Optimal Joint Routing and Scheduling in Millimeter-Wave Cellular Networks

Transcription

1 Optimal Joint Routing and Sheduling in Millimeter-Wave Cellular Networks Dingwen Yuan, Hsuan-Yin Lin, Jörg Widmer and Matthias Hollik SEEMOO, Tehnishe Universität Darmstadt Bergen Norway Institute IMDEA Networks Abstrat Millimeter-wave (mmwave) ommuniation is a promising tehnology to ope with the expeted exponential inrease in data traffi in 5G networks. mmwave networks typially require a very dense deployment of mmwave base stations (mmbs). To redue ost and inrease flexibility, wireless bakhauling is needed to onnet the mmbss. The harateristis of mmwave ommuniation, and speifially its high diretionality, imply new requirements for effiient routing and sheduling paradigms. We propose an effiient sheduling method, so-alled shedule-oriented optimization, based on mathing theory that optimizes QoS metris jointly with routing. It is apable of solving any sheduling problem that an be formulated as a linear program whose variables are link times and QoS metris. As an example of the shedule-oriented optimization, we show the optimal solution of the maximum throughput fair sheduling (MTFS). Pratially, the optimal sheduling an be obtained even for networks with over 00 mmbss. To further inrease the runtime performane, we propose an effiient edge-oloring based approximation algorithm with provable performane bound. It ahieves over 80% of the optimal max-min throughput and runs 5 to 00 times faster than the optimal algorithm in pratie. Finally, we extend the optimal and approximation algorithms for the ases of multi-rf-hain mmbss and integrated bakhaul and aess networks. I. INTRODUCTION 5G ellular systems are embraing millimeter wave (mmwave) ommuniation in the GHz band where abundant bandwidth is available to ahieve Gbps data rates. One of the main hallenges for mmwave systems is the high propagation loss at these frequeny bands. Although it an be partially ompensated by diretional antennas [], [], the effetive ommuniation range of a mmwave base station (mmbs) remains around 00 meters at best. Thus, base station deployment density in 5G will be signifiantly higher than in 4G [3], [4]. This leads to high infrastruture ost for the operators. Besides the ost of site lease, bakhaul link provisioning is in fat the main ontributor to this ost beause the mmwave aess network may require multi-gbps bakhaul links to the ore network. Currently, suh a high data rate an only be aommodated by fiber-opti links whih have high installation ost and are inflexible with respet to reloation. Reent studies show that mmwave self-bakhauling is a ost-effetive alternative to the wired bakhauling. This approah is partiularly interesting in a heterogeneous network setting where the existing ellular base stations (enbs) at as a gateway for the mmbss. Fig. illustrates suh a setup in whih the enb an reah mmbss diretly or via other mmbss. Moreover, diretionality of mmwave ommuniation enb mmbs mmwave bakhaul Fiber bakhaul Bakhaul and Aess Network S-GW P-GW Core Network Fig.. mmwave self-bakhauling setup. Internet redues or removes the wireless bakhaul interferene and allows simultaneous transmissions of multiple links over the same hannel as long as their beams do not overlap. However, the number of simultaneous links a base station an have is limited by the number of its RF hains. To date, muh of the researh on mmwave ommuniation has been dediated to issues faed by the mobile users (UEs) in the aess networks. How to maximize performane suh as throughput and energy effiieny in mmwave bakhaul and aess networks has reeived less attention. Here, two important issues need to be addressed: (i) routes to be taken, (ii) the sheduling of the transmission overs the links. A naive sheduling whih lets the enb serve all the mmbss in a round robin fashion is neither pratial nor effiient. If mmbss links to the enb are weak ompared to their links to other nearby mmbss (whih in turn have high-apaity links to the enb), a shedule allowing multi-hop routing is muh more favorable sine it alleviates the bottlenek at the enb. At the same time, the limited interferene at mmwaves makes it effiient to maximize spatial reuse and operate as many links simultaneously as possible. The goal of the paper is to design a sheduler that exploits these harateristis to optimize mmwave bakhaul effiieny. The paper is organized as follows. We disuss related work in Se. II. Se. III provides the system model. The relation between a shedule and mathings is studied in Se. IV. In Se. V, we present our optimal shedule-oriented optimization method, through an example maximum throughput fair sheduling. We then propose a fast edge-oloring based approximation algorithm in Se. VI. In Se. VII, we extend our algorithms to more general senarios. Se. VIII shows the numerial evaluation and Se. IX onludes the paper.

2 II. RELATED WORK mmbs 4 mmbs 3 Few works on mmwave bakhaul and aess network sheduling exist [5] [8]. These works share the assumptions that (i) the traffi demand is measured in disrete units of slots or pakets, and (ii) a flow has to be sheduled sequentially, i.e., a hop loser to the soure should be sheduled earlier than a hop farther away. The resulting optimization problems are all formulated as mixed integer programming (MIP) problems. As MIPs are in general NP-omplete, optimal solutions an only be omputed for small networks with a few nodes. For pratial use, these works all rely on heuristis, whih are based on the ideas suh as greedy edge oloring [5], [7] or finding the maximum independent set in a graph [6], [8]. Furthermore, [5], [6], [8] assume that routing is pre-determined, whih does not fully exploit the freedom given by a reonfigurable mmwave bakhaul, and may limit performane. In ontrast, our work relaxes the onstraint of sequential flow sheduling (i.e., if needed, pakets are queued for a short time) whih does not harm the long-term throughput, and allows the slots in a shedule to be of any length. Based on these assumptions, we propose a polynomial time optimal sheduling method whih is shown by simulation to be pratial for mmwave ellular networks. Moreover, the sheduling takes QoS optimization goals or QoS requirements as input and finds an optimal routing automatially. The first attempt to solve the problem of joint routing and sheduling in a network with Edmonds mathing formulation goes bak to [9]. Hajek et al. s polynomial time sheduling algorithm is different from ours in that it minimizes the shedule length. Furthermore, we use a one-step shedule-oriented approah while they first ompute the optimal link time and then ompute the minimum length shedule given the link time. Following [9], reent researh on sheduling fouses on optimization of delay [0] and queue length [], as well as investigating more realisti interferene models []. Another interesting line of researh is the on-line node-based sheduling algorithms whih ahieve good performane bound in throughput and evauation time [3]. III. SYSTEM MODEL The system model onsiders a bakhaul network whih has a single enb, equipped with multiple mmwave RF hains, and multiple mmbss, eah equipped with a single RF hain. Later in Se. VII, we will show that our optimization method applies equally to (i) the ase where eah node has multiple RF hains, and (ii) a network model that inludes UEs. We onsider an enb maro ell together with a number of mmwave base stations in a heterogeneous TDMA ellular network. The enb ats as the bakhaul gateway for mmbss. In addition to an LTE radio interfae, the enb is equipped with R mmwave RF hains. There are W single-rf-hain mmbss in the maro ell. We assume analog or hybrid beamforming with R RF hains whih allows up to R simultaneous links at the enb. We use direted graphs to model the links between the different nodes in the network. Fig. illustrates a toy example of a bakhaul network. The following analysis fouses on enb mmbs mmbs Fig.. A bakhaul network example. The edges shown are the potential links for downlink shedule. A bidiretional edge represents two opposite links. downlink ommuniation. The same analysis an be applied to the uplink senario. Let G = (V, E) be a direted graph with the vertex (node) set V and edge (link) set E. Eah edge represents a potential link between two verties. The apaity of eah edge e is denoted e. The reeived power is given by p rx = p tx +g x P L, where p tx is the transmission power, g x is the diretivity gain, and P L is path loss between the transmitter and the reeiver. P L(d) = α + 0β log 0 d + ξ, where α, β are onstants that depend on the frequeny and line-of-sight onditions. d is the distane between the transmitter and reeiver. ξ represents the shadowing effet and is a normal distributed random variable with zero mean and σ standard deviation. We an observe in Fig. that there are many ways to shedule downlink ommuniation among the enb and mmbss. Our goal is to obtain the optimal unit length shedule with respet to a QoS metri, while satisfying given QoS requirements and the onstraints on simultaneous transmissions. In pratie, the unit time is the duration of the radio frame. We observe that eah feasible shedule S an always be divided into N slots numbered as to N. We define t i as the length of the i-th slot. It is required that N i= t i = and t i > 0. Moreover, in the i-th slot, a set of links E i E (an be empty) are ative for the whole slot. IV. PRELIMINARY: SCHEDULE POLYHEDRON This setion first shows the relation between a feasible shedule and mathings in a graph. Based on the relation, we mathematially formulate the set of all feasible shedules as the shedule polyhedron that is desribed by linear onstraints. Suppose that a set of links E i are sheduled in the i-th slot, and e, e E i are two different links. Then e and e an not share a ommon mmbs node sine a mmbs has one RF hain and is therefore half-duplex. On the other hand, e and e may share the enb node given that R >. However, the number of links in E i that are inident to the enb annot be more than R. We enfore this onstraint through the enb expansion. A. enb expansion In graph G, we replae the enb with R expanded enbs: enb,..., enb R. If the enb is onneted to a set of mmbss, then eah enb i is onneted to the same set of mmbss with the same respetive link apaities as the enb. The resulting graph is equivalent to the original graph with respet to sheduling. Yet eah expanded enb has one RF hain. In the rest of the paper, the graph G = (V, E) is assumed to be expanded if not expliitly stated otherwise. An example of the proess desribed above is shown in Fig. 3. Let R and W denote the set of expanded enbs and mmbss, respetively. As a result, we an ensure that the ative link set of a slot orresponds to a mathing in the expanded graph. A mathing

3 enb 3 mmbs mmbs mmbs 3 Expansion enb 3 enb 3 mmbs mmbs mmbs 3 Fig. 3. enb expansion with two RF hains. in a graph is defined as a set of edges in the graph that share no ommon verties. B. The shedule polyhedron We define the link time t e [0, ] as the total ative time of a link e in a shedule. Correspondingly, t is the link time vetor, eah element of whih is a link time t e, e E. A link time vetor t is feasible if t an be sheduled in unit time. We first define the shedule polyhedron P and then prove that eah point in P is one-to-one mapped to eah feasible link time vetor. Definition (Shedule Polyhedron). Given a graph G = (V, E), the shedule polyhedron of G is defined as the set of link time vetors t that satisfy the following linear onstraints. t e v V, (a) e E(O) t e O odd set O V, (b) t e 0 e E, () where δ(v) is the set of links inident to node v. An odd set O has odd number of nodes. E(O) is the set of links whose endpoints are both ontained in O. The following lemma summarizes the relation between the shedule polyhedron and the unit length shedules. Lemma. () Eah point in the shedule polyhedron P is a feasible link time vetor t, and () eah feasible link time vetor t is a point in P. Proof. The proof uses the Edmonds mathing polyhedron theorem [4]. A feasible shedule S onsists of N slots. Eah slot ontains a set of links from G that is a mathing and therefore orresponds to a vertex of the mathing polyhedron Q. Sine Q has the same formulation as P, exept the variables are binary, P and Q has the same set of verties. Sine S has a length of, the link time vetor t of S is a onvex ombination of the verties of P. So it is a point in P. On the other hand, a point in P an be written as a onvex ombination of all verties of P where eah vertex orresponds to a slot. Therefore, eah point in P orresponds to a feasible unit time shedule. See the details in the extended version [5]. V. MAXIMUM THROUGHPUT FAIR SCHEDULING Having established the relation between a shedule and mathings, we now investigate the problem of maximum throughput fair sheduling (MTFS) for bakhaul networks. The goal of the problem is to maximize the downlink network throughput under the ondition that the max-min fairness [6], b a α 8 β 6 γ 3 δ ε 4 d α θ γ δ ɛ α, ɛ a b d Fig. 4. Node-mathing matrix, a to d are nodes, α to ɛ are edges. The numbers are apaities. [7] in throughput is ahieved at the mmbss. The MTFS problem serves as one example of our method for sheduling optimization in mmwave bakhaul networks. Definition (Maximum Throughput Fair Shedule). Given a bakhaul network G and a unit time shedule S, let the throughput vetor of S be h S = [h S v v W], where h S v denotes the downlink throughput of an mmbs node v. (i) A feasible unit time shedule S f is said to satisfy the maxmin fairness riteria if min v W h S f v min v W h S v for any feasible unit time shedule S. Suh min v W h S f v is alled the max-min throughput. (ii) A feasible unit time shedule S is a solution of the MTFS problem if S has ahieved the maximum network throughput v W hs f v among all possible feasible unit time shedule S f satisfying the max-min fairness riteria in (i). In the following, we present our general optimization method shedule oriented optimization. A. Shedule oriented optimization The shedule oriented optimization solves a linear optimization problem, the solution to whih is diretly the optimal shedule. For the mathematial formulation of the optimization problem, we onstrut the node-mathing matrix. Definition 3 (Node-Mathing Matrix). Given a direted graph G = (V, E). Suppose the number of all possible mathings of G is K. Then the node-mathing matrix A = [a i,j ] is a V K matrix, whose elements is defined as follows: a i,j =, if there is a link with apaity entering node i in the j-th mathing; a i,j =, if there is a link with apaity leaving node i in the j-th mathing. Otherwise, a i,j = 0. As we will see, the node-mathing matrix helps in formulating the throughput onstraints at individual nodes. Fig. 4 gives an example of node-mathing matrix for a graph. Let A be the node-mathing matrix of the bakhaul network G, we define A W as the submatrix of A, whih onsists only of the rows of A related to the nodes in W (mmbss). As we pointed out in Se. IV, the link set sheduled in eah slot of a shedule must be a mathing in G. We define t S as a K slot length vetor, eah element of whih is the length of a potential slot. Let the minimum throughput among all mmbss be θ. Then we an solve the MTFS problem in two steps: (i) maximizing θ (suh θ is the max-min throughput) and (ii) omputing the optimal shedule S that offers the highest network throughput, given the max-min throughput θ. Linear programs for the MTFS problem. The linear program to maximize θ for step (i) of the MTFS problem 3

4 an be formulated as follows, maximize θ (a) subjet to A W t S θ (b) T t S = and t S 0, () where and 0 represent the all-one and all-zero olumn vetor, respetively. The supersript T " denotes the vetor transposition. (b) is the onstraint that the throughput at eah mmbs should be at least θ. () is the onstraint that the shedule length should be unit time. The feasibility of the shedule is impliitly guaranteed by the formulation in terms of all possible mathings. After we have omputed θ from (), we an formulate the linear program that maximizes the network throughput, i.e., the total throughput of all mmbss under the ondition that eah mmbs has throughput at least θ. maximize T t S (3a) subjet to (b), and (). (3b) Here, is the apaity vetor whose element j is the umulative apaity of all enb-to-mmbs links in the j-th mathing M j, i.e., j = (v,v ) M j,v R (v,v ), where (v, v ) denotes the link from node v to node v. Note that θ is a variable in (), but it is a onstant in (3). The apparent diffiulty in solving () and (3) is the huge number of elements in t S (same as the number of mathings in G, whih is exponential to the number of verties in G). Yet, we will show that we an still solve it in polynomial time. Theorem. The MTFS problem an be solved in polynomial time with the ellipsoid algorithm [8]. Proof. This proof uses a similar tehnique to the proof of Theorem in [9], whih states that the frational edge oloring an be solved in polynomial time by the ellipsoid algorithm. See the details in the extended version [5]. Although polynomial, in pratie the ellipsoid algorithm almost always takes longer than the simplex algorithm. In the following, we propose algorithms based on the revised simplex algorithm [0] whih does not require the generation of all olumns of A W. Coneptually, the algorithms first reate a feasible shedule. Then in eah iteration, to improve the optimization objetive, we replae one slot in the shedule by another mathing (a set of simultaneous links) while keeping the shedule feasible, until the optimum is reahed. The maximum weighted mathing algorithm [4] is used to hoose the mathing (olumn) to enter the basis (shedule). B. Solving the MTFS problem To optimize θ, we need an initial basi feasible solution to (). Suppose that the bakhaul network G is onneted, otherwise there are mmbss unreahable from the enb. We perform a breath-first-searh (BFS) starting from an arbitrary expanded enb, say enb. The result is a tree T that spans enb and all mmbss. T has exatly W edges. The initial shedule S 0 is onstruted as follows: S 0 has W slots, eah of whih ontains one link in T. Moreover, it is required that Algorithm : Compute the max-min throughput θ Set the basis B = B 0 orresponding to the initial shedule S 0; while True do 3 Compute the dual variable p T = fbb T ; 4 Set weight w (vi,v j ) to eah link (v i, v j) of G as follows. { w (vi,v j ) = (vi,v j )(p j p i) if v i, v j W (vi,v j )p j otherwise v i R, v j W Do max weighted mathing on G. Let the optimal mathing be M, ompute η = e M we pw +; 5 Compute η = + W k= p k; 6 Compute η 3 = min k W p k ; 7 Compute η = min(η, η, η 3) and let the orresponding olumn be u η U; 8 if η 0 then 9 return the optimal θ and B θ = B; 0 else Update B by replaing a olumn of B with u η aording to the simplex algorithm; end 3 end the throughputs of all mmbss are the same and the shedule takes exatly unit time. It is obvious that the initial solution is unique. We onvert the linear program () to the standard form (4) by introduing W surplus variables s i as follows. minimize f T x (4a) subjet to Ux = g and x 0, (4b) [ ] A where U [U U U 3 ] W I, f T 0 0 T = [ T 0T 0 T], x T [ (t S ) T θ s T], and g T [ 0 T ]. Alg. shows the omputation of the max-min throughput θ. The basis B is a square matrix that onsists of W + olumns from U. f B is the elements of f orresponding to the basis B. The lines 4, 5 and 6 ompute the minimum redued ost of a olumn in the matries U, U and U 3 respetively. To derease θ, we need to find a olumn of U, u k that has negative redued ost f k p T u k < 0 to enter the basis. In the algorithm, we find the olumn u η in U that produes the minimum redued ost η. If η 0, then no olumns an be used to derease θ, thus we have reahed the optimum. Let the final basis in omputing θ be B θ. To diretly use B θ as the initial basis to the solution of step (ii) of the MTFS problem, we add an artifiial salar variable y 0 to (3) and replae the onstraint A W t S θ with A W t S y θ. Sine θ is the max-min throughput, the feasible y must be 0. Hene, the optimal solution (maximum network throughput) to (3) is unaffeted. Again, we onvert (3) into the standard form of (4), whih is solvable with the revised simplex algorithm. In the standard form, U remains unhanged, we redefine f T [ T 0 0 T], x T [ (t S ) T y s T], and g T [ θ T ]. The optimization algorithm is similar to Alg. and is outlined in Alg.. Sine the basis B is a square matrix of W + dimension, it follows that the optimal shedule S ontains no more than W + slots. Additionally, sine the links on a flow from the enb to a destination mmbs may not be sheduled in 4

5 Algorithm : Solving the MTFS problem Set the basis B = B θ ; while True do 3 Compute the dual variable p T = fbb T ; 4 Set weight w (vi,v j ) to eah link (v i, v j) of G as follows. { w (vi,v j ) = (vi,v j )(p j p i) if v i, v j W (vi,v j )(p j + ) otherwise v i R, v j W Do max weighted mathing on G. Let the optimal mathing be M, ompute η = e M we pw +; 5 η = W k= p k; 6 η 3 = min k W p k ; 7 Compute η = min(η, η, η 3) and let the orresponding olumn be u η U; 8 if η 0 then 9 return the optimal shedule S orresponding to B; 0 else Update B by replaing a olumn of B with u η; end 3 end sequential order, some transmission opportunities of the flow in the first few frames may be wasted. Therefore, maximum throughput is ahieved in the long-term. C. Generalization The sheduled-oriented optimization method illustrated by the optimal MTFS algorithm is quite general. It an solve any sheduling problem that an be formulated as a linear program whose variables are link times and QoS metris. For example, it an optimize for the onstraint that eah mmbs has a minimum throughput requirement. Another example is that the proposed method an optimize the energy onsumption as it an be translated into the minimization of total transmission time in a shedule. We do not further elaborate on them due to the spae limitation. Moreover, in Se. VII, we extend the optimization method to bakhaul and aess networks, as well as to multi-rf hains at eah node. VI. EDGE-COLORING BASED APPROXIMATION ALGORITHM In Se. V, we proposed an optimal joint routing and sheduling algorithm for mmwave bakhaul networks. Although it is optimal, it may have a high runtime (.f. the evaluation in Se. VIII) when the number of mmbs nodes is large. Hene, we propose a run-time effiient edge-oloring (EC) based approximation algorithm that has a provable performane bound. The EC algorithm follows a two-step approah of (i) omputing the link time and (ii) sheduling within unit time. A. Step (i): omputing link time To preisely ompute the link time, we need to inlude all the onstraints of the shedule polyhedron (). Sine the number of odd set onstraints (b) is huge, whih leads to a high runtime for the optimization, instead we use a small set of onstraints that is a neessary but not suffiient ondition for a feasible unit time shedule. The seletion of the new set of onstraints is based on the following observation. Let G = (V, E) be the bakhaul network before the enb expansion (Se. IV-A) and G W be the subgraph of G, whih ontains only the mmbss W and the links among them. We define ν = W as an upper bound of the maximum number of mmbs-to-mmbs links that an be ative simultaneously. We assume the number of RF hains at the enb satisfies R L, where L is the number of mmbss that are diretly onneted to the enb, beause L RF hains is enough to serve the mmbss. We have the following observation. Observation. If k mmbs-to-mmbs links are ative at a time t, then at most min(r, W k) enb-to-mmbs links an be ative at t, eah using one RF hain of the enb. Hene, at least R min(r, W k) = max(0, R W + k) RF hains of the enb are idle at t. For a shedule of unit time, we define t k as the time in whih exatly k mmbs-to-mmbs links are ative. Eah feasible shedule should be subjet to the following onstraints. ν t k, and t k 0 k {,,..., ν} (5a) k= ν k t k = k= e δ(enb) t e R e {(v,v ) E : v,v W} t e ν max(0, R W + k)t k k= (5b) (5) t e v enb, and t e 0 e E. (5d) (5b) formulates the total mmbs-to-mmbs transmission time in terms of the variables t k and t e (link time of e), respetively. (5) shows that the total enb-to-mmbs transmission time should be no more than R minus the minimum idle time of the RF hains at the enb. (5d) expresses the single RF hain onstraint on mmbss. We substitute the preise onstraint set to link times () with the onstraints in (5). The advantage is the low runtime and small memory omplexity of linear programming due to the following reason. The total number of onstraints in () and (5) are O( W +R ) (exponential) and O(W ) (polynomial), respetively. Moreover, with the polynomial number of onstraints, the omputation of link time an be arried out by off-the-shelf linear optimization tools. However, the link time vetor that satisfies (5) may be infeasible in unit time, beause satisfying the onstraints in (5) is a neessary but not suffiient ondition for a feasible unit time shedule. Speifially, for the MTFS problem, the linear program for omputing the max-min throughput θ is maximize θ (6a) subjet to e t e e t e θ v W (6b) e δ (v) and (5), e δ + (v) The ative links form a mathing in G W. Aording to the definition of mathing, the number of ative links is upper-bounded by W. 5

6 Algorithm 3: EC-based sheduling. Redue graph. Given the link time vetor t, we remove edges in G with zero link time and all the subgraph G r; Expand enb. We perform the enb expansion on G r. Let the expanded enbs be enb,..., enb R. For a given link (enb, v) in G r with link time t (enb,v), we set the link time of the links (enb k, v), k =,..., R to t (enb,v). We all the graph after enb R expansion G v = (V v, E v); 3 Create multigraph and assign link time. We reate the oloring graph G m = (V m, E m), whih has the same vertex set as V v, and its edges is defined as follows. For eah e E v between two nodes v and v with link time t e, we install t e t g edges between v and v in G m. Among these edges, t e t g edges are assigned t g link time, and the left edge is assigned mod(t e, t g ) link time (mod is the modulo operation); 4 Coloring and sheduling. We perform edge oloring on G m. Suppose that G m an be edge-olored with κ olors. For those edges olored by the i-th olor, i =,..., κ, we shedule the orresponding links in the i-th slot (a slot has the length t g ); 5 Sale. The shedule is now of length κt g. If κt g >, we sale the total time length with the fator κt g ; where δ (v) and δ + (v) are the set of links oming into node v and the set of links leaving v, respetively. With the optimal θ, we ompute the link time for the MTFS. maximize e t e e δ + (enb) subjet to onstraints in (6). (7) B. Step (ii): edge-oloring based sheduling After we obtain the link time, the next step is to generate a unit time shedule. The approximation algorithm is based on the idea of edge-oloring of multigraphs (graphs allowing multiple edges between two nodes). A proper edge-oloring assigns a olor to eah edge in a graph suh that any two adjaent edges (sharing one or two ommon nodes) are assigned different olors. Obviously, the set of edges E λ of a olor λ must be a mathing. Hene, E λ orresponds to a slot and an edge oloring sheme orresponds to a shedule. The ECbased sheduling takes a parameter granularity t g (0, ], whih is the quantization of the link time. A smaller t g typially leads to better shedules at the ost of longer runtime. Alg. 3 shows the proess of the EC-based sheduling. C. Performane analysis of the EC-based sheduling The following lemma shows that Alg. 3 (step (ii) of the EC algorithm) redues the performane metri µ and link times t e by a fator of κt, if the κt g >. Therefore, a high quality g edge-oloring heuristi (small κ) [] and a small t g improve the shedule performane. Lemma. Suppose that after step (i) of the EC algorithm, eah link e of G v has link time t e, and the performane metri is µ 0. Moreover, assume that if a shedule is saled by ρ > 0, then µ is also saled by ρ. Therefore, after step (ii), the final link time is t e = min( te κt, t g e ) and the final performane metri is µ = min( µ κt, µ). g µ is the optimization goal suh as throughput, energy onsumption, et. Proof. If κt g, then G v an be sheduled in unit time, and t e = t e and µ = µ. On the other hand, if κt g >, then G v needs κt g time to shedule. To fit in the unit time shedule, we perform the saling. Afterwards, t e = te κt and µ = µ g κt. g Sine minimum edge oloring of an arbitrary graph is NPomplete [], we have to employ approximation algorithms. We hoose a simple multigraph edge-oloring algorithm by Karloff et al [3]. It uses at most 3 (G)/ olors, where (G) is the maximal node degree of a multigraph G. The following lemma gives the upper bounds on (G m ), the number of verties V m and edges E m of G m. Lemma 3. ( (G m ) W + R + /t g ). V m W + R and E m < W + (R )W + W +R t. g Proof. G v ontains W + R nodes due to the enb expansion. For a expanded enb node, the maximum degree is no more than W. For a mmbs, the maximum degree is no more than W + R sine the direted graph G v ontains no yles as all yles an be eliminated by shortening the link time. So (G v ) W + R. G m is transformed from G v by installing t e t g edges for eah edge e in Gv. Therefore, the degree of a node v in G m is deg(v) = te < v V v t g v V v ( ) te t g + t g + W + R In addition, we have V m = V v W + R. Using the degree sum formula, the number of edges in G m is E m = deg(v) < ( ) te t g + v V m v V v ( W + (R )W + W + R ) t g. Sine R W (W RF hains is suffiient to serve all mmbss), from the above Lemma, we have (G m ) = O(W + t g ). V m = O(W ) and E m = O(W + W t g ). In the following, we show the quality and time omplexity of the step (ii) of the EC algorithm. Pratially, step (i) is always muh faster than step (ii). Theorem. Let the performane metri after step (i) be µ 0. Then step (ii) ahieves the performane metri µ > 3[(W +R+)t g +] µ and it has time omplexity of O([ W + ] W t log(w + g t )). g Proof. The Karloff s algorithm uses κ 3 (G m )/ olors. Due to Lemma 3, κ 3 (W + R + t )/. Hene, g ( W + R + ) κt g < 3t g t g + = 3 (W + R + )tg + 3. Aording to Lemma, the final performane metri µ = min( κt g, )µ > 3[(W + R + )t g + ] µ The time omplexity of step (ii) is determined by Karloff s edge oloring algorithm, whih has the same time omplexity as the perfet edge oloring of a bipartite graph of max- 6

7 imum degree (G m )/ and number of edges O( E m ). Sine the time omplexity of the perfet edge oloring of a graph with E edges and degree is O( E log ) [4], the time omplexity of step (ii) is O( E m log( (G m )/ )) = O( [ W t + W ] log(w + g t )). g For the MTFS problem, let the optimal max-min throughput be θ. Sine step (i) of the EC algorithm uses a looser onstraint set than the preise set of the shedule polyhedron, it gives a θ θ. From Theorem, we have that the final maxmin throughput of the EC algorithm θ > 3[(W +R+)t g +] θ. In a typial bakhaul network, we have W R, so 3[(W +R+)t g +] 3(W t g +). This means to keep a onstant performane quality, we an hoose t g to be inversely proportional to W, i.e., a bigger network requires a smaller t g. If t g is so seleted, then the time omplexity is O(W log(w ), whih is quite salable with the number of mmbss, and thus feasible at the enb in pratie. Moreover, by setting the granularity t g 0, the performane metri approahes µ > 3 µ. For the MTFS problem, this means θ > 3 θ. VII. EXTENSION TO MORE GENERAL SCENARIOS In this setion, we show that our shedule-oriented optimization method proposed in Se. V and the edge-oloring based approximation algorithm in Se. VI an be extended to more general senarios of (i) bakhaul and aess networks and (ii) multiple RF hains at eah node. A. Extension to bakhaul and aess networks The bakhaul and aess networks add an additional layer of UEs to the bakhaul networks. Eah UE has a single RF hain and is allowed to have links with one or more mmbss. Let U denote the set of UEs. For the downlink traffi, only one-diretional links from mmbss to UEs exist. We illustrate as an example the solution of the MTFS problem. The max-min fairness in throughput is now defined for the UEs, as they are the destinations. Let G be the bakhaul and aess network after enb expansion, and A be the node-mathing matrix of G. We define A W and A U as the submatries of A related to the nodes in W (mmbss) and in U (UEs), respetively. The linear program () to ompute max-min throughput θ needs to be modified as follows: (b) should be replaed by the onstraints of (8). A U t S θ and A M t S = 0 (8) here, (8) expresses the onstraint that the throughput at a UE must be at least θ and eah mmbs is a pure relay. Obviously, the new MTFS problem an be solved with the same optimization tehnique as proposed in Se. V. As for the EC algorithm, we need to replae the data flow onstraint of (6b) with the following (9). e t e e t e = 0 v W (9a) e δ (v) e δ (v) e t e e δ + (v) e δ + (v) e t e θ v U (9b) v v Expansion v v v v v v v 3 Fig. 5. Node expansion. v has RF hains and v has 3 RF hains. B. Extension to multiple RF hains at eah node Now we remove the restrition that all nodes exept the enb have single RF hain. The tehnique to deal with this problem is the so-alled node expansion whih extends the enb expansion. Node expansion. Let G be a direted graph representing the network. We reate a expanded graph G as follows: for eah node v in G, we reate R v expanded nodes v,..., v Rv in G where R v is the number of RF hains at v. The expanded nodes of v are olletively alled a super node v in G. Moreover, if there is a link of apaity between node v and node v in G, then we install R v R v links between all ombinations of v i, i =,..., R v and v j, j =,..., R v. Eah link (v i, v j ) is assigned the apaity. An example is shown in Fig. 5. After the node expansion, the onstraint of the RF hains is impliitly guaranteed by the mathings in G. For the MTFS problem, the definition of the node-mathing matrix A W needs adaptation beause now multiple links an be inident to a super node in a mathing in G. A row of A W orresponds to an mmbs super node (a olletion of expanded nodes) and a olumn of A W orresponds to a mathing in G. Therefore, an element of the matrix a W i,j has the value of the sum apaity of all links entering the i-th mmbs super node in the j-th mathing minus the sum apaity of all links leaving the i-th mmbs super node in the j-th mathing. As for the EC algorithm, we need to modify the neessary shedule onstraints in (5). Sine the maximum number of simultaneous mmbs-to-mmbs links inreases for R times when eah mmbs has R RF hains. This leads to R times more variables of t k in (5), whih may lead to long omputation time in linear program. So, we delete the onstraints on t k. The following simple set of neessary shedule onstraints is used to replae (5), t e R v v V, and t e 0 e E. Another modifiation is to replae the expand enb step in Alg. 3 with the following step. Expand node. We perform the node expansion on G r. Let the resulting graph be G v. For a given link (v, v ) in G r with link time t (v,v ), we assign the link time t (v,v ) R vr to the links (v v i, v j ) in G v for all ombinations of i =,..., R v and j =,..., R v. VIII. NUMERICAL EVALUATION In this setion, we evaluate the optimal MTFS algorithm and EC-based approximation algorithm in terms of max-min throughput, network throughput and runtime effiieny. A. Evaluation setting We simulate a mmwave bakhaul network, where n n mmbss are plaed on the intersetions of a n n grid and 7

8 TABLE I SIMULATION PARAMETERS. Parameter Value 0.8 Distane between mmbss, d g Path loss parameters α, β, σ in P L(d) = α + 0β log 0 d + ξ Transmission power, p tx Diretivity gain, g x Bandwidth, b Noise N 0 = kt 0 +F +0 log 0 b Minimum SINR threshold, τ 00 m LOS: α = 6.4, β =, σ = 5.8 NLOS: α = 7, β =.9, σ = db 30 db GHz kt 0 = 74 dbm/hz, F = 4 db 5 db number of mmbs's (a) max-min throughput θ the enb is plaed in the enter of the grid. The distane between two neighboring mmbss is d g. The apaity of eah link is alulated with the hannel model desribed in Se. III. We assume a arrier frequeny of 8 GHz. The hannel state between any two nodes is simulated aording to the statistial model derived from the real-world measurement [5]. The hannel state has three possibilities LOS (line-of sight), NLOS (non line-of-sight) or outage. The simulation parameters are listed in Tab. I. The proposed algorithms are implemented in MATLAB, with the exeption that the optimal MTFS algorithm uses a C++ implementation for maximum weighted mathing [6]. run time (seonds) number of mmbs's (b) run time B. mmbs with single RF hain We evaluate the MTFS sheduling by varying the number of mmbss from 4 4 to 6 6. The enb has R = 0 RF hains and all the other nodes have single RF hain. For eah network size, 30 instanes of link apaities are randomly generated and then the network is sheduled for the MTFS problem. The performane results are shown in Fig. 6. As expeted, the optimal MTFS algorithm (OPT-MTFS) always attains the highest max-min throughput (Fig. 6(a)). In ontrast, the maxmin throughput of the EC-based approximation algorithm (EC) is smaller. However, the value is signifiantly better than the theoretial lower bound of Theorem (dashed lines in the figure). It goes up with an inrease in granularity (orresponding to a lower t g ). Pratially, t g in the range of 0.0 to 0.00 is ideal for the network size of up to 00 nodes, as on average, the EC algorithm ahieves 70% to 90% of the optimal max-min throughput. The prie for the high max-min throughput is a derease in runtime effiieny. For a network of 56 mmbss, the EC algorithm with t g = 0.00 runs 00x faster than the optimal MTFS algorithm (Fig. 6(b), note the log sale). Moreover, with the inrease of the number of nodes, the runtime of the EC algorithm grows more slowly than the optimal MTFS algorithm, whih shows the better salability of the former for large networks. As the goal of the MTFS is to maximize the network throughput under the fairness ondition, we also ompare the network throughput of the optimal MTFS algorithm, the EC algorithm and the unonditional maximum network throughput (MAX-TPUT). The MAX-TPUT ahieves the maximum network throughput for a given network. It is obtained when the min(r, L) (L is the number of mmbss diretly onneted to the enb) enb-to-mmbs links with the highest apaities Fig. 6. Performane of the MTFS algorithm (OPT-MTFS) and the EC-based approximation algorithm (EC) under the ondition of single-rf-hain mmbss. The figures show the average performane and ± standard deviation. are ative throughout the unit shedule and all the other links are inative. If the number of mmbss W > L, some of the mmbss will have zero throughput. This is the worst ase with respet to max-min fairness in throughput. Our evaluation results show that, on average, max-min fairness limits the network throughput to be approximately half of the maximum value. Sine the simulated bakhaul network is well-onneted, the network throughput is in most ases equal to the max-min throughput times the number of mmbss. Therefore, the relative performane of the network throughput between the optimal MTFS algorithm and the EC algorithm at different granularities is almost the same as that of the maxmin throughput in Fig. 6(a). C. mmbs with multiple RF hains We now evaluate the performane for the situation that eah mmbs is equipped with multiple RF hains. For that purpose, we simulate a bakhaul network with 0 0 mmbss and evaluate the ases that the enb has 0 RF hains and eah mmbs has R W RF hains, with R W varying from to 0. For eah given R W, 30 instanes of random link apaities are generated. As shown in Fig. 7(a), the max-min throughput goes up steadily with R W. The optimal max-min throughput at R W = 0 is over 4 times higher than the value at R W =. The differene in performane is due to the larger number of simultaneous links in the setting of multi-rf-hain mmbss. Evaluation results show that the number of simultaneous links is almost proportional to R W, as node expansion has inreased the number of nodes for R W times. Therefore, to attain higher 8

9 run time (seonds) (a) max-min throughput θ (b) run time Fig. 7. Performane omparison for different number of RF hains at mmbss. The figures show the average performane and ± standard deviation. throughput at eah mmbs, an option is to equip mmbss with multiple RF hains. However, the run time of the optimal MTFS algorithm also inreases with R W (Fig. 7(b)). The extra time is spent in the maximum weighted mathing in an expanded network with roughly R W times more nodes and RW times more edges. By using the EC-algorithm with t g = 0.00, we ahieve 85% to 90% of the optimal max-min throughput while using % to 0% of the time. IX. CONCLUSIONS The paper presents an optimal joint routing and sheduling method shedule-oriented optimization for mmwave ellular networks based on mathing theory. It an solve any problem that an be formulated as a linear program whose variables are link times and QoS metris. The method is demonstrated to be effiient in pratie, apable of solving the maximum throughput fair sheduling (MTFS) problem within a few minutes for over 00 mmbss. For better runtime effiieny, an edge-oloring based approximation algorithm is presented, whih runs 5 to 00 times faster than the optimal algorithm while ahieving over 80% of the optimal performane. In summary, the proposed optimal and approximation algorithms are highly pratial for mmwave ellular networks. ACKNOWLEDGEMENT This work has been supported by the German Researh Foundation (DFG) in the Collaborative Researh Center (SFB) 053 MAKI: Multi-Mehanism-Adaptation for the Future Internet and by LOEWE NICER. It has also been partially supported by the Minister of Siene and Tehnology (MOST) of Taiwan under Grants MOST 03-9-I-0-55 and MOST 04-9-I Lin s work was ompleted during his visit to the Center for Advaned Seurity Researh Darmstadt (CASED), Tehnishe Universität Darmstadt, Germany, during 04 to 06. REFERENCES [] T. S. Rappaport et al., Millimeter wave mobile ommuniations for 5G ellular: It will work! IEEE Aess, vol., pp , 03. [] S. Rangan, T. S. Rappaport, and E. Erkip, Millimeter-wave ellular wireless networks: Potentials and hallenges, in Proeedings of the IEEE, 04. [3] S. Singh, M. N. Kulkarni, A. Ghosh, and J. G. Andrews, Tratable model for rate in self-bakhauled millimeter wave ellular networks, IEEE Journal on Seleted Areas in Communiations, vol. 33, no. 0, pp. 96, 05. [4] A. Ghosh et al., Millimeter-wave enhaned loal area systems: A high-data-rate approah for future wireless networks, IEEE Journal on Seleted Areas in Communiations, vol. 3, no. 6, pp. 5 63, 04. [5] Y. Niu et al., Exploiting devie-to-devie ommuniations in joint sheduling of aess and bakhaul for mmwave small ells, IEEE Journal on Seleted Areas in Communiations, vol. 33, no. 0, pp , 05. [6] Y. Zhu et al., QoS-aware sheduling for small ell millimeter wave mesh bakhaul, in 06 IEEE ICC, 06, pp. 6. [7] W. Feng et al., Millimetre-wave bakhaul for 5g networks: Challenges and solutions, Sensors, vol. 6, no. 6, p. 89, 06. [8] Y. Li et al., A joint sheduling and resoure alloation sheme for millimeter wave heterogeneous networks, in WCNC, 07, pp. 6. [9] B. Hajek and G. Sasaki, Link sheduling in polynomial time, IEEE Transations on Information Theory, vol. 34, no. 5, pp , 988. [0] P. K. Huang, X. Lin, and C. C. Wang, A low-omplexity ongestion ontrol and sheduling algorithm for multihop wireless networks with order-optimal per-flow delay, IEEE/ACM Transations on Networking, vol., no., pp , 03. [] V. Angelakis et al., Minimum-time link sheduling for emptying wireless systems: Solution haraterization and algorithmi framework, IEEE Transations on Information Theory, vol. 60, no., pp , 04. [] S. Hariharan and N. B. Shroff, On sample-path optimal dynami sheduling for sum-queue minimization in trees under the k-hop interferene model, in IEEE INFOCOM, 0, pp [3] B. Ji, G. R. Gupta, and Y. Sang, Node-based servie-balaned sheduling for provably guaranteed throughput and evauation time performane, in IEEE INFOCOM, 06. [4] J. Edmonds, Maximum mathing and a polyhedron with 0, verties, J. of Res. the Nat. Bureau of Standards, vol. 69 B, pp. 5 30, 965. [5] Extended version of the paper, [6] J. Tang, G. Xue, and W. Zhang, Maximum throughput and fair bandwidth alloation in multi-hannel wireless mesh networks, in IEEE INFOCOM, 006, pp. 0. [7] L. Tassiulas and S. Sarkar, Maxmin fair sheduling in wireless networks, in IEEE INFOCOM, vol., 00, pp [8] L. Khahiyan, Polynomial algorithms in linear programming, USSR Computational Mathematis and Mathematial Physis, vol. 0, no., pp. 53 7, 980. [9] G. L. Nemhauser and S. Park, A polyhedral approah to edge oloring, Oper. Res. Lett., vol. 0, no. 6, pp. 35 3, 99. [0] G. B. Dantzig et al., The generalized simplex method for minimizing a linear form under linear inequality restraints, Paifi Journal of Mathematis, vol. 5, no., pp , 955. [] S.-i. Nakano, X. Zhou, and T. Nishizeki, Edge-oloring algorithms. Berlin, Heidelberg: Springer Berlin Heidelberg, 995, pp [] I. Holyer, The NP-ompleteness of edge-oloring, SIAM Journal on Computing, vol. 0, no. 4, pp , 98. [3] H. J. Karloff and D. B. Shmoys, Effiient parallel algorithms for edge oloring problems, J. Algorithms, vol. 8, no., pp. 39 5, 987. [4] R. Cole, K. Ost, and S. Shirra, Edge-Coloring Bipartite Multigraphs in O(E log D) Time, Combinatoria, vol., no., pp. 5, 00. [5] M. R. Akdeniz et al., Millimeter wave hannel modeling and ellular apaity evaluation, IEEE Journal on Seleted Areas in Communiations, vol. 3, no. 6, pp , 04. [6] V. Kolmogorov, Blossom V: a new implementation of a minimum ost perfet mathing algorithm, Mathematial Programming Computation, vol., no., pp ,