Spectral Graph Theory Based Resource Allocation for IRS-Assisted Multi-Hop Edge Computing
aa r X i v : . [ c s . I T ] F e b Spectral Graph Theory Based Resource Allocationfor IRS-Assisted Multi-Hop Edge Computing
Huilian Zhang † , Xiaofan He † , Qingqing Wu ‡ , and Huaiyu Dai ∗ † School of Electronic Information, Wuhan University, Wuhan 430072, China, ‡ State Key Laboratory of Internet of Things for Smart City, University of Macau, Macao 999078, China, ∗ Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27606, USAE-mails: { huilianzhang, xiaofanhe } @whu.edu.cn, [email protected], [email protected] Abstract —The performance of mobile edge computing (MEC)depends critically on the quality of the wireless channels. Fromthis viewpoint, the recently advocated intelligent reflecting sur-face (IRS) technique that can proactively reconfigure wirelesschannels is anticipated to bring unprecedented performance gainto MEC. In this paper, the problem of network throughputoptimization of an IRS-assisted multi-hop MEC network isinvestigated, in which the phase-shifts of the IRS and the resourceallocation of the relays need to be jointly optimized. However,due to the coupling among the transmission links of differenthops caused by the utilization of the IRS and the complicatedmulti-hop network topology, it is difficult to solve the consideredproblem by directly applying existing optimization techniques.Fortunately, by exploiting the underlying structure of the networktopology and spectral graph theory, it is shown that the networkthroughput can be well approximated by the second smallesteigenvalue of the network Laplacian matrix. This key findingallows us to develop an effective iterative algorithm for solvingthe considered problem. Numerical simulations are performed tocorroborate the effectiveness of the proposed scheme.
I. I
NTRODUCTION
To support the various computation-intensive services in thenew information era, mobile-edge computing (MEC) [1] hasemerged as a promising computing paradigm. The core idea ofMEC is to push the computing resource to the network edgeand allow mobile devices (MDs) to offload their computationtasks to the nearby edge servers for further processing [2]–[4].As a result, its performance depends critically on the quality ofthe communication links between the MDs and edge servers.As a recently emerging communication technology, an in-telligent reflecting surface (IRS) [5]–[10] is composed of alarge number of passive reflection elements, and each of theelements can induce phase-shift of the incident signals. By co-ordinating the reflections of all elements, the reflected signalscan add constructively at a desired point, thus improving thetransmission link quality. Driven by these appealing features,there is a recent surge of interests in studying IRS-assistedMEC systems [11]–[14] where an IRS is deployed to assisttask offloading by mitigating the signal propagation inducedimpairments and improving the quality of the correspondingtransmission links. For example, in [11] and [12], IRS is
This work was supported in part by the NSFC Grant No. 61901305, theWuhan University Start-up Grant No. 1501–600460001, the grants SRG2020-00024-IOTSC and FDCT 0108/2020/A, as well as the NSF grants ECCS-1444009 and CNS-1824518. leveraged to assist the task offloading from single-antannaMDs to an edge server co-located with a multi-antenna accesspoint (AP). In [13], IRS is employed in a wireless-poweredMEC network to improve the links both for wireless energytransfer and task offloading. In addition, IRS is adopted toimprove the wireless channel quality of a millimeter-wave-MEC system in [14] for a lower task offloading latency.Nevertheless, the research on IRS-assisted MEC is still in itsinfancy, and the existing relevant works are mainly focused onsingle-hop MEC. In many scenarios, multi-hop MEC is neededto achieve more satisfactory performance. For example, in adisaster site or urban areas with poor cellular coverage, usersmay need to request computing services from a remote edgeserver through a multi-hop ralay network. However, multi-hopMEC is more sensitive to propagation induced impairments ascompared to the single-hop case. To the best of our knowledge,how to leverage IRS to improve the performance of multi-hopMEC still remains largely unexplored.Motivated by the above, an IRS-assisted multi-hop MECnetwork is studied in this paper. Specifically, to maximizethe network throughput, the phase-shifts of the IRS as wellas the power and bandwidth allocation of the relays in theedge network need to be jointly optimized. However, thisis highly non-trivial. In particular, the utilization of the IRSbrings coupling among the transmission links of differenthops. Besides, the complicated multi-hop network topologymakes it more difficult to derive a closed-form expressionof the network throughput, which thus prevents us fromdirectly applying existing optimization techniques to solve theconsidered problem.To overcome this technical challenge, the network through-put optimization of the considered IRS-assisted multi-hopMEC network is converted into a max-flow problem in adirected-graph. By exploiting the underlying structure of thenetwork topology, it is shown that the max-flow in this directedgraph is equivalent to that of its undirected counterpart. Basedon prior results from spectral graph theory [15]–[18], a directconsequence of this key finding is that the network throughputcan be well approximated by the second smallest eigenvalue ofthe network Laplacian matrix, whose gradient can be readilycomputed. This in turn allows us to effectively increase thenetwork eigenvalue (and thus approximately the throughput)by adjusting the IRS phase-shifts as well as the power and
Copyright © 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, includingreprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of anycopyrighted component of this work in other works by sending a request to [email protected]. andwidth allocation via the generic gradient descent method.To the best of our knowledge, this work is among the firstto investigate IRS-assisted multi-hop MEC and analyze itsthroughput from the perspective of spectral graph theory.II. S
YSTEM M ODEL AND P ROBLEM F ORMULATION
In this section, the system model will be presented first,followed by the problem formulation.
IRS yx z L l l l l Task OffloadingLocal Computing r h D SourceRelayDestination r y Fig. 1. An IRS-assisted multi-hop MEC system.
A. System Model
Consider an IRS-assisted multi-hop MEC system, as depictedin Fig. 1. In this scenario, a terrestrial MD (i.e., the sourcenode) intends to offload its tasks to a remote edge server (i.e.,the destination node). In addition, it is assumed that thereare multiple MDs in between and they can form a multi-hopedge relay network. For the ease of description, the sourcenode, the relay nodes, and the destination node are numberedfrom to N ; define N ∆ = { , , ..., N − } . In contrast tothe conventional relays [19] that merely perform forwarding,the edge relays (i.e., the intermediate MDs) considered in thiswork can also help execute part of their received computationtasks by local computing. Specifically, node- i ( i ∈ N ) canlocally compute a portion of the tasks and offload the rest tonode- ( i + 1) . To facilitate task offloading, an off-the-shelf IRSwith M reflection elements is deployed in the environment(e.g., on the facade of nearby building).Denote by L the source-to-destination (s-d) distance. Asdepicted in Fig. 1, the source node, the relay nodes, and thedestination node are assumed to be located in a line. Thelocations of the source and the destination nodes are denotedby ( D, , and ( D, L, , respectively. The IRS is assumedto be a uniform rectangular array (URA) with M z rows and M y columns, and is located in the y-z plane. Without lossof generality, the bottom-left reflection element of the IRSis taken as the reference point to represent the location ofthe IRS, which is assumed to be h r meters (m) above theground. The horizontal coordinate of node- i ( i ∈ N ) is givenby q i = [ D, ( i − · l ] T , with l = L/ ( N − . The horizontalcoordinate of the IRS is q r = [0 , y r ] T , with < y r < L .
1) Task Offloading Model:
Assisted by the IRS, the channel H i ∈ C from node- i to node- ( i + 1) is given by H i = h Hr,i +1 Θh i,r + h i,i +1 , (1)with h i,i +1 ∈ C denoting the coefficient of the direct channelfrom node- i to node- ( i + 1) , h i,r ∈ C M × denoting thechannel vector from node- i to the IRS, and h Hr,i +1 ∈ C × M being the channel vector from the IRS to node- ( i + 1) . The diagonal matrix Θ = diag( e jθ ,e jθ , . . . ,e jθ M ) is the phase-shift matrix of the IRS, with θ m ∈ [0 , π ) being the phase-shift of the m -th ( m ∈ M , M ∆ = { , , ..., M } ) element ofthe IRS, and j ∆ = √− being the imaginary unit. To avoidinter-channel interference, frequency-division multiple accessis adopted for task offloading. Denote by B the total availablesystem bandwidth, and η i ∈ (0 , the fraction of bandwidthallocated to the link from node- i to node- ( i + 1) . Then, itfollows that P N − i =1 η i = 1 . The transmission data rate R oi fortask offloading from node- i to node- ( i + 1) is given by R oi = η i B log (cid:16) µ i P i | H i | ) / ( η i BN ) (cid:17) , (2)with N denoting the noise power spectral density, P i denotingthe total power for task offloading and local computing ofnode- i , and µ i ( < µ i < ) denoting the fraction of powerallocated for task offloading. Therefore, the power/bandwidthallocation vectors of all the MDs can be compactly writtenas µ ∆ = [ µ , µ , . . . , µ N − ] T and η ∆ = [ η , η , . . . , η N − ] T ,respectively.
2) Channel Model:
The wireless links between the IRS andthe MDs are assumed to be dominated by LoS components dueto the flexible IRS deployment. Rician fading is assumed forthe direct link between eack pair of MDs, since there are usu-ally more scatters and obstacles near the ground [6]. Denotethe distances between node- i and the IRS and that between theIRS and node- ( i +1) by d i,r and d r,i , respectively. The channelvectors from node- i to the IRS and that from the IRS to node- ( i + 1) are given respectively by h i,r = q ρ ( d i,r /d ) − α ¯h i,r and h Hr,i = q ρ ( d r,i /d ) − α ¯h Hr,i , with ρ the path-loss at thereference distance d = 1 (m) and α the path-loss exponent ofthe LoS links. Here, ¯h i,r ∈ C M × and ¯h Hr,i ∈ C × M representrespectively the corresponding normalized LoS paths [6]. Thechannel coefficient from node- i to node- ( i + 1) is given by h i,i +1 = χ · s β β h LoS i,i +1 + r
11 + β h
NLoS i,i +1 ! , (3)with χ = q ρ ( d i,i +1 /d ) − α capturing the path-loss effect.The notations α and β denote the corresponding path lossexponent and the Rician factor, respectively, and d i,i +1 isthe distance between this pair of nodes. h LoS i,i +1 and h NLoS i,i +1 denote respectively the normalized LoS component with unitmodulus and the circularly-symmetric-complex-Gaussian dis-tributed (with zero mean and unit variance) normalized NLoS(Rayleigh fading) component.
3) Local Computing Model:
Based on the above discus-sions, the power for local computing at node- i is given by P li = (1 − µ i ) P i . It follows that P li = κf i , with f i the corre-sponding CPU frequency, and κ a coefficient depending on thechip architecture [2], [20] . Accordingly, it has f i = p P li /κ .Assume that ε CPU cycles are required to process one bit oftask locally [2], and hence the data rate of local computing atnode- i is given by R li = q P li /κ/ε = p (1 − µ i ) P i /κ/ε. (4) . Problem Formulation To maximize the throughput (i.e., the number of task processedin unit time) of the IRS-assisted multi-hop MEC network, amax-flow problem [16] is formulated as follows. In particular,the task-handling procedure of the considered IRS-assistedmulti-hop MEC network can be represented by a directedgraph G d as depicted in Fig. 2. In G d , the solid lines witharrows represent single-hop task-offloading, while the dottedlines with arrows represent local computing. The associatededge weights R oi and R li are determined by the task offloadingrate and the local computing rate specified in (2) and (4),respectively. The rationale behind the virtual edges representedby the dotted lines is as follows: When a task is locallycomputed by a certain node- i , it is equivalent to, in termsof network throughput, the case that this task is sent to theserver with negligible transmission delay and then processedthere. i N - N oi R l R li R o R o R o lN N R R - - + l R Fig. 2. The directed graph G d for the considered multi-hop MEC network. From the above description, it is not difficult to verify thatthe throughput of this IRS-assisted multi-hop MEC networkis essentially the max-flow F s,d from the source node (i.e.,node- ) to the destination node (i.e., node- N ) in the directedgraph G d . Hence, the objective of this work is to maximizethe max-flow F s,d by jointly optimizing the phase-shifts Θ ofthe IRS, the power allocation µ of the MDs, and the systembandwidth allocation η . Mathematically, the problem can beformulated as(P1): max Θ , µ , η F s,d (5a)s.t. ≤ θ m ≤ π, ∀ m ∈ M , (5b) ≤ µ i , η i ≤ , ∀ i ∈ N , (5c) X N − i =1 η i = 1 . (5d)One of the main difficulties in solving the above problem isthat there exists no closed-form expression of F s,d , whichmakes the conventional optimization methods inapplicable.To overcome this difficulty, a spectral graph theory basedoptimization method will be proposed in the next section.III. T HE P ROPOSED S CHEME
In this section, a spectral graph theory based method isdeveloped to solve the optimization problem (P1). The basicidea of the proposed solution is to first use an appropriatespectral graph quantity to bound the objective function in (P1),and then use the gradient descent method to gradually optimizethis spectral graph quantity, hoping to achieve a reasonablygood solution of (P1). Similar to existing literature [21], [22], it is assumed in this work that thetime of sending the computation results back to the source is negligible.
In particular, according to prior results on spectral graphtheory [18], the max-flow of an undirected graph can beproperly approximated by the so-called weighted Cheeger’sconstant. However, the graph G d in the considered problem isa directed one. To this end, consider another graph ˜ G that isidentical to G d , except that all the edges in ˜ G are undirected .As shown in Proposition 1, due to its special topologicalstructure, the max-flow F s,d in G d is equal to the max-flow ˜ F s,d in ˜ G . Proposition 1
For any directed graph G d with the structureas shown in Fig. 2, its max-flow from the source to thedestination equals that of the undirected graph ˜ G . Proof
Please see Appendix A in [23]. (cid:4)
Remark 1
The main merit of this result is to allow us toimprove the max-flow F s,d (or, equivalently, the networkthroughput) in the directed graph G d by using spectral graphtheoretic methods, as elaborated below. A. Preliminary on the Weighted Cheeger’s Constant
For the undirected graph ˜ G , its adjacency matrix [17] isdefined as A ∆ = [ a ij ] Ni,j =1 , where a ij ∆ = R oi , ∀ i ∈ N \ { N − } , j = i + 1 ,R li , ∀ i ∈ N \ { N − } , j = N ,R oN − + R lN − , i = N − , j = N , , otherwise . (6)The elements in the lower triangular part of A can be similarlyobtained by interchanging the subscripts i and j . Given theadjacency matrix A , the max-flow ˜ F s,d of ˜ G is given by [16] ˜ F s,d = min { S | s ∈ S,d ∈ ¯ S } X i ∈ S,j ∈ ¯ S a ij , (7)where S is a subset of nodes in ˜ G and ¯ S denotes itscomplement. Due to the combinational nature of (7), it is quitedifficult to directly optimize ˜ F s,d (by adjusting the phase-shifts Θ of the IRS, the power allocation µ of the MDs, and thesystem bandwidth allocation η ).Prior results from spectral graph theory [15], [18] reveal thatby properly assigning large weights to the source node s andthe destination node d , the corresponding weighted Cheeger’sconstant C can be used as a good estimate of ˜ F s,d . Specifically, C is defined as: C = min S P i ∈ S,j ∈ ¯ S a ij min (cid:8) | S | W , (cid:12)(cid:12) ¯ S (cid:12)(cid:12) W (cid:9) , (8)where | S | W = P i ∈ S w i is the weighted cardinality, with w i ≥ the weight assigned to node- i . In addition, the followingweighted Cheeger’s inequality holds [18] λ/ ≤ C ≤ p δ max λ/w min , (9)where w min ∆ = min i w i . The second smallest eigenvalue λ ofthe weighted Laplacian matrix L W is defined as λ ∆ = inf g ⊥ W / g T L W gg T g , (10)here L W ∆ = W − / LW − / , with the diagonal matrix W ∆ =diag { w , . . . , w N } ; L ∆ = D − A is the Laplacian matrix of theundirected graph with D ∆ = diag { δ , . . . , δ N } the generalizeddegree matrix and δ i ∆ = P { j | j = i } a ij . Remark 2
The above results reveal that ˜ F s,d can be approx-imated by C and both the upper and the lower bounds of C are increasing in λ . Consequently, the max-flow ˜ F s,d in ˜ G canbe improved by increasing the corresponding second smallesteigenvalue λ of the weighted Laplacian matrix. B. Joint Phase-Shifts, Power Allocation, and Bandwidth Allo-cation Optimization
Based on the above discussions, an effective algorithm isdeveloped in the following to obtain a good ˜ F s,d by increasing λ through jointly optimizing the phase-shifts Θ of the IRS,the power allocation µ of the MDs, and the system bandwidthallocation η .Let θ = diag { Θ } be a column vector consisting of theelements on the main diagonal of the maxtrix Θ . It followsfrom (1), (2), (4), (6), and (10) that λ is a function of thevariables Θ , µ , and η , and for the ease of presentation, write λ = λ ( x ) , where x = [ θ T , µ T , η T ] T . According to [18], if λ is differentiable with respect to (w.r.t.) the variables Θ , µ ,and η , λ can be improved by properly increasing the variablesalong the gradient direction ∇ λ ( x ) = { ∂λ/∂x k } M +2( N − k =1 ;here, { x k } Mk =1 , { x k } M + N − k = M +1 , and { x k } M +2( N − k = M + N correspondto θ , µ , and η , respectively. In the following, the computationof ∇ λ ( x ) is given in detail. Specifically, the partial derivativeof ∂λ/∂x k is given by [18] ∂λ∂x k = v T ∂ L W ∂x k v = X i,j (cid:18) v i √ w i − v j √ w j (cid:19) ∂a ij ∂x k , (11)where v is the eigenvector of the matrix L W correspondingto λ , v i ( v j ) is the i -th ( j -th) component of v . Therefore, toobtain the spatial gradient direction of λ , one need to compute { ∂a ij /∂x k } M +2( N − k =1 in advance. By observing Fig. 2, it isnot difficult to see that this boils down to compute the gradientof the offloading rate { ∂R oi /∂x k } M +2( N − k =1 and that of thelocal computing rate { ∂R li /∂x k } M +2( N − k =1 .Note that the direct channel cofficient from node- i to node- ( i + 1) is given by h i,i +1 = ξe jω , with ξ and ω the amplitudeand the phase of the complex element h i,i +1 , respectively.Similarly, it has h i,r = [ γ e jϕ , γ e jϕ , . . . , γ M e jϕ M ] T and h Hr,i +1 = [ δ e jψ ,δ e jψ , . . . ,δ M e jψ M ] . Therefore, the channel H i ∈ C from node- i to node- ( i + 1) can be written as H i = P Mm =1 δ m γ m e j ( θ m + ψ m + ϕ m ) + ξe jω = X Mm =1 δ m γ m sin( θ m + ψ m + ϕ m ) + ξ sin ω | {z } I i + j ( X Mm =1 δ m γ m cos( θ m + ψ m + ϕ m ) + ξ cos ω ) | {z } Q i . (12) According to (2) and (12), the partial derivative of R oi w.r.t.the phase-shift θ m of the m -th reflection element is given by ∂R oi ∂θ m = η i Bµ i P i ( η i BN + µ i P i | H i | ) · ln 2 · ∂ | H i | ∂θ m , (13)where ∂ | H i | ∂θ m = 2 I i δ m γ m cos( θ m + ψ m + ϕ m ) − Q i δ m γ m sin( θ m + ψ m + ϕ m ) . (14)Similarly, the other relevant partial derivatives are given by ∂R oi ∂µ i = η i BP i | H i | ( η i BN + µ i P i | H i | ) · ln 2 , (15) ∂R oi ∂η i = B log (cid:16) µ i P i | H i | η i BN (cid:17) − Bµ i P i | H i | ( η i BN + µ i P i | H i | ) · ln 2 , (16)and ∂R li /∂µ i = − P i ((1 − µ i ) P i /κ ) − / / (3 κε ) . (17)Moreover, according to the local computing model specifiedin Section II-A, it has ∂R li /∂θ m = 0 and ∂R li /∂η i = 0 .After obtaining the set of gradients { ∂λ/∂x k } M +2( N − k =1 by using (11)-(17), the value of λ can be improved byadjusting Θ , µ , and η along their gradient directions ∇ λ ( x ) = { ∂λ/∂x k } M +2( N − k =1 . Specifically, according to the first orderTaylor expansion λ ( x + ∆ x ) = λ ( x ) + ∇ λ ( x ) T ∆ x + o ( || ∆ x || ) , (18)for a given point x , the value of λ ( x +∆ x ) can be improvedby maximizing ∇ λ ( x ) T ∆ x . This indicates that λ can beoptimized by solving the following linear programming(P2): max ∆ Θ , ∆ µ , ∆ η M X m =1 ∂λ∂θ m · ∆ θ m + N − X i =1 ∂λ∂µ i · ∆ µ i + N − X i =1 ∂λ∂η i · ∆ η i , (19a)s.t. ≤ θ m +∆ θ m ≤ π, | ∆ θ m | ≤ ∆¯ θ m , ∀ m ∈ M , (19b) ≤ µ i + ∆ µ i ≤ , | ∆ µ i | ≤ ∆¯ µ i , ∀ i ∈ N , (19c) ≤ η i + ∆ η i ≤ , | ∆ η i | ≤ ∆¯ η i , ∀ i ∈ N , (19d) X N − i =1 ∆ η i = 0 , (19e)where ∆ Θ , ∆ µ , and ∆ η denote the adjustment vectors of thephase-shifts of the IRS, the power allocation of the MDs, andthe system bandwidth allocation, respectively. Constraint (19e)ensures that the total bandwidth remains unchanged. ∆ ¯Θ = { ∆¯ θ m } Mm =1 , ∆ ¯ µ = { ∆¯ µ i } N − i =1 , and ∆ ¯ η = { ∆¯ η i } N − i =1 aredefined as the upper bounds of adjustments to smooth theupdating procedure.Based on the above description, a joint phase-shifts, powerallocation, and bandwidth allocation optimization (JPPBO)algorithm is proposed as summarized in Algorithm 1. In Agorithm 1, ∆ ¯Θ , ∆ ¯ µ , and ∆ ¯ η are the predefined lower bounds of ∆ ¯Θ , ∆ ¯ µ , and ∆ ¯ η , respectively, and τ ( < τ < ) is a predefined scalingfactor.
10 20 30 40 50
Iteration index T h r oughpu t ( bp s ) Multi-IRSSingle-IRSBaseline 1Baseline 2Baseline 3
Fig. 3. Throughput v.s. number of iterations.
Number of hops T h r oughpu t ( bp s ) Multi-IRSSingle-IRS Baseline 1Baseline 2 Baseline 3
Fig. 4. Throughput v.s. number of hops.
Total bandwidth (MHz) T h r oughpu t ( bp s ) Multi-IRSSingle-IRS Baseline 1Baseline 2 Baseline 3
Fig. 5. Throughput v.s. total bandwidth.
Algorithm 1
The proposed JPPBO algorithm Input: Θ , µ , η , ∆ ¯Θ , ∆ ¯ µ , ∆ ¯ η , ∆ ¯Θ , ∆ ¯ µ , ∆ ¯ η . Compute the initial value of ˜ F s,d . for t = 1 , , . . . , T max do Compute the gradient { ∂λ/∂x k } M +2( N − k =1 . Compute ∆ Θ , ∆ µ , and ∆ η by solving (19). Generate temporary variables: ˆΘ ← Θ +∆ Θ , ˆ µ ← µ +∆ µ , and ˆ η ← η +∆ η . Precompute the magnitude of max-flow ˜ F pres,d . if ˜ F pres,d ≥ ˜ F t − s,d then Update the variables: Θ ← ˆΘ , µ ← ˆ µ , η ← ˆ η . Update ˜ F ts,d ← ˜ F pres,d , and record ˜ F ts,d . else Adjust the step size: ∆ ¯Θ ← max { τ ∆ ¯Θ , ∆ ¯Θ } , ∆ ¯ µ ← max { τ ∆ ¯ µ , ∆ ¯ µ } , ∆ ¯ η ← max { τ ∆ ¯ η , ∆ ¯ η } . Go to step 5. end if end for
Output: Θ , µ , η .IV. S IMULATION R ESULTS
In this section, numerical experiments are conducted to cor-roborate the effectiveness of the proposed scheme.The simulation parameters are as follows. The s-d distanceis set to L = 1000 (m), the distance from the s-d line to the y-z plane is set to D = 50 (m). For the MDs, the total power forlocal computing and task offloading is set to P i = 1 (W). Thealtitude of the reference element of the IRS is set to h r = 50 (m), and the total number of reflection elements of the IRS isset to M = 10000 , with M z = M y = 100 . Since the IRS isassumed to be deployed on a favorable altitude such that thechannels between the IRS and any of the MDs are dominatedby LoS paths, the attenuation factor of these LoS channelsis set to α = 2 . The attenuation factor for the Rician fadingchannels, i.e., the channels among the MDs, is set to α = 3 . ,since there are usually masses of obstacles and scatters on theground. Other parameters are set as ρ = − (dB), β = 3 (dB), ε = 700 , and κ = 1 × − . The number of nodesand the total system bandwidth are set to N = 10 and B = 1 (MHz), respectively, unless otherwise stated. For each scheme,1000 Monte-Carlo runs are conducted. In the simulations, both the single- and multi-IRS casesare considered with identical number of reflection elements.Specifically, for the single-IRS case, the y-coordinate of thereference element of the IRS is set to y r = L/ . For themulti-IRS case, it is assumed that there are K = 10 IRSs,with each being a × UPA. These IRSs are assumed tobe uniformly located above the projection of the s-d line in they-axis. Specifically, for the k -th ( k = 1 , , ..., K ) IRS, the y-coordinate of its reference element is y kr = ( k − L/ ( K − .In addition, to demonstrate the advantages of the proposedschemes, three baseline schemes are considered for compar-ison. Specifically, in baseline 1, the power allocation of theMDs and the system bandwidth allocation are jointly opti-mized without the aid of IRS. In baseline 2, no local computingis performed and the network nodes form a conventional multi-hop relay network assisted by a single-IRS. In baseline 3, thenetwork nodes form a conventional multi-hop relay networkwithout IRS.First, the throughput performance of the proposed schemeis compared with that of the baseline schemes in Fig. 3. Itis observed that the proposed scheme can achieve a largerthroughput as compared to the baseline schemes. The reasonis that proper phase-shifts of the IRS and proper bandwidthallocation can improve the transmisnion rate for task offloding,while proper power allocation of the MDs can help strike abalance between local computing and task offloading. Specifi-cally, the proposed scheme can achieve a throughput of about . × (bps) for the multi-IRS case, and about . × (bps) for the single-IRS case. Since there is no IRS deployedin baseline 1, the transmission rate for task offloading is notsatisfactory, and hence it can only achieve a throughput ofabout . × (bps). For baseline 2, although an IRS isdeployed to improve the qualities of the transmission links, itcan achieve a throughput of only about . × (bps), aslocal computing is not performed in beseline 2. Baseline 3gives the worst performance of about . × (bps), sinceneither local computing nor IRS is considered.The impact of the number of hops on the network through-put is shown in Fig. 4. Specifically, all parameters are keptunchanged, except for the number of hops. As it can be seenfrom Fig. 4, when the number of hops is , the single-IRSscheme can achieve the largest throughput, bacause the single-IRS is located above the middle point of the projection ofhe s-d line in the y-axis, resulting in the shortest distancebetween the IRS and the relay node. The throughput achievedby the multi-IRS scheme is slightly smaller than that of thesingle-IRS scheme, because these multiple IRSs are deployeduniformly in a line, leading to relatively longer distances be-tween the IRSs and the relay node. In addition, the achievablethroughput of baseline 1 increases in the number of hops.This is because of that the number of nodes performing localcomputing is positively correlated with the number of hops.In contrast, the achievable throughput of baseline 2 decreasesas the number of hops increases, as the nodes exhaust all theiravailable power for task offloading in this scheme. In this case,the network throughput hinges on the transmission rates ofall of the hops, which however decreases when more relaysshare the total bandwidth. For baseline 3, when the number ofhops is small, the bandwidth allocated to each hop is relativeabundant but the distance of each hop is relatively long. Asthe number of hops increases, less bandwidth is availableto each hop but the distance of each hop becomes shorter.As a result, the throughput of baseline 3 remains unchangedat a relatively low value. In contrast, the proposed schemecan achieve substantially higher throughputs than that of thebaselines, by concertedly exploiting the advantages of bothlocal computing and IRS-assisted transmission.Finally, the impact of the total bandwidth on the networkthroughput is investigated in the setting of an 8-hop network.As shown in Fig. 5, for all the schemes, the network through-puts increase as the total bandwidth increases. In particular,when the total bandwidth is sufficiently large, e.g., B = 10 (MHz), the performance of the proposed scheme and that ofbaseline 2 are quite close. The reason is that the achievablethroughput of the proposed scheme (single-IRS case) is domi-nated by the task offloading under this circumstance. However,when the available total bandwidth is limited, the throughputsof the proposed schemes are significantly larger than that ofthe baseline schemes. V. C ONCLUSION
In this paper, throughput optimization of an IRS-assistedmulti-hop MEC network is investigated. Due to the cou-pling among the transmission links of different hops andthe complicated multi-hop network topology, it is difficult toderive a closed-form expression of the network throughput.To tackle this challenge, the original problem is transformedinto a max-flow problem in directed graph. By exploiting thespecial structure of the directed graph and the results fromspectral graph theory, a joint phase-shifts, power allocation,and bandwidth allocation optimization algorithm is proposedto obtain a high-quality solution. Numerical results show thatthe proposed algorithm can achieve a substantially highernetwork throughput as compared to the baselines.R
EFERENCES[1] Y. Mao, C. You, J. Zhang, K. Huang, and K. B. Letaief, “A surveyon mobile edge computing: The communication perspective,”
IEEECommun. Surv. Tut. , vol. 19, no. 4, pp. 2322–2358, Aug. 2017. [2] X. He, R. Jin, and H. Dai, “Peace: Privacy-preserving and cost-efficienttask offloading for mobile-edge computing,”
IEEE Trans. WirelessCommun. , vol. 19, no. 3, pp. 1814–1824, Dec. 2019.[3] Y. Wang, M. Sheng, X. Wang, L. Wang, and J. Li, “Mobile-edge com-puting: Partial computation offloading using dynamic voltage scaling,”
IEEE Trans. Commun. , vol. 64, no. 10, pp. 4268–4282, Aug. 2016.[4] S. E. Mahmoodi, R. Uma, and K. Subbalakshmi, “Optimal joint schedul-ing and cloud offloading for mobile applications,”
IEEE Trans. CloudComput. , vol. 7, no. 2, pp. 301–313, Apr. 2019.[5] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment:Intelligent reflecting surface aided wireless network,”
IEEE Commun.Mag. , vol. 58, no. 1, pp. 106–112, Nov. 2019.[6] ——, “Intelligent reflecting surface enhanced wireless network viajoint active and passive beamforming,”
IEEE Trans. Wireless Commun. ,vol. 18, no. 11, pp. 5394–5409, Aug. 2019.[7] ——, “Beamforming optimization for wireless network aided by intelli-gent reflecting surface with discrete phase shifts,”
IEEE Trans. Commun. ,vol. 68, no. 3, pp. 1838–1851, Dec. 2019.[8] G. Zhou, C. Pan, H. Ren, K. Wang, M. Di Renzo, and A. Nallanathan,“Robust beamforming design for intelligent reflecting surface aidedMISO communication systems,”
IEEE Wireless Commun. Lett. , vol. 9,no. 10, pp. 1658–1662, Jun. 2020.[9] B. Di, H. Zhang, L. Song, Y. Li, Z. Han, and H. V. Poor, “Hybridbeamforming for reconfigurable intelligent surface based multi-usercommunications: Achievable rates with limited discrete phase shifts,”
IEEE J. Sel. Areas Commun. , vol. 38, no. 8, pp. 1809–1822, Jun. 2020.[10] S. Li, B. Duo, X. Yuan, Y.-C. Liang, and M. Di Renzo, “Reconfigurableintelligent surface assisted UAV communication: Joint trajectory designand passive beamforming,”
IEEE Wireless Commun. Lett. , vol. 9, no. 5,pp. 716–720, Jan. 2020.[11] T. Bai, C. Pan, Y. Deng, M. Elkashlan, A. Nallanathan, and L. Hanzo,“Latency minimization for intelligent reflecting surface aided mobileedge computing,”
IEEE J. Sel. Areas Commun. , vol. 38, no. 11, pp.2666–2682, Jul. 2020.[12] Y. Liu, J. Zhao, Z. Xiong, D. Niyato, Y. Chau, C. Pan, and B. Huang,“Intelligent reflecting surface meets mobile edge computing: Enhancingwireless communications for computation offloading,” arXiv preprintarXiv:2001.07449 , 2020.[13] T. Bai, C. Pan, H. Ren, Y. Deng, M. Elkashlan, and A. Nallanathan,“Resource allocation for intelligent reflecting surface aided wirelesspowered mobile edge computing in OFDM systems,” arXiv preprintarXiv:2003.05511 , 2020.[14] Y. Cao and T. Lv, “Intelligent reflecting surface enhanced resilientdesign for MEC offloading over millimeter wave links,” arXiv preprintarXiv:1912.06361 , 2019.[15] F. R. Chung and F. C. Graham,
Spectral graph theory . Providence, RI,USA: Amer. Math. Soc., 1997.[16] L. R. Ford and D. R. Fulkerson, “Maximal flow through a network,”
Can. J. Math. , vol. 8, pp. 399–404, 1956.[17] S. Bhattacharya and T. Bas¸ar, “Graph-theoretic approach for connectivitymaintenance in mobile networks in the presence of a jammer,” in
Proc.IEEE Conf. Decision Control (CDC) , Atlanta, Georgia, Dec. 2010.[18] X. He, H. Dai, and P. Ning, “Dynamic adaptive anti-jamming viacontrolled mobility,”
IEEE Trans. Wireless Commun. , vol. 13, no. 8,pp. 4374–4388, Apr. 2014.[19] H. Ju, S. Lim, D. Kim, H. V. Poor, and D. Hong, “Full duplexityin beamforming-based multi-hop relay networks,”
IEEE J. Sel. AreasCommun. , vol. 30, no. 8, pp. 1554–1565, Aug. 2012.[20] W. Zhang, Y. Wen, K. Guan, D. Kilper, H. Luo, and D. O. Wu, “Energy-optimal mobile cloud computing under stochastic wireless channel,”
IEEE Trans. Wireless Commun. , vol. 12, no. 9, pp. 4569–4581, Aug.2013.[21] J. Zhang, X. Hu, Z. Ning, E. C.-H. Ngai, L. Zhou, J. Wei, J. Cheng, andB. Hu, “Energy-latency tradeoff for energy-aware offloading in mobileedge computing networks,”
IEEE Internet Things J. , vol. 5, no. 4, pp.2633–2645, Dec. 2017.[22] C. Wang, C. Liang, F. R. Yu, Q. Chen, and L. Tang, “Computationoffloading and resource allocation in wireless cellular networks withmobile edge computing,”