the optima power contro and rate adaptation probem described previousy, the scheduing probem is formuated as an Integer Programming (IP) probem where the number of variabes is exponentia in the number of the inks. To sove the arge-scae IP formuation for the scheduing probem, Branch and Price Method and Coumn Generation Method based heuristics are proposed. IV. POWER CONTROL AND RATE ADAPTATION PROBLEM The power contro and rate adaptation probem aims to minimize the ength of the time sot given the number of packets to be transmitted by each active ink in that time sot. Given the power aocation of the inks in a time sot, the feasibe and achievabe rate region for a ink given in equations (9), (12) and (15) for constant, continuous and discrete rate transmission modes respectivey is independent of the rate of the concurrenty transmitting inks. Since higher rate resuts in smaer transmission deay, the optima transmission rate is the maximum achievabe rate for the minimization of the time sot ength. Power contro and rate adaptation probem can therefore be reduced to a pure power contro probem where the transmission rates of the inks are formuated as a function of the transmit powers of the inks. Let us define the set of active inks in a time sot S. We assume that none of the inks in the set S have common end point to satisfy the requirement given in (10). Let t be the ength of the time sot to be aocated to the transmission of α packets by each ink within the set S. The objective of the optima power contro probem under a transmission modes is to minimize t. We remove the superscript of the variabes, which represent the time sot in Section III, to simpify the expressions in this section. A. Constant Rate Transmission Mode The optima power contro probem for constant rate transmission mode is formuated as minimize subject to variabes t (16) α R t, x S (17) p p max, S (18) p g N 0 + k p β, kg k S (19) x = r, S (20) t 0, p 0, S (21) The constraints given in equations (17) (20) correspond to the constraints given in (5), (7), (8), and (9), respectivey. The power contro probem for constant rate transmission mode first requires testing the feasibiity of the concurrent transmission of the inks in S. For the feasibiity of the concurrent transmission of the inks in S, there shoud exist a set of transmit power aocations to the inks in the set S such that the SINR constraint of each ink S as given in (19) is satisfied whie meeting the maximum transmit power constraint in (18). The existence of such a set of transmit power aocations is determined by testing Perron-Frobenius conditions [32] described as foows: Let B S be S S reative channe gain matrix such that the eement in the -th row and k-th coumn of B S takes vaue g k /g for k and 0 for = k. LetD S be S S diagona matrix with the -th diagona entry equa to β.letv S be S 1 normaized noise power vector with the

6 SADI AND ERGEN: MINIMUM LENGTH SCHEDULING WITH PACKET TRAFFIC DEMANDS th eement equa to β N 0 /g. Perron-Frobenius conditions state that there exists a feasibe power vector if and ony if the argest rea eigenvaue of D S B S is ess than 1 and every eement of the component-wise minimum power vector (I D S B S ) 1 V S is ess than or equa to p max. If the concurrent transmission of the inks within the set S is feasibe, then the optima vaue of t is equa to max S (α R /r). Since the compexity of evauating Perron-Frobenius conditions for S inks is O( S 3 )[32], the compexity of soving the power contro probem for constant rate transmission mode is O( S 3 ). B. Continuous Rate Transmission Mode The optima power contro probem for continuous rate transmission mode is very simiar to that formuated for constant rate transmission mode except equations (19) and (20) are repaced by the constraint given in equation (12) for the inks in the set S. This power contro probem is a non-convex optimization probem for which there is no known poynomia time agorithm [31]. In the foowing, we first state the theorem on the equaity of the optima time duration required for the transmission of a the inks assigned to the same time sot then provide an iterative agorithm that gets cose to the optima time sot ength within a certain accuracy exponentiay fast. Theorem 2: Let t denote the optima ength of the time sot aocated to the transmission of α packets by each ink within the set S such that t =max S t where t is the transmission time required for ink. Then, the transmission times of a the inks within S are equa; i.e., t = t S. Proof: We wi prove the theorem by contradiction. Suppose that t t for an arbitrary S. Lett k =max j S t j. Then t k >t. If we decrease the transmission power of ink by an arbitrariy sma amount such that t is sti ess than t k, the transmission time of a the inks except ink decreases due to the decreasing amount of interference created by node.note that the transmission rate of a ink j is an increasing function of the transmission power of ink j and decreasing function of the transmission power of ink i j. As a resut, time sot ength required for the transmission of a the inks, t =max j S t j decreases. Hence, as ong as there exists a node j such that t j < max S t, we can improve the soution by decreasing the transmission power of ink j. Hence, at the optima soution, the transmission time of every ink in the set S must be equa. If we fix the vaue of the ength of the aocated time sot, the optima power contro probem reduces to determining whether there exists a set of transmit power assignment to the inks in the set S such that the SINR constraint of each ink S as given in p g N 0 + k,k S p β (22) kg k where β = W ( ) 2 α R /(Wt) 1 (23) is satisfied whie meeting the maximum transmit power constraint in (18). The existence of such a set of transmit power vector is determined by testing Perron-Frobenius conditions as expained in Section IV-A. Fast-Iterative Power Contro Agorithm (FIPCA) is based on performing an inteigent search for the minimum feasibe vaue of t. FIPCA agorithm determines an interva for the optima vaue of the ength of the time sot corresponding to the transmission of α packets by each ink within the feasibe set S and then reduces the ength of this interva by haf in each iteration. The ower and upper end points of this interva determined as the owest infeasibe vaue and highest feasibe vaue for the time sot ength t are denoted by t w and t up respectivey. The agorithm starts by setting the ower bound to the maximum vaue of the time sot engths required when ony one ink transmits at maximum transmit power p max and upper bound to the ength of the time sot when every ink S transmits at maximum transmit power p max (Lines 1 2). In each iteration, the agorithm checks the feasibiity of the midde point of the interva (t w,t up ) (Lines 4 5). If the vaue t =(t w + t up )/2 is feasibe then any vaue greater than t is a worse feasibe soution than t thus requiring the search for the optima time sot over interva (t w, (t w + t up )/2) by updating the upper bound (Lines 6 7). On the other hand, if the vaue t =(t w + t up )/2 is infeasibe then any vaue ess than t is aso infeasibe requiring the search for the optima time sot over interva ((t w + t up )/2,t up ) by updating the ower bound (Lines 8 9). The termination criteria of the agorithm is achieving a predetermined reative error bound eve represented by ɛ (Line 3). The vaue of the time sot ength is then set to t up (Line 12). Since the optima time sot ength t opt is between t w and t up, the reative error bound of the optima time sot ength given by (t up t opt )/t opt is ess than (t up t w )/t w so ess than ɛ meaning that the vaue picked by the agorithm t up ies in the interva [t opt, (1 + ɛ)t opt ]. Agorithm 1 Fast Iterative Power Contro Agorithm (FIPCA) 1: t w =max S (α R /(W og(1+(1/w ) (p max g /N 0 ))); 2: t up =max S (α R /(W og(1+(1/w ) (p max g /(N 0 + k,k S p maxg k )))) 3: whie ((t up t w )/t w ) >ɛdo 4: t =(t up + t w )/2; 5: check feasibiity for time-sot ength t; 6: if t is feasibe then 7: t up = t; 8: ese 9: t w = t; 10: end if 11: end whie 12: t = t up ; Lemma 1: For a fixed predetermined reative error bound eve ɛ, the compexity of the FIPCA agorithm is O(K S 3 ), where K = og((t up init tw init )/ɛtw init ), tup init and tw init are the initia vaues of the upper and ower bounds for the optima time sot ength set by the agorithm (Lines 1 2).

7 3744 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 7, JULY 2014 Proof: The FIPCA agorithm reduces the initia ratio (t up init tw init )/tw init by a factor of at east two in each iteration since t up t w is reduced by haf by setting either t up or t w to the midde point of the interva [t up,t w ] (Lines 4 9). Hence, the number of iterations required to decrease the ratio (t up t w )/t w init )/ɛtw init ). Since the agorithm checks the feasibiity of the existence of a transmit power vector satisfying SINR and maximum power constraints in each iteration evauating the Perron-Frobenius conditions to ɛ is K = og((t up init tw (Line 5) of O( S 3 ) compexity, the compexity of the agorithm is O(K S 3 ). C. Discrete Rate Transmission Mode The optima power contro probem for discrete rate transmission mode is very simiar to that formuated for constant rate transmission mode except Equations (19) and (20) are repaced by the constraints given in Equations (13) (15) for the inks in the set S. This power contro probem is a Mixed Integer Non-inear Programming thus difficut to sove for the goba optimum [31]. Note that unike the continuous rate transmission mode, the optima time duration required for the transmission of the inks assigned to the same time sot are not necessariy equa in this case due to the discrete nature of the rates. A straightforward search agorithm enumerates a possibe rate aocations of the inks in the set S and check their feasibiity by testing the existence of a power vector satisfying the SINR requirements corresponding to these rate aocations and maximum power constraints by evauating Perron-Frobenius conditions. The optima soution is then the minimum of the time sot engths of the feasibe rate aocations. The compexity of this search is O(M S S 3 ). We wi now propose a fast fathoming-based smart enumeration agorithm based on the foowing two emmas. In this agorithm, we use a tree structure for the rate index vector x r =(i 1,i 2,...,i S ) where the inks are enumerated from 1 to S and the j-th ink is assigned to the i j -th rate eve where i j [1,M]. The root of the tree is (M,M,...,M). The chidren of the rate index vector x r =(i 1,i 2,...,i S ) are obtained by updating the j-th eement of x r as i j 1 and keeping the remaining eements the same for every j [1, S ] such that i j > 2. A rate index vector (i 1,i 2,...,i S ) is a descendant of another rate index vector (m 1,m 2,...,m S ) if i j m j for every j [1, S ]. Lemma 2: If a rate index vector x r =(i 1,i 2,...,i S ) is feasibe with the corresponding time sot ength t r then the time sot ength corresponding to any descendant of x r is greater than or equa to t r. Proof: Let us denote the descendant of x r by y r = (p 1,p 2,...,p S ). Since p j i j for j [1, S ], the transmission duration of ink j at rate r p j is greater than or equa to that at rate r i j for every j [1, S ]. Since the time sot ength is the maximum of the transmission duration of a the inks in the set S, the time sot ength corresponding to any descendant of x r is greater than or equa to t r. Lemma 3: If a rate index vector x r =(i 1,i 2,...,i S ) is infeasibe and max j [1, S ] (α j R j /r i j ) is greater than or equa to the time sot ength corresponding to a feasibe rate index vector denoted by t f, then the time sot ength corresponding to any feasibe descendant of x r is greater than or equa to t f. Proof: Let us denote the descendant of x r by y r =(p 1, p 2,...,p S ). Since p j i j for j [1, S ], the time sot ength corresponding to y r that is given by max j [1, S ] (α j R j /r p j ) is greater than or equa to max j [1, S ] (α j R j /r i j ), which is greater than or equa to t f. Agorithm 2 Fast Tree-Based Fathoming Agorithm (FTFA) 1: A = {(M,M,...,M)}; 2: F = ; 3: C = ; 4: t opt = ; 5: x opt r = ; 6: whiea do 7: x r =(i 1,i 2,...,i S )=first rate index vector in A; 8: discard x r from set A; 9: if x r is not descendant of a node in F and not inside C then 10: t =max j [1, S ] (α j R j /r i j ); 11: if x r is feasibe then 12: if t<t opt then 13: t opt = t; 14: x opt r = x r ; 15: end if 16: add x r to set F ; {due Lemma 2} 17: ese 18: if x r =(2, 2,...,2) then 19: break; {no feasibe soution exists} 20: end if 21: if t>= t opt then 22: add x r to set F ; {due Lemma 3} 23: ese 24: incude chidren of x r at the beginning of set A; 25: incude x r in set C; 26: end if 27: end if 28: end if 29: end whie Based on the foregoing emmas, we propose the Fast Tree- Based Fathoming Agorithm (FTFA) as described next. x opt r and t opt correspond to the optima rate index vector and optima time sot ength in each iteration respectivey. x opt r and t opt are initiaized to and respectivey (Lines 4 5). Three sets are updated in each iteration of the agorithm. A is the set of rate index vectors to be evauated in order. A is initiaized to {(M,M,...,M)} (Line 1) and extended to incude the chidren of the infeasibe rate index vector for which t =max j [1, S ] (α j R j /r i j ) <t opt since these chidren are potentia candidates for optima rate index vectors (Line 24). F is initiaized to (Line 2) and extended to incude feasibe rate index vectors since any descendant of these vectors

8 SADI AND ERGEN: MINIMUM LENGTH SCHEDULING WITH PACKET TRAFFIC DEMANDS 3745 rate index vectors by evauating Perron-Frobenius conditions with compexity O( S 3 ) in the worst case. However, the average runtime of FTFA is much smaer due to the fathoming mechanism as wi be demonstrated via extensive simuations in Section VIII. Fig. 2. FTFA agorithm iustration on the tree structure for the case where S =3and M =4. The rate vectors are evauated in the foowing order: (4,4,4), (3,4,4), (2,4,4), (1,4,4), (2,3,4), (1,3,4), (2,2,4), (2,3,3), (2,4,3), (3,3,4), (2,3,4), (3,2,4), (3,3,3), (3,4,3), (4,3,4), and (4,4,3). Green-coored rate index vectors are the fathomed vectors that are evauated as feasibe and the descendants of which are pruned by the agorithm based on Lemma 2 (Line 16). Greycoored rate index vectors are the vectors that are not evauated since they are either a descendant of a node in F or inside C (Line 9) or fathomed since they are infeasibe rate index vectors for which t is greater than or equa to the current optima time sot ength t opt based on Lemma 3 (Lines 21 22). Red-coored rate index vectors are the vectors that are evauated as infeasibe for which t is ess than the current optima time sot ength t opt and whose chid nodes are generated as a consequence (Lines 24 25). For exampe, vector (2,4,3) is fathomed since the corresponding t vaue is higher than the current optima time sot ength associated with the best feasibe rate vector (2,2,4), vector (1,3,4) is not evauated since it is a descendant of a previousy fathomed vector (1,4,4) and vector (2,3,4) which is generated as a chid node of rate vector (3,3,4) is not evauated since it is inside C. is aso feasibe and cannot give a time sot ength smaer than the current time sot ength t due to Lemma 2 (Line 16) and infeasibe rate index vectors for which t is greater than or equa to the current optima time sot ength t opt since any descendant of these vectors cannot give a time sot ength smaer than t opt due to Lemma 3 (Lines 21 22). C is the set of rate index vectors that are evauated but not fathomed. C is initiaized to (Line 3). The chidren of the rate index vector paced into C in each iteration are incuded in A to be checked in the foowing iterations (Line 25). In each iteration of the agorithm, the first rate index vector from the set A is chosen and evauated for feasibiity by checking Perron-Frobenius conditions if it is not descendant of a node in F and not inside C (Lines 7 9). If the rate index vector evauated in the current iteration is feasibe, its corresponding time sot ength is checked for optimaity (Lines 11 12). If the time sot ength is ess than the current optima, and t opt are updated (Lines 13 14). If there are no rate index vectors to be evauated in A (Line 6) or the smaest rate index vector where a the nodes in the set S are active, i.e., x r =(2, 2,...,2), is infeasibe (Lines 18 20), the agorithm stops. Fig. 2 iustrates the FTFA agorithm through an exampe. The compexity of FTFA agorithm is O(M S S 3 ) since the agorithm checks the feasibiity of a of the M S possibe vaue t opt, x opt r V. O PTIMAL SCHEDULING PROBLEM FORMULATION Once the minimum time sot ength of a possibe simutaneous packet transmission scenarios, i.e., the transmission of α packets by ink where α [0,G ] for [1,L], is determined by soving the power contro and rate adaptation probem described in Section IV, the minimum ength scheduing probem can be formuated by assigning a variabe to each of these scenarios as described next. Let E = {E k :1 k E } denote the set of a feasibe simutaneous packet transmission scenarios of the ink set L = {1, 2,...,L}. Note that E L i=1 (G i +1)with equaity for the case where no two inks share any common node. Let Q denote an L E matrix such that the eement in the -th row and k-th coumn of Q denoted by q k is the number of packets transmitted by ink in the k-th simutaneous packet transmission scenario in E, i.e., E k. The minimum ength scheduing probem is then formuated as an Integer Programming (IP) probem [33] as minimize subject to variabes t T c (24) Qc G (25) c (k) {0, 1}, k [1, E ] (26) where t is E 1 vector whose k-th eement t (k) is the minimum time sot ength corresponding to the k-th simutaneous packet transmission scenario E k, G is L 1 vector whose k- th eement is the packet requirement of the k-th ink, i.e., G k. The variabe of the IP probem is the E 1 vector c whose k-th eement denoted by c (k) takes vaue 1 if the transmission scenario E k exists in the optima schedue and 0 otherwise. Equation (25) represents the packet requirements of the inks in the network. There arise two fundamenta difficuties in soving this IP probem. First, it requires an exponentia effort to determine the time sot ength required for each possibe transmission scenario. Second, even if the time sot engths are determined, there are exponentia number of integer variabes in the foregoing IP formuation which makes the probem intractabe. Since there are L i=1 (G i +1) possibe transmission scenarios, the compexity of determining the time sot ength corresponding to a transmission scenarios is O( L i=1 (G i +1)f p ) where f p is the compexity of the power contro agorithm used depending on the transmission rate mode and equa to O( L 3 ), O( L 3 ) and O(M L L 3 ) for constant, continuous and discrete rate transmission modes as derived in Sections IV-A IV-C respectivey. The compexity of soving the IP probem with L i=1 (G i +1) variabes given the corresponding time sot engths on the other hand is O(2 L i=1 (G i+1) ). The compexity of soving the

9 3746 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 7, JULY 2014 optima scheduing probem is therefore O( L i=1 (G i +1)f p + L 2 i=1 (G i+1) ). VI. BRANCH AND PRICE METHOD BASED HEURISTIC ALGORITHM FOR SCHEDULING Branch and Price Method Based Heuristic Agorithm (BPH) deveoped to overcome the intractabiity probem of the IP formuation in Section V is based on using Coumn Generation Method based heuristic agorithm to sove the LP reaxation of the IP probem at each node of the Branch-and-Bound tree. BPH starts with the exponentia IP formuation given in (24) (26). The LP reaxation of the exponentia IP formuation is soved using the Coumn Generation Method (CGM). If the soution is fractiona, one of the fractiona variabes is picked and two new IP probems are formed by setting the vaue of the corresponding variabe to 0 and 1, which we ca branching forming new nodes in the tree. After branching, each resuting node has an associated IP formuation. Note that at each node, the associated IP formuation does not contain some of transmission scenarios in E since the associated variabes to these transmission scenarios are either set to 0 or 1 through the branch on which the particuar node is ocated. We iterativey sove the LP reaxation of the IP formuations at the nodes in the tree using CGM. If the CGM does not provide a feasibe soution at a certain node, we stop branching (fathom) that node. If the soution of the CGM is integra, we update the best integra soution and fathom that node. If the soution is fractiona and the soution is worse than the best integra soution, we aso fathom that node; otherwise we continue branching by using one of the fractiona variabes and setting its vaue to 0 and 1 resuting in two new IP probems, i.e., nodes. This process continues unti every node is fathomed. Coumn Generation Method (CGM), which is an eegant method generay used for soving arge-scae LP probems [33], is used to sove the LP-reaxation of the IP probem associated with the nodes of the Branch and Price tree. CGM decomposes the resuting LP probem into two sub-probems: Restricted Master Probem and Pricing Probem. A. Restricted Master Probem The Restricted Master Probem (RMP) is simiar to the origina probem except that ony a sma subset of the transmission scenarios E (s) Eis considered as minimize subject to variabes ( t (s)) T c (27) Q (s) c G (28) 0 c (k) 1, k [1, E s ] (29) where t (s) and Q (s) ony contain the timesot engths in vector t and the coumns of the transmission scenario matrix Q corresponding to the subsets in E s respectivey. To start CGM, we need to choose an initia set E (s) that guarantees the feasibiity of the RMP. A proper choice of E (s) contains the transmission scenarios where ony one ink becomes active at a time transmitting a its packets resuting in t (s) as a vector containing the corresponding minimum time sot engths and Q (s) as an L L diagona matrix whose diagona eements are the packet requirements of the inks. We can sove RMP and its dua probem with simpex method in poynomia time to obtain its prima optima soution c p and dua optima soution c d. Since we consider ony an arbitrary subset E (s) E, c p is not necessariy the optima soution to the origina probem given in (24) (26). In the origina probem, the cost coefficient of each variabe c (k) in the objective function is t (k) for k [1, E ]. Thus the reduced cost of a coumn q (k) in the matrix Q but not in Q (s) is r (k) = t (k) (c d ) T q (k) (30) where t (k) is the minimum ength of the time sot for the transmission scenario E k corresponding to the coumn q (k). If there exists a coumn q (k) the reduced cost of which denoted by r (k) is ess than 0, the objective function of the RMP can be further reduced by incuding the coumn q (k) in the matrix Q (s). Otherwise, the current soution c p is the optima soution to the origina probem. Instead of cacuating the reduced cost of each coumn q (k) in the matrix Q but not in Q (s), which requires an exponentia effort, we can sove the foowing Pricing Probem which wi generate the coumn with the minimum reduced cost among a coumns in Q matrix. B. Pricing Probem The goa of formuating the Pricing Probem (PP) is to find a transmission scenario that can improve the objective of the RMP when incuded in Q (s) matrix as a coumn. The objective of the PP formuation is given as t L c d α (31) =1 where c d is the -th eement of the dua optima soution of the RMP c d ; α is the number of packets transmitted by ink ; t, the ength of the time sot corresponding to the transmission scenario represented by the number of packets α transmitted by each ink where [1,L]. The constraints and variabes of the optimization probem are the same as those of the power contro probem provided in Section IV for a transmission modes with the additiona variabe α, [1,L]. The PP formuation is a Mixed-Integer Programming probem for which there is no known poynomia-time agorithm [33]. We therefore propose the foowing heuristic agorithm to efficienty and rapidy sove PP. C. Heuristic Agorithm for Pricing Probem The heuristic agorithm caed Reduced Cost Minimization Agorithm (RCMA) is based on iterativey incuding inks and

10 SADI AND ERGEN: MINIMUM LENGTH SCHEDULING WITH PACKET TRAFFIC DEMANDS 3747 corresponding number of packet transmissions one at a time such that the reduced cost of the resuting set is minimized. Let S and α S denote the set of inks and the number of packet transmissions by the inks in the set S respectivey such that each ink S transmits α packets, i.e. α S = {α S(1),α S(2),...,α S( S ) } where S(i) is the i-th eement of the set S. The reduced cost of the transmission scenario represented by the pair (S, α S ) is then defined as r (S,αS) = t (S,αS) S c d α (32) where t (S,αS) is the ength of the time sot corresponding to the transmission scenario where the nodes in the set S transmit the number of packets in the set α S. RCMA agorithm iterativey incudes the inks and corresponding number of packet transmissions in the initiay empty sets S and α S respectivey (Line 1) such that the ink and corresponding number of packet transmissions minimize the reduced cost of the resuting transmission scenario after their addition (Lines 3 5). The agorithm stops when the set S incudes a the inks in L (Line 2) or the addition of a ink does not decrease the reduced cost of the transmission scenario represented by (S, α S ) (Lines 6 8). Agorithm 3 Reduced Cost Minimization Agorithm (RCMA) 1: S =, α S =, D = L; 2: whie S L do 3: if min D,α [1,G ] r (S+{},αS +{α }) <r (S,αS ) then 4: (k, α k )=argmin D,α [1,G ] r (S+{},αS +{α }) ; 5: S = S + {k}, D = D {k}, α S = α S + {α k }; 6: ese 7: break; 8: end if 9: end whie D. Compexity of BPH BPH investigates a the nodes in the branch-and-price tree in the worst case. The tota number L of the nodes in the branchand-price tree is ess than 2 (Gi+1)+1 i=1 since each node is obtained by setting a subset of L i=1 (G i +1) variabes to either 0 or 1. On the other hand, at each node of the branchand-price tree, CGM is used to sove the LP reaxation of the associated IP formuation. The maximum number of iterations in the CGM is equa to the maximum number of coumns CGM can generate, which is equa to the tota number of transmission scenarios, i.e., L i=1 (G i +1). In each iteration of CGM, the RMP soves an LP probem of at most L i=1 (G i +1)variabes resuting in the worst case compexity of O(L( L i=1 (G i + 1)) 4 d) using Interior Method [34] where d is the maximum number of bits used to represent the coefficients of the objective function and constraints of the optimization probem, i.e., t k, q k and G,for [1,L] and k [1, L i=1 (G i +1)].The RCMA agorithm soving the PP of the CGM in each iteration has compexity of O( L i=1 (G i)) provided that the time sot ength corresponding to a transmission scenarios is generated once in the worst case with compexity O( L i=1 (G i +1)f p ). Therefore, the overa compexity of BPH is O( L i=1 (G i + L 1)f p +2 (Gi+1)+1 i=1 (L( L i=1 (G i +1)) 5 d)). Note that the worst case compexity of BPH is much higher than that of the optima IP formuation. However, since the fathoming mechanism prevents the generation of a the nodes in the branch-andprice tree and CGM ony generates a subset of the coumns, the average runtime of BPH is ess than that of the optima IP formuation as wi be demonstrated via extensive simuations in Section VIII. VII. COLUMN GENERATION METHOD BASED HEURISTIC ALGORITHMS FOR SCHEDULING BPH described in Section VI expores the whoe branchand-bound tree by soving the LP reaxation of the IP probem at each node using the Coumn Generation Method (CGM) based heuristic agorithm. We now propose heuristic methods of ower compexity that sti expoit the CGM soution given in Sections VI-A and VI-B. A. Restricted Master Heuristic Restricted Master Heuristic (RMH) first soves the LP reaxation of the optima IP formuation generating the coumns of the Q matrix required for the optima fractiona soution then formuates the IP probem with these coumns. The agorithm starts with the exponentia IP formuation given in (24) (26). The LP reaxation of the exponentia IP formuation is soved using the Coumn Generation Method (CGM). In CGM, the set of transmission scenarios E (s) is initiaized to guarantee the feasibiity of the RMP and extended by incuding the transmission scenario that minimizes the reduced cost by the use of the PP at each iteration of CGM. The resuting set E (s), the corresponding time sot ength vector t (s) and transmission scenario matrix Q (s) are then input to the origina IP formuation given in (24) (26) by setting Q = Q (s) and t = t (s). RMH requires soving CGM in L i=1 (G i +1) iterations with at most L i=1 (G i +1) variabes and the optima IP formuation with L i=1 (G i +1) variabes in the worst case. The overa compexity of RMH is therefore O( L i=1 (G i + 1)f p + L( L i=1 (G i +1)) 5 L d +2 (Gi+1) i=1 ). Simiar to BPH, the worst case compexity of RMH is much higher than that of the optima IP formuation. However, since in most cases CGM terminates in much smaer number of iterations than the tota number of transmission scenarios given by L i=1 (G i +1),the average runtime of the RMH is much smaer than that of the optima IP formuation as wi be demonstrated via extensive simuations in Section VIII. B. Rounding Heuristic Rounding Heuristic (RH) expores ony one branch of the branch-and-price tree. The agorithm starts by soving the

11 3748 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 7, JULY 2014 TABLE I COMPLEXITY COMPARISON OF THE ALGORITHMS LP-reaxation of the exponentia IP formuation given in (24) (26) using the CGM method. Among the fractiona vaues in the soution, the argest one is chosen setting its vaue to 1, in contrast to exporing both vaues of 0 and 1 as in the Branch-and-Price Method. The packet requirements of the inks are then updated by considering that the transmission scenario corresponding to the variabe is set to 1. In each iteration, the argest fractiona vaue in the soution of the CGM for the LP reaxation of the resuting probem is set to 1 updating the packet requirements of the inks accordingy. The agorithm stops when the packet requirements of a the inks are met. RH requires soving CGM on the nodes in one branch of the branch-and-price tree. As the agorithm proceeds on each node of the investigated branch, one of the transmission scenarios is incuded by setting the corresponding variabe to 1, resuting in the transmission of at east one packet. The maximum number of the nodes on the branch is therefore L i=1 (G i). Therefore, the worst case compexity of RH is O( L i=1 (G i +1)f p + L i=1 (G i)(l( L i=1 (G i +1)) 5 d)), which is much smaer than that of BPH. The worst case compexities of the proposed agorithms and the optima IP formuation (OPT) are summarized in Tabe I. VIII. SIMULATIONS AND PERFORMANCE EVALUATION The goa of this section is to evauate the performance of the proposed scheduing agorithms incuding BPH, RMH and RH together with the power contro and rate adaptation agorithms for constant rate, continuous rate and discrete rate transmission modes, and compare their performance to that of the Traditiona Scheduing (TS) agorithm for constant and discrete rate transmission modes. TS agorithm is described as foows: The CGM based scheduing agorithm proposed in [22] is impemented to obtain the optima schedues negecting the packet based transmissions. Since the resuting optima schedues divide the packet transmissions into severa data chunks each of which is transmitted in different time sots, the schedues are adapted for packet based transmissions by updating the time sot engths such that each ink transmits an integer number of packets, which is the smaest number greater than or equa to the ratio of the number of bits assigned to that ink to the packet ength of the ink, if the ink did not satisfy its packet traffic requirement unti that time sot. The performance of the TS agorithm is not incuded for continuous rate transmission mode since no agorithm has been proposed to minimize the schedue ength by using the continuous rate transmission mode in the iterature and CGM cannot be appied to the continuous rate transmission mode since the number of coumns corresponding to the feasibe transmission rates of the inks is infinite. We used MATLAB on a computer with a 2.5 GHz CPU and 4 GB RAM to run the simuations. Simuation resuts are obtained based on 1000 independent random network topoogies where various numbers of inks with 1 m fixed ength are uniformy distributed over a square area of 3 3 meters. The attenuation of the inks are determined using the path oss mode given by PL(d) =PL(d 0 ) 10n og 10 (d/d 0 )+ Z, where d is the distance between the transmitter and receiver, d 0 is the reference distance, PL(d) is the path oss at distance d, n is the path oss exponent and Z is a Gaussian random variabe with zero mean and σ 2 z variance. The parameters used in the simuations are n =4, σ 2 z =2dB 2, PL(d 0 )=30dB, d 0 =1m, N 0 =10 8 W/Hz, p max =10mW, W = 100 MHz, R = 100 bits, G is randomy determined from the set {1, 2,3} for each [1,L], β =10dB for a [1,L] in constant rate transmission mode, ɛ = in continuous rate transmission mode and γ =(, 10, 20, 30) db in discrete rate transmission mode. The transmission rate corresponding to these SINR vaues are cacuated by using (2). Fig. 3(a) (c) show 95% confidence interva bars depicted around the mean of the approximation ratio of the proposed scheduing agorithms incuding BPH, RMH and RH agorithms and the TS agorithm for constant rate, continuous rate and discrete rate transmission modes respectivey. The approximation ratio is defined as the ratio of the schedue ength obtained by the heuristic agorithm to that obtained by OPT. Note that the number of the inks is imited to 20 for constant and continuous rate modes, and to 10 for discrete rate modes since soving the OPT is intractabe for networks with higher number of inks. For constant and discrete rate transmission modes, the proposed agorithms outperform TS agorithm. We observe simiar behavior for a the proposed agorithms under a transmission modes: The approximation ratio of BPH is very cose to 1, i.e., optima soution, and robust to the increase in the number of the inks whereas the approximation ratio of RMH and RH is worse than that of BPH and increases as the number of the inks increases. Moreover, the performance of RH is sighty better than RMH since RH continues to search for the best set of transmission scenarios in each iteration whereas RMH uses ony the set of transmission scenarios determined at the output of the first CGM. Fig. 4(a) (c) iustrate the average runtime performance of BPH, RMH, RH and OPT scheduing agorithms for constant rate, continuous rate and discrete rate transmission modes respectivey. For constant and continuous rate transmission modes, the average runtime of the RMH and RH scheduing agorithms increases amost ineary in the number of the inks due to the greedy structure of these agorithms and inear increase in the runtime of the corresponding power contro agorithms. On the other hand, the runtime of the BPH and OPT agorithms is exponentia in the number of the inks. However, the exponent of the runtime of the OPT and BPH agorithms are around 2 and 1.3 respectivey, which creates a significant runtime difference as the number of the inks grows arger. Athough the runtime of the OPT agorithm is ower than that of the BPH agorithm for up to 20 inks, when projected, it is expected that the BPH agorithm woud outperform OPT agorithm for 22 inks and the runtime of the BPH woud be

12 SADI AND ERGEN: MINIMUM LENGTH SCHEDULING WITH PACKET TRAFFIC DEMANDS 3749 Fig. 3. Approximation ratio of BPH, RMH, RH, and TS scheduing agorithms for a) constant rate, b) continuous rate, c) discrete rate transmission modes. Fig. 4. Average runtime of BPH, RMH, RH, and OPT scheduing agorithms for a) constant rate, b) continuous rate, c) discrete rate transmission modes. negigibe compared to the OPT agorithm for 30 inks. Note that it is impossibe to sove the OPT agorithm for higher than 20 inks using MATLAB whie the BPH agorithm provides soutions for arger number of the inks. The reason for this intractabiity difference is that the OPT agorithm soves the scheduing probem as a pure IP probem whereas the BPH agorithm searches for the optima soution over a BPH tree by expoiting the fathoming and the efficiency of the CGM based method together with the proposed heuristic method to sove the pricing probem. Furthermore, the runtime behavior of the scheduing agorithms for the discrete rate transmission mode is very different from that of constant and continuous rate modes. The average runtime of the OPT agorithm increases dramaticay as the number of inks increases and becomes 100 times greater than that of the BPH agorithm for 10 inks. This is a direct resut of the exponentia compexity of the FTFA power contro agorithm: The OPT agorithm uses FTFA to determine the time sot ength for a possibe subsets of the inks whereas the BPH agorithm uses FTFA for smaer number of inks since the average number of concurrenty transmitting inks in the optima soution is smaer than the number of the inks in the network. Fig. 5 shows the ength of the schedue for constant rate, continuous rate and discrete rate transmission modes normaized by that of the continuous rate mode. As expected, the ength Fig. 5. Effect of transmission mode on the schedue ength. of the schedue is owest and highest for continuous rate and constant rate modes respectivey. The schedue engths of the discrete rate and constant rate transmission modes get coser to that of the continuous rate mode as the number of the inks increases mainy due to exponentiay increasing number of possibe ink rate vectors in the number of the inks eading to the seection of a better ink rate vector.

13 3750 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 7, JULY 2014 IX. CONCLUSION We formuate the joint optimization of power contro, rate adaptation and scheduing with the goa of generating minimum ength schedue given traffic demands on the inks in practica packet based wireess ad hoc networks. In contrast to the traditiona approach where the optimization probem ignores the packet eve detais assigning the transmission of any amount of data to the time sots at different rates, we assume that ony an integer number of packets can be transmitted in a time sot. We first formuate the optima power contro and rate adaptation probem with the goa of minimizing the ength of the time sot given the number of packets transmitted by the subset of the inks and propose both optima and fast heuristic agorithms. Given the minimum time sot engths corresponding to a the subset of the inks concurrenty transmitting any possibe number of packets, the joint optimization of power contro, rate adaptation and scheduing probem can then be reduced to a pure scheduing probem. The scheduing probem is an IP probem in which the number of variabes is exponentia in the number of the inks in the network. In order to sove this arge-scae IP, we propose Branch and Price method and two Coumn Generation method based heuristic agorithms, and compare their performance to that of the Traditiona Scheduing agorithm that ignores the constraints of the packet transmissions via extensive simuations. The proposed agorithms outperform the traditiona scheduing agorithm. Moreover, Branch and Price method based heuristic agorithm (BPH) provides performance very cose to the optima soution and robust to the increasing number of the inks with much smaer runtime than that of the optima agorithm enabing the usage of the method in arge networks. The Coumn Generation method based heuristic agorithms on the other hand provide performance sighty worse than the BPH but degrading more as the number of the inks increases with runtime much smaer than the BPH. In the future, we are panning to extend the proposed framework for different objectives such as throughput maximization and energy minimization to provide reaistic soutions for various packet based wireess network appications. REFERENCES [1] J. Tang, G. Xue, C. Chander, and W. Zhang, Link scheduing with power contro for throughput enhancement in mutihop wireess networks, IEEE Trans. Veh. Techno., vo. 55, no. 3, pp , May [2] Z. Yang, L. Cai, and W. 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14 SADI AND ERGEN: MINIMUM LENGTH SCHEDULING WITH PACKET TRAFFIC DEMANDS 3751 Yacin Sadi (M 13) received the B.S. and the M.S. degrees in eectrica and eectronics engineering from Koc University, Istanbu, Turkey, in 2010 and He is currenty a Ph.D. student in eectrica and eectronics engineering in Koc University, Istanbu and working in Wireess Networks Laboratory under supervision of Prof. Sinem Coeri Ergen. His research interests are communication design in wireess networks and intra-vehicuar wireess sensor networks. He received Tubitak (The Scientific and Technoogica Research Counci of Turkey) Feowship in 2010 and Koc University Fu Merit Schoarship from Koc University, Istanbu, in Sinem Coeri Ergen (M 98) received the B.S. degree in eectrica and eectronics engineering from Bikent University, Ankara, Turkey, in 2000 and the M.S. and Ph.D. degrees in eectrica engineering and computer sciences from University of Caifornia Berkeey, Berkeey, CA, USA, in 2002 and 2005, respectivey. She worked as a Research Scientist in Wireess Sensor Networks Berkeey Lab under sponsorship of Pirei and Teecom Itaia from 2006 to Since September 2009, she has been Assistant Professor in the department of Eectrica and Eectronics Engineering at Koc University, Istanbu, Turkey. Her research interests are in wireess communications and networking with appications in sensor networks and transportation systems. Dr. Ergen received Science Academy Young Scientist Award (BAGEP) in 2014, Turk Teekom Coaborative Research Award in 2011 and 2012, Marie Curie Reintegration Grant in 2010, Regents Feowship from University of Caifornia Berkeey, Berkeey in 2000 and Bikent University Fu Schoarship from Bikent University, Ankara, in 1995.