research articledownloads.hindawi.com/journals/wcmc/2019/8416592.pdfmapping algorithm can be divided...

Research ArticleEPVNE An Efficient Parallelizable Virtual NetworkEmbedding Algorithm

Yuanzhen Li and Yingyu Zhang

School of Computer Science Liaocheng University Liaocheng Shandong 252059 China

Correspondence should be addressed to Yuanzhen Li liyuanzhen163com

Received 6 August 2019 Revised 22 October 2019 Accepted 31 October 2019 Published 22 November 2019

Academic Editor Hui Cheng

Copyright copy 2019 Yuanzhen Li and Yingyu Zhang is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

Virtual network embedding (VNE) problem is a key issue in network virtualization technology and much attention has been paidto the virtual network embedding However very little research work focuses on parallelized virtual network embedding problemswhich assumes that the substrate infrastructure supports parallel computing and allows one virtual node to be mapped to multiplesubstrate nodes Based on the work of Liang and Zhang we extend the well-known VNE to parallelizable virtual networkembedding (PVNE) in this paper Furthermore to the best of our knowledge we give the rst formulation of the PVNE problemA new heuristic algorithm named ecient parallelizable virtual network embedding (EPVNE) is proposed to reduce the cost ofembedding the VN request and increase the VN request acceptance ratio EPVNE is a two-stage mapping algorithm which rstperforms node mapping and then performs link mapping In the node mapping phase we present a simple and ecient virtualnode and physical node sorting formula and perform the virtual node mapping in order When mapping virtual nodes we mapvirtual nodes to physical nodes that just meet the CPU requirements Substrate nodes with more CPU resources will be retainedfor subsequent virtual network mapping requests In the link mapping phase Dijkstrarsquos algorithm is used to nd a substrate pathfor each virtual link Finally simulations are carried out and simulation results show that our algorithm performs better than theexisting heuristic algorithms

1 Introduction

As we all know the Internet has become one of the in-frastructures of todayrsquos social communications informationexchange economic and commercial operations andmultimedia services [1] e existing Internet architecturehas played an important role in promoting the rapid de-velopment of the Internet Various new applications emergein an endless stream and dierent applications have dif-ferent requirements for the network environment in termsof quality of service security and scalability [2] Howeverthe existing network architectures and protocols are dicultto meet the needs of new application development and thereis a certain degree of ossication As a new technology forbuilding a new Internet architecture network virtualizationcan eectively solve this bottleneck of the current Internet

In recent years network virtualization technology hasbeen widespread concern in industry and academia [3ndash5]

Network virtualization technology allows multiple virtualheterogeneous networks to coexist on top of a shared un-derlying physical network infrastructure Each virtual net-work is a piece of resources of the underlying physicalnetwork which consists of virtual nodes (for examplevirtual routers and cloud computing center) and virtual links[6] Additionally with the programmability of the un-derlying physical network virtual networks can run IP ornon-IP protocols in the selected network As a result net-work virtualization technologies can deploy new networkarchitectures protocols and applications without aectingexisting networks eectively supporting innovation innetwork technologies It not only provides a feasible way toevolve from the current Internet to the future network butalso provides one of the key features that the Internet shouldhave in the future [7]

In the virtualization network environments the vir-tualized network model divides the role of Internet service

HindawiWireless Communications and Mobile ComputingVolume 2019 Article ID 8416592 10 pageshttpsdoiorg10115520198416592

provider (ISP) into two separate entities infrastructureprovider (InP) and service provider (SP) [8 9] In such anetwork environment an infrastructure provider manages asubstrate network (SN) which is composed of substratenodes and physical communication links Infrastructureproviders use virtualization technology to abstract the un-derlying substrate resources into functional entities providea unified programming interface shield the underlyingphysical network resources from the upper layers and splitthe substrate resources into multiple virtual slices for use bydifferent service providers In addition infrastructure pro-viders can communicate and collaborate with each other tocreate a complete interdomain infrastructure for serviceproviders Service provider rents slices of substrate resources(eg nodes and links) and then creates his own customizedvirtual network (VN) to provide value-added services (egcontent distribution) to end users Service provider deployscustomized protocols in virtual networks and is responsiblefor running managing and maintaining virtual networksService provider generates a virtual network requestaccording to the needs of the service (eg CPU resourcesmemory resources bandwidth resources latency re-quirements and geographic location) and forwards therequest to the infrastructure provider to establish a virtualnetwork Regardless of whether the virtual network spansmultiple infrastructure providersrsquo domains the serviceprovider has a unified view of the entire virtual network

)e problem of allocating infrastructure network re-sources according to virtual network requests with node andlink resource constraints is called virtual network embed-ding (VNE) problem [10] )e virtual network embedding isan important challenge in the realization process of networkvirtualization technology and has become a hotspot formany scholars )e VNE problem is an NP-hard problem[11] Even after all virtual nodes have been mapped map-ping virtual links with bandwidth resource constraints is stillNP-hard )e virtual network embedding problem is one ofthe main challenges faced by network virtualization tech-nology It has become a hot issue in this research field andhas received extensive attention from researchers Manyresearches on VNE have been reported [5] However there isvery few focus on parallelizable virtual networks embedding[12] In the parallelizable virtual network embedding(PVNE) environment a virtual node could be mapped intomultiple substrate nodes With parallelization severalsubstrate nodes can parallely accomplish the computation towhich the virtual node is dedicated With parallelization asubstrate network can carry more virtual networks Forexample in cloud computing MapReduce processes large-scale data in parallel [13 14] Another important advantageof parallelization is its robustness that the computation canquickly migrate to other substrate nodes in the event of asubstrate node crash [15] With the development of cyber-physical system and cloud computing [16] the need forparallelization will increase Parallelization also providesnew ideas for the implementation of cyber-physical systemand cloud computing [17]

Consider the example in Figure 1 where Figure 1(a)shows a virtual network request and Figure 1(b) shows a

substrate network )e corresponding number near eachvertex (or edge) is the CPU (or bandwidth) capacitycon-straint )is virtual network request contains four virtualmachines (VMs) linked by four links If parallelization is notsupported this request would be rejected as there is nosubstrate node that has more than 50 units of available CPUresources which the virtual node d requires In PVNE afeasible mapping scheme is shown in Figure 1(c))emastermapping is a⟶B b⟶H c⟶C d⟶D and theslave mapping is a⟶Φ b⟶Φ c⟶Φ d⟶ F )elink mapping is ab⟶ BH Ac⟶ BC bed⟶HD CD⟶ CD In the mapping scheme ofFigure 1(c) it can be seen that the virtual nodes a b and c aremapped to one substrate node respectively However nophysical node can satisfy the resource requirements of thevirtual node d In this case a parallelized mapping scheme isrequired that is a virtual node is mapped to multiplesubstrate nodes Finally the virtual node d is mapped to thephysical nodes D and F

In this paper we study parallelizable virtual networkembedding (PVNE) and propose an efficient parallelizablevirtual network embedding (EPVNE) algorithm We sum-marize the main contributions here as follows

(i) Based on optimization theory and resource in-tegration technology an optimized mathematicalmodel for parallel virtual network embedding isproposed

(ii) A new heuristic algorithm named EPVNE is pro-posed to reduce the cost of embedding the VNrequest and increase the VN request acceptanceratio Firstly the substrate nodes and virtual nodesneed to be ordered by a metric proposed in thispaper which is simple and can be calculated effi-ciently Secondly when mapping each virtual nodenv our algorithm will search for a substrate nodewhose CPU value is just larger than nv Substratenodes with more CPU resources will be retained forsubsequent virtual network mapping requests

(iii) Extensive simulations have been performed toevaluate the performance of our new algorithm)eresults demonstrate that our algorithm performsbetter than the existing heuristic algorithms

)is paper is organized as follows the related work isintroduced in Section 2 In Section 3 we describe systemmodel and problem formulation We then present theproposed algorithm in Section 4 Section 5 introduces theexperimental conditions and related parameter settings andfocuses on the experimental results Finally Section 6concludes the paper

2 Related Work

Researchers have conducted in-depth research on virtualnetwork embedding algorithms from different perspectivesand a series of research results have emerged Several papers[3ndash5] summarize the virtual network mapping algorithmFischer A [5] summarized the research content of VNE

2 Wireless Communications and Mobile Computing

provided the classication scheme of the VNE algorithmand discussed further research works Haotong Cao [4]summarized the existing typical VNE exact solutions andproposed future research directions Since the VNE problemis NP-hard in terms of computational complexity manyVNE algorithms proposed in the literature are heuristicHaotong Cao investigated analyzed and summarized somerepresentative heuristic solutions in the paper [3] In generalthe virtual network mapping algorithm can be classied asfollows

21 Consider Both Node and Link Resource Constraints vsIgnore Node or Link Resource Constraints Depending onwhether the node and link resource constraints of the un-derlying network or virtual network are fully considered thevirtual network mapping algorithm is divided into a map-ping algorithm that considers both node and link resourceconstraints and a mapping algorithm that ignores node orlink resource constraints For example in the virtual net-work mapping process the literature [11 18] considers theresource constraints of nodes and links at the same timewhile the literature [19 20] ignores the resource constraintsof nodes or links e path splitting path migration andcustomized embedding algorithms can be used to obtainbetter mapping performance [11]

22 Static vs Dynamic Depending on the underlying net-work resource allocation method the virtual network

embedding algorithm can be divided into static embeddingalgorithms [11 21] and dynamic embedding algorithms [19] e static embedding algorithm refers to statically assigningxed underlying network resources to the virtual networkand the dynamic mapping algorithm dynamically allocatesthe underlying network resources according to the virtualnetworkrsquos own resource requirements e dynamicreconguration mapping algorithm refers to dynamicallyadjusting the mapping scheme without changing the virtualnetwork resource requirements [19]

23 Online vs Oine According to the dierent processingmethods for virtual network requests the virtual networkembedding algorithm can be divided into oyenine embeddingalgorithms [22 23] and online embedding algorithms[11 18 21 24] Oyenine embedding means that all virtualnetwork request information is known before the embed-ding algorithm is implemented while the online embeddingalgorithm does not make any assumptions about the arrivaltime duration and topology information of the virtualnetwork request e oyenine embedding algorithm iscomputationally intensive and it is dicult to nd thesolution to the problem in polynomial time

24 Centralized vs Distributed Depending on the calcula-tionmethod of virtual networkmapping the virtual networkmapping algorithm can be divided into a centralizedmapping algorithm and a distributed mapping algorithm

a

c d

1

b

8

3

7

1010

20 50

(a)

A

B H

C D

9

34

13 17 F

35

77

20 40

4830

10 20

(b)

A

B H

C D

9

34

13 17 F

35

77

a b

dc

20 40

4830

10 20

dprime

(c)

Figure 1 Parallelizable virtual network embedding (a) Virtual network request (b) Substrate network (c) Substrate network withparallelization

Wireless Communications and Mobile Computing 3

)e centralized virtual network mapping algorithm allocatescorresponding resources for virtual network requestsaccording to the underlying network resource status by thecentral decision-making organization [11 21 24] Central-ized mapping algorithms can generally find the optimalsolution for VNE problems in small-scale networks How-ever this type of method has a common problem with asingle point of failure If the central node is attacked or failsthe entire map will fail In addition in large-scale networksthe scalability of the algorithm is a problem that needs to besolved )e distributed virtual network mapping algorithmsgenerally do not have a computing center and the virtualnetwork mapping process is completed through the un-derlying nodes [23] )e advantage of the distributedmapping algorithm over centralized algorithms is betterscalability However there are some disadvantages )eamount of calculation will be very large )ere is also anincrease in overhead )ere is a tradeoff between feasibilityand optimality

25 One Stage vs Two Stage According to the mappingorder of virtual nodes and virtual links the virtual networkmapping algorithm can be divided into one-stage mappingalgorithm and two-stage mapping algorithm )e nodemapping and link mapping of the one-stage mapping al-gorithm are completed in the same stage In other words thevirtual link in the one-stage mapping algorithm maps whenmapping virtual nodes [20 25] )e node mapping phase ofthe two-stage mapping algorithm is separate from the linkmapping phase )is type of algorithm generally maps allvirtual nodes first and then performs virtual link mapping[11 24]

26 Concise vs Redundant In the real substrate networkvarious failures often occur In a network virtualizationenvironment failure of a single physical network entity willaffect all virtual networks mapped to that physical networkentity In order to solve the problem a very realistic idea is toprovide redundancy or backup )e redundant virtualnetwork mapping refers to the provision of redundant re-sources in the virtual network mapping including noderesources and link resources to deal with node failures Incontrast the concise virtual network mapping algorithmdoes not provide redundant resources )e concise solutionuses only as few substrate resources as possible to meet theneeds of the proposed virtual network without leaving ad-ditional resources for any failures )is also means thatvirtual networks cannot be guaranteed to recover fromaccidental failures although more substrate resources can bereserved to embed more virtual networks

)e redundant VNE methods [26 27] reserve additionalresources for virtual networks to prevent certain substratecomponents from failing during operation )is type ofapproach can improve the reliability of mapping virtualnetworks However a tradeoff must be made between thereliability of the solution and its cost of embedding )ehigher the degree of reliability the more the substrate re-sources are consumed and the fewer the number of

embedded virtual networks In general redundant VNEalgorithms are more reliable and more popular than concisealgorithms

27 Heuristic Solutions vs Exact Solutions vs MetaheuristicSolutions Recently Cao et al divided the virtual networkmapping algorithm into exact solution [4] and heuristicsolution [3] )e exact solution solves the virtual networkmapping problem based on the optimization theory [28])eVNE problem can generally be abstracted into an integerlinear programming problem or a mixed integer pro-gramming problem )is problem can be solved usingbranch and bound branch and cut and branch and price Ofcourse there are also some software tools available forsolving this problem such as GLPK ALEVIN CPLEX andMatlab

Virtual network mapping problems can be solved usingthe heuristic algorithm [11 21 24] In the study of virtualnetwork mapping problems in order to obtain acceptableexecution time system performance can be appropriatelyreduced )at is the heuristic algorithm obtains a sub-optimal solution in an acceptable time )ese heuristic al-gorithms include the greedy method the k-shortest pathmethod the multicommodity solution and the subgraphisomorphism detection method

Some metaheuristic algorithms perform well in solvingsome problems such as flowshop scheduling problems[29ndash32] )erefore some researchers use the metaheuristicalgorithm [33ndash35] to solve the virtual network mappingalgorithm )ese researchers have made useful attempts andachieved some results

)e above is a description of the classification of virtualnetwork mapping algorithms It should be noted that thedifferent classifications are independent)at is any selectedVNE algorithm can be for example static centralized andconcise Let us discuss the results of recent research As theresearch progressed people came up with some novel ideasand were not able to classify them according to the previousclassification Ni et al [36] proposed a multidomain virtualnetwork embedding algorithm based on particle swarmoptimization (PSO) As Internet traffic has grown expo-nentially network energy consumption has increased dra-matically worldwide Some researchers have consideredenergy consumption when studying virtual network map-ping problems [37 38] Energy efficiency concurrency andtopology awareness are all considered when designing vir-tual network mapping algorithms [39] With the rapiddevelopment of artificial intelligence in the world peopleconsider using artificial intelligence to solve the problem ofvirtual network mapping Learning actively and makingonline decisions based on previous experiences are used inpaper [40] Haeri and Trajkovic [8] formalized the virtualnode mapping problem by using the Markov decisionprocess (MDP) framework )e Monte Carlo tree searchalgorithm is used to design an action strategy (node map-ping) for the proposed MDP Yao et al [41] used re-inforcement learning to study virtual network mappingproblems )ey designed and implemented a strategy


network based on reinforcement learning to develop nodemapping decisions In order to cope with the dynamic re-source requirements a fitness-based dynamic virtual net-work embedding (DYVINE) algorithm is proposed with thegoal to maximize the resource utilization by maximizing theacceptance rate [42] A self-adaptive VNE algorithm isproposed in paper [43]

)e paper mentioned above does not discuss the casewhere virtual nodes can be mapped to multiple physicalnodes )e literature [12] first proposed to map a virtualnode to multiple physical nodes In this article we willfurther study the problem and propose our solution

3 System Model and Problem Formulation

In this section we will first model the substrate network ofInPs and VN of SPs and give the VN embedding problemdescription followed by the definition of objectives

Similar to the papers of other scholars the two pa-rameters of CPU and bandwidth are mainly studied in thispaper )e subscript s indicates the substrate network andthe subscript v indicates the virtual network In the fol-lowing we will model the substrate network and the virtualnetwork as undirected weighted graphs where the verticesrepresent nodes and the edges represent links Each vertex isassociated with CPU capacityconstraint and each edge isassociated with bandwidth capacityconstraint

31NetworkModel We represent the substrate network as aweighted undirected graph Gs (Ns Ls An

s Als) where Ns

represents the set of the substrate nodes and Ls is the set ofthe substrate links )e notation An

s represents the attributeof the substrate nodes and the notation Al

s represents theattribute of the physical links )e attributes of a node couldbe CPU processing power storage capacity and geographiclocation of the node )e attributes of a link could bebandwidth delay and delay jitter In this paper we onlyconsider the available CPU processing power of the nodeand the available bandwidth properties of the link

Similarly the topology of the virtual network can also berepresented by a weighted undirected graphGv (Nv Lv An

v Alv) where Nv is the set of virtual nodes

and Lv is the set of virtual links In addition Anv and Al

vrepresent the CPU requirement constraints on the virtualnode and the bandwidth requirements on the virtual linkrespectively For the last two symbols in the substrate net-work and virtual network representation An

s represents theamount of resources that the substrate node can provideand An

v represents the resource demand of the virtual net-work node Similarly Al

s represents the amount of resourcesthat the substrate link can provide and Al

v represents theresource requirements of the virtual network link

Each VN request may be represented byVNR(i)(Gv ta td) where the variables ta and td denote thearrival time of the VN request and the duration of the VNstaying in the substrate network respectively When the ithVN request arrives the infrastructure provider should al-locate the substrate network resources to the service

provider while satisfying the resource requirements of thevirtual nodes and links If there are no enough substrateresources available to satisfy the requirements of the virtualnetwork request the VN request should be rejected orpostponed When the virtual network has completed its taskand no longer uses the substrate network resources theallocated substrate resources are released

Figure 1 shows an example of these notations In thisarticle letters i and j represent substrate nodes and u and v

denote virtual nodes)e CPU value of nodem is denoted asCPU(m) and the bandwidth value of link l is denoted asBW(l) Figure 1(b) shows a substrate network )ere are 6nodes and 8 links in the substrate networkNs ABCDE F and Ls ABBHHF FDCDAC

BCDH )e number inside the rectangle indicates theavailable CPU resources of the node it is close to for ex-ample CPU(A) 10 and CPU(B) 20 )e number on thelink indicates the bandwidth of the link for exampleBW(AB) 5 and BW(AC) 7 A virtual network request isshown in Figure 1(a) )ere are four virtual nodes and fourvirtual links in the virtual network request Nv a b c d and Lv ab ac bd cd )e number in the rectangular boxindicates the resource requirements of the virtual networknode and the number on the link indicates the linkbandwidth requirement As mentioned earlier if paralleli-zation is not supported the virtual network request inFigure 1(a) will be rejected or delayed If parallelization issupported Figure 1(c) shows one of the mapping schemes

32ParallelizableVirtualNetworkEmbedding Yu Liang andSheng Zhang divided the parallelizable virtual networkembedding into three components master mapping salvemapping and link mapping)emaster mappingMms mapsa virtual node nv to a substrate node denoted by Mms(nv)and the slave mapping Msl maps a virtual node nv to a subsetof the neighbors of Mms(nv) denoted by Msl(nv)

As shown in Figure 1 in a parallel virtual networkmapping a virtual node can be mapped to multiple physicalnetworks which can improve the mapping successful ratioHowever please note that parallelization may result in ad-ditional CPU and bandwidth costs To reflect these additionalcomputation and bandwidth consumptions caused by par-allelization Liang and Zhang [12] proposed the penalty factorpf and the additional bandwidth consumption Z In theaforementioned example in Figure 1 if pf 12 then 60(50lowast 12 60) units of CPU should be allocated to d inD andF For additional bandwidth consumption an additional Z

unit which is a constant should be allocated between each pairof master and slave nodes Mapping a virtual node to toomany substrate nodes cannot significantly improve compu-tational performance Liu Yang and Zhang Sheng proposed aparameter named speedup which means that a virtual nodecan only map to speedup substrate nodes at most

33 Problem Formulation and Objectives Based on the in-troduction of Section 32 mapping the virtual node tomultiple substrate nodes will lead to extra CPU andbandwidth consumption )e main idea of this paper is that


the virtual node should not be mapped to multiple substratenodes as much as possible Please note that in the followingformulas (1) (3) and (4) 1113936iisinNs

lceilxui CPU(u)rceil 1 represents

that the virtual node u is mapped to a unique substrate nodeand 1113936iisinNs

lceilxui CPU(u)rceil gt 1 indicates that the virtual node u

is mapped to multiple substrate nodesVariables

(i) fuvij a binary variable where it is 1 if virtual link luv

is routed on physical link lij and 0 otherwise(ii) xiu a real variable such that xiugt 0 if virtual node u

is mapped to the substrate node i and xiu unit CPUresource of the substrate node i is allocated tovirtual node u xiu 0 if virtual node u is notmapped to the substrate node i

Objective

Min 1113944luvisinLv

1113944lijisinLs

fuvij times BW luv( 1113857

+ 1113944uisinNvlceil1113936iisinNslceil xu

i( 1113857(CPU(u))rceil minus 1speedup rceil times(pf minus 1) + 1

⎛⎝ ⎞⎠

times CPU(u) + 1113944uisinNv

Z times 1113944iisinNslceil xu

i

CPU(u)rceil minus 1⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(1)

where lceilxrceil is the least integer that is larger than x

Constraints

(i) Capacity constraints

foralllij isin Lsforallluv isin Lv fuvij times BW luv( 1113857leBW lij1113872 1113873 (2)

foralli isin Ns 1113944uisinNslceil xu

i

CPU(u)rceil le 1 (3)

forallu isin Nv

1113944iisinNs

xui CPU(u) if 1113944

iisinNslceil xu

i

CPU(u)rceil 1

1113944iisinNs

xui pf times CPU(u) if 1113944

iisinNslceil xu

i

CPU(u)rceil gt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)

(ii) Speedup constraints

forallu isin Nv 1113944iisinNslceil xu

i

CPU(u)rceil le speedup (5)

(iii) Domain constraints

foralli j isin Ns u v isin Nv fuvij isin 0 1 (6)

forallu isin Nv i isin NsCPU(u)ge xui ge 0 (7)

Remarks

(i) )e objective function tries to minimize the cost ofembedding the VN request )e objective functionconsists of three parts )e first part is the cost ofmapping the virtual links on the substrate links)e second part is the cost when the virtual node ismapping )e third is the cost of the links betweenthe slave nodes and the master node

(ii) Constraint set (2) refers to virtual link constraints)e substrate link (i j) must meet the bandwidthrequirement for the virtual link (u v) In thispaper path splitting is unsupported and sup-porting path splitting will be our future workConstraint set (3) refers to that no more than onevirtual node is placed on a substrate node Con-straint set (4) ensures that the CPU assigned to thevirtual node u by the substrate nodes is equal to itsrequirement

(iii) Constraint set (5) enforces the number of substratenodes which the virtual node u is mapped to is lessthan or equal to speedup

(iv) Finally constrain sets (6) and (7) denote the binaryand real-domain constrains on the variables fuv

ij

and xui xi

u respectively

4 Our Proposed EPNVE Scheme

)e proposed heuristic EPVNE employs a greedy approachto deal with node mapping and the link mapping utilizesDijkstrarsquos algorithm to find the shortest path that meets theresource demands EPVNE is shown in Algorithm 1 itconsists of three phases

In the initialization phase (lines (1)ndash(3) in Algorithm 1)all virtual nodes are sorted and placed in a queue in thedecreasing order of CPU(nv) Similarly all substrate nodesare sorted in the decreasing order of ACPU(nv) which can bederived from the following equation

forallns isin NsACPU ns( 1113857 CPU ns( 1113857 + 1113944

misinVunusednei ns( )

CPU(m)

(8)

where Vnei(ns) denotes the set of direct neighbors of node nsand Vunused

nei (ns) denotes the set of direct neighbors of node nswhich has not been used )ere is a similar calculationformula (2) in paper [12] to calculate RCnei(ns) whosefunction is similar to ACPU(ns) in this paper )e calcu-lation formula (2) in paper [12] is complicated )e value itcalculates is of little significance for a mapping guide )eresult of its calculation is of little significance for onemapping )erefore a simpler calculation method is used inour paper

In the node mapping phase (lines (4)ndash(9) in Algo-rithm 1) EPVNE maps each virtual node to one substratenode (using Mapping-one-to-one shown in Algorithm 2) asmuch as possible If no substrate node CPU resource canmeet the resource requirements of the virtual node the


virtual node is mapped to multiple substrate nodes (usingMapping-one-to-multi described in Algorithm 3)

In the link mapping phase (lines (10)-(11) in Algo-rithm 1) Dijkstrarsquos algorithm is used to find a substrate pathfor each virtual link

Mapping-one-to-one shown in Algorithm 2 will searchfor a substrate node whose CPU value is just larger thanQ[i]for mappingQ[i] Substrate nodes with more CPU resourceswill be retained for subsequent virtual network mappingrequests

Mapping-one-to-multi described in Algorithm 3 findsup to speedup-1 nodes from the one-hop neighbors of P[0] tomap Q[i] When searching it proceeds in the ascendingorder first looking for the smallest one-hop neighbor thenthe second smallest one and so on |Neigh| in line (5) inAlgorithm 3 represent the cardinality of Neigh

In the following we analyze the computational com-plexity of Algorithms 1ndash3 )e complexities of Algorithms 2and 3 are O(|Ns|) and O(|Ns|lowast speed) respectively )ecomplexity of Algorithm 3 is larger than that of Algorithm 2|Ns| denotes the number of physical nodes and |Nv| is thenumber of virtual nodes

In the initialization phase of Algorithm 1 the complexity ofvirtual nodes sorting isO(|Nv|2) and the complexity of substratenodes sorting is O(|Ns|2) In the node mapping phase of Al-gorithm 1 the complexity is |Nv| multiplied by the complexityof Algorithm 3 that is |Nv|lowastO(|Ns|lowast speed)O(|Nv|lowast |Ns|lowastspeed))e complexity of the Dijkstra algorithm isO(|Ns|2) Inthe link mapping phase of Algorithm 1 the complex is |Lv|lowastO(|Ns|2) and |Lv| is the number of virtual links Finally thecomplexity of Algorithm 1 is the sum of the complexity of theabove three phases that is O(|Nv|2) +O(|Ns|2) +O(|Nv|lowast |Ns|lowastspeed)+ |Lv|lowastO(|Ns|2) )e complexity of EPVNE isacceptable

5 Performance Analysis

In this section we first introduce our simulation setup andthen present performance comparison results )e substratenetwork topology is configured to have 20 nodes and eachpair of nodes is connected with probability 04)e CPU andbandwidth resources of the substrate nodes and links are realnumbers uniformly distributed between 50 and 100 )e

number of virtual nodes in each virtual network follows auniform distribution between 2 and 10 Each pair of virtualnodes is connected with probability 05 )e bandwidthrequirements at virtual links are generated randomly fromthe range [50 70] )e CPU requirements of the virtualnodes are also randomly generated and the ranges are [5090] [70 110] [90 130] [110 150] [130 170] [150 190] and[170 210] )e three parameters described in Section 32 areset as follows pf 12 Z 20 and speedup 5 In thissection we concentrate on comparing EPVNE ProactiveP[12] LazyP [12] and RW-MaxMatch [21] ProactiveP alwaysmaps a virtual machine to speedup nodes and LazyP doesnot map a virtual machine to multiple nodes unless there arenot enough substrate resources RW-MaxMatch is a virtualnode-first mapping algorithm based on a novel topology-aware node ranking measure Since the RW-MaxMatchalgorithm is widely quoted we also compare our algorithmwith it 2500 experiments were performed for each sceneand the average was taken as the result

)e following metrics are used for performance com-parison (i) Acceptance ratio which is the ratio of thenumber of accepted virtual network requests to all requests(ii) CPU ratio which is the ratio of the amount of occupiedCPU resources in the substrate network to overall CPUresources in the virtual network and (iii) Z ratio which isthe ratio of the additional bandwidth consumption causedby parallelization to the constant Z )e latter two metricsare indicators that measure parallelism RW-MaxMatch doesnot support parallelization In fact the CPU ratio of RW-MaxMatch is equal to 100 and the Z ratio of RW-Max-Match is zero

Figure 2 shows the comparison of the acceptance ratioIn general EPVNE achieves a high acceptance ratio thanLazyP ProactiveP and RW-MaxMatch As the CPU re-quirements of the virtual nodes increase the acceptance rateshows a downward trend )e reason is very obvious )eCPU requirements of the virtual network nodes are smallthe substrate nodesrsquo resources are relatively abundant andthe acceptance rate is higher Because the RW-MaxMatchalgorithm does not support parallelization that is it cannotmap a virtual node to multiple physical nodes its perfor-mance is the worst )e ProactiveP algorithm always maps avirtual node to multiple physical nodes resulting in more

(1) initialization phase(2) Q⟵ Sorted virtual nodes in nonincreasing order according to their CPU requirement(3) P⟵ Sorted substrate nodes in nonincreasing order according to their ACPU(4) node mapping phase(5) Mapping the virtual nodes in queue Q in turn

i⟵ 0 to Qmiddotlength-1(6) If CPU(Qmiddot[i])leCPU(P[0]) then(7) Mapping Q[i] to one substrate nodes using Algorithm 2(8) Else(9) Mapping Q[i] to multiple substrate nodes using Algorithm 3(10) link mapping phase(11) Dijkstrarsquos algorithm is used to find a substrate path for each virtual link

ALGORITHM 1 EPVNE


overhead)erefore its performance is second to last LazyPand EPVNEmap a virtual node to multiple nodes only whenone physical node cannot meet the virtual node re-quirements so the two algorithms perform better LazyPuses a more complicated calculation formula which does not

truly reflect the situation of the remaining physical re-sources resulting in the mapping failure In the experimentswe found that ProactivePmaps each virtual node to multiplesubstrate nodes As a result in the late mapping too manysubstrate nodes are used and the available substrate nodesare lacking resulting in mapping failure )is is the reasonwhy the ProactiveP acceptance ratio curve in Figure 2 issignificantly lower than the other two

)e comparison of the CPU ratio is depicted in Figure 3Because ProactiveP always maps a virtual node to multiplephysical nodes its CPU ratio is highest and is close to 12 Inaddition because the mapping success rate of EPVNE isrelatively high that is EPVNE can successfully map the casewhich LazyP cannot map the CPU ratio of EPVNE is alsorelatively high When the CPU requirement virtual node isrelatively small the node can be mapped to a single substratenode without causing additional CPU consumption Withthe increase of CPU requirement a single substrate nodecannot meet the CPU resource requirements of the virtualnodes and multiple substrate nodes must complete themapping together In the end the CPU requirement in-creases and the CPU ratio tends to be 12 which is the valueof pf

Figure 4 which shows the comparison of the Z ratio issimilar to the situation in Figure 3 In fact the Z ratiorepresents the average number of additional links or numberof slave nodes per virtual network )e curves of ProactiveP

(1) For j 0 to Pmiddotlength-1 do(2) If CPU(Pmiddot[j])geCPU(Q[i]) and CPU(Pmiddot[j+ 1])ltCPU(Q[i]) then(3) Break(4) End if(5) End for(6) Map Q[i] to P[j](7) Remove P[j] from P

ALGORITHM 2 Mapping-one-to-one

(1) Neigh⟵ P[0](2) Sum⟵CPU(P[0])(3) For j Pmiddotlength-1 to 1 do(4) If P[j] isinVnei

unused(P[0]) do(5) If |Neigh| speedup do(6) Remove the node with the lowest CPU value from Neigh(7) Sum Sum-CPU (the node with the lowest CPU value)(8) End if(9) add P[j] to Neigh(10) Sum Sum+CPU(P[j])(11) End if(12) If SumgeQ[i] do(13) Break(14) End if(15) End for(16) Map Q[i] to the elements in Neigh(17) Remove the elements in Neigh from P

ALGORITHM 3 Mapping-one-to-multi

[50ndash90] [70ndash110] [90ndash130] [110ndash150] [130ndash170] [150ndash190] [170ndash210]The CPU requirements of the virtual nodes

01

02

03

04

05

06

07

08

09

Acc

epta

nce r

atio

EPNVELazyP

ProactivePRW-MaxMatch

Figure 2 Acceptance ratio


in Figures 3 and 4 are smoother and higher than the othertwo because the algorithm maps each virtual node tomultiple substrate nodes regardless of the actual situation ofnetwork resources)e reason why the two curves of EPVNEin Figures 3 and 4 are higher than that of LazyP is that theacceptance rate of EPVNE is high EPVNE successfully mapssome of the virtual network requests that LazyP failed tomap and these virtual network requests require more extralinks and additional CPU resources

6 Conclusion

Network virtualization technology allows multiple het-erogeneous virtual networks to be built on top of a sharedunderlying physical network Network virtualization

technology enables network operators to deploy newnetwork architectures or protocols without affecting theexisting Internet providing a viable path for evolutionfrom the current network to the future network As one ofthe main challenges facing network virtualization thevirtual network mapping problem is NP-hard It hasbecome a research hotspot in the field of networkvirtualization

In this paper we studied parallelizable virtual networkembedding Firstly the formulation of PVNE is presentedSecondly an efficient parallelizable virtual network em-bedding (EPVNE) algorithm is proposed Finally extensivesimulations are done to evaluate the performance of EPVNE)e results demonstrate that our algorithm performs betterthan the existing heuristic algorithms Our future work is totake path splitting into account

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

References

[1] W Fang X Liang S Li L Chiaraviglio and N XiongldquoVMPlanner optimizing virtual machine placement and trafficflow routing to reduce network power costs in cloud datacentersrdquo Computer Networks vol 57 no 1 pp 179ndash196 2013

[2] Y Zhou Y Zhang H Liu N Xiong and A V Vasilakos ldquoAbare-metal and asymmetric partitioning approach to clientvirtualizationrdquo IEEE Transactions on Services Computingvol 7 no 1 pp 40ndash53 2014

[3] H Cao H Hu Z Qu and L Yang ldquoHeuristic solutions ofvirtual network embedding a surveyrdquo China Communica-tions vol 15 no 3 pp 186ndash219 2018

[4] H Cao L Yang Z Liu and M Wu ldquoExact solutions of VNEa surveyrdquo China Communications vol 13 no 6 pp 48ndash622016

[5] A Fischer J F Botero M T Beck H de Meer andX Hesselbach ldquoVirtual network embedding a surveyrdquo IEEECommunications Surveys amp Tutorials vol 15 no 4pp 1888ndash1906 2013

[6] X Liu Z Zhang X Li and S Su ldquoOptimal virtual networkembedding based on artificial bee colonyrdquo EURASIP JournalonWireless Communications and Networking vol 2016 no 1p 273 2016

[7] P Zhang H Yao C Fang and Y Liu ldquoMulti-objectiveenhanced particle swarm optimization in virtual networkembeddingrdquo EURASIP Journal on Wireless Communicationsand Networking vol 2016 no 1 p 167 2016

[8] S Haeri and L Trajkovic ldquoVirtual network embedding viaMonte Carlo tree searchrdquo IEEE Transactions on Cyberneticsvol 48 no 2 pp 510ndash521 2018

[9] H Cao Y Zhu G Zheng and L Yang ldquoA novel optimalmapping algorithm with less computational complexity forvirtual network embeddingrdquo IEEE Transactions on Networkand Service Management vol 15 no 1 pp 356ndash371 2018

1

105

11

115

12

125

CPU

ratio


EPNVELazyP


Figure 3 CPU ratio

0

2

4

6

8

10

12

14

Z ra

tio


EPNVELazyP


Figure 4 Z ratio


[10] X Chen C Li and Y Jiang ldquoOptimization model and al-gorithm for energy efficient virtual node embeddingrdquo IEEECommunications Letters vol 19 no 8 pp 1327ndash1330 2015

[11] M L Yu Y Yi J Rexford andM Chiang ldquoRethinking virtualnetwork embedding substrate support for path splitting andmigrationrdquo ACM SIGCOMM Computer CommunicationReview vol 38 no 2 pp 19ndash29 2008

[12] Y Liang and S Zhang ldquoEmbedding parallelizable virtualnetworksrdquo Computer Communications vol 102 no 4pp 47ndash57 2017

[13] K Z Gao P N Suganthan Q K Pan M F Tasgetiren andA Sadollah ldquoArtificial bee colony algorithm for scheduling andrescheduling fuzzy flexible job shop problem with new job in-sertionrdquoKnowledge-Based Systems vol 109 no 6 pp1ndash16 2016

[14] Q Qi J Wang Z Ma et al ldquoKnowledge-driven serviceoffloading decision for vehicular edge computing a deepreinforcement learning approachrdquo IEEE Transactions onVehicular Technology vol 68 no 5 pp 4192ndash4203 2019

[15] H-Y Sang J-Q Duan and J-Q Li ldquoAn effective invasiveweed optimization algorithm for scheduling semiconductorfinal testing problemrdquo Swarm and Evolutionary Computationvol 38 no 1 pp 42ndash53 2018

[16] J-Q Li and Q-K Pan ldquoSolving the large-scale hybrid flowshop scheduling problem with limited buffers by a hybridartificial bee colony algorithmrdquo Information Sciences vol 316pp 487ndash502 2015

[17] J Xia G Chen and W Sun ldquoExtended dissipative analysis ofgeneralizedMarkovian switching neural networks with two delaycomponentsrdquo Neurocomputing vol 260 pp 275ndash283 2017

[18] NMM K ChowdhuryM R Rahman and R Boutaba ldquoVirtualnetwork embedding with coordinated node and linkmappingrdquo inProceedings of the 28th Conference on Computer Communicationspp 783ndash791 IEEE Rio de Janeiro Brazil April 2009

[19] Y Zhu and M Ammar ldquoAlgorithms for assigning substratenetwork resources to virtual network componentsrdquo in Pro-ceedings of the 25th IEEE International Conference on Com-puter Communications Barcelona Spain April 2006

[20] J Shamsi andM Brockmeyer ldquoQoSMap QoS aware mappingof virtual networks for resiliency and efficiencyrdquo in Pro-ceedings of the IEEE Globecom Workshops 2007 WashingtonDC USA November 2007

[21] X Cheng S Su Z Zhang et al ldquoVirtual network embeddingthrough topology-aware node rankingrdquo ACM SIGCOMMComputer Communication Review vol 41 no 2 pp 39ndash47 2011

[22] J Fan and M H Ammar ldquoDynamic topology configuration inservice overlay networks a study of reconfiguration policiesrdquo inProceedings of the 25th IEEE International Conference on Com-puter Communications pp 1ndash12 Barcelona Spain April 2006

[23] I Houidi W Louati and D Zeghlache ldquoA distributed virtualnetwork mapping algorithmrdquo in Proceedings of the IEEEInternational Conference on Communications pp 5634ndash5640Beijing China April 2008

[24] M Chowdhury M R Rahman and R Boutaba ldquoViNEYardvirtual network embedding algorithms with coordinated nodeand link mappingrdquo IEEEACM Transactions on Networkingvol 20 no 1 pp 206ndash219 2012

[25] H Cao Y Guo Y Hu S Wu H Zhu and L Yang ldquoLocationaware and node ranking value-assisted embedding algorithmfor one-stage embedding in multiple distributed virtual net-work embeddingrdquo IEEE Access vol 6 pp 78425ndash78436 2018

[26] J Ai H Chen Z Guo and G Cheng ldquoDefending against linkfailure in virtual network embedding using a hybrid schemerdquoChina Communications vol 16 no 1 pp 129ndash138 2019

[27] X Zheng J Tian X Xiao X Cui and X Yu ldquoA heuristicsurvivable virtual network mapping algorithmrdquo Soft Com-puting vol 23 no 5 pp 1453ndash1463 2019

[28] L F S Moura L P Gaspary and L S Buriol ldquoA branch-and-price algorithm for the single-path virtual network embed-ding problemrdquo Networks vol 71 no 3 pp 188ndash208 2017

[29] Y-Y Han J J Liang Q-K Pan J-Q Li H-Y Sang andN N Cao ldquoEffective hybrid discrete artificial bee colonyalgorithms for the total flow time minimization in theblocking flow shop problemrdquo Ce International Journal ofAdvanced Manufacturing Technology vol 67 no 1ndash4pp 397ndash414 2013

[30] H-Y Sang Q-K Pan P-Y Duan and J-Q Li ldquoAn effectivediscrete invasive weed optimization algorithm for lot-streaming flow shop scheduling problemsrdquo Journal of In-telligent Manufacturing vol 29 no 6 pp 1337ndash1349 2018

[31] J Li Q Pan and S Xie ldquoAn effective shuffled frog-leapingalgorithm for multi-objective flexible job shop schedulingproblemsrdquo Applied Mathematics and Computation vol 218no 18 pp 9353ndash9371 2012

[32] J-Q Li J-D Wang Q-K Pan et al ldquoA hybrid artificial beecolony for optimizing a reverse logistics network systemrdquo SoftComputing vol 21 no 20 pp 6001ndash6018 2017

[33] I Pathak A Tripathi and D P Vidyarthi ldquoA model forvirtual network embedding using artificial bee colonyrdquo In-ternational Journal of Communication Systems vol 31 no 10p e3573 2018

[34] X Cheng S Su Z Zhang et al ldquoVirtual network embeddingthrough topology awareness and optimizationrdquo ComputerNetworks vol 56 no 6 pp 1797ndash1813 2012

[35] P Zhang H Yao M Li and Y Liu ldquoVirtual network em-bedding based on modified genetic algorithmrdquo Peer-to-PeerNetworking and Applications vol 12 no 2 pp 481ndash492 2019

[36] Y Ni G Huang S Wu C Li P Zhang and H Yao ldquoA PSObased multi-domain virtual network embedding approachrdquoChina Communications vol 16 no 4 pp 105ndash119 2019

[37] M Zhu Q Sun S Zhang P Gao B Chen and J GuldquoEnergy-aware virtual optical network embedding in slice-able-transponder-enabled elastic optical networksrdquo IEEEAccess vol 7 pp 41897ndash41912 2019

[38] P Zhang ldquoIncorporating energy and load balance into virtualnetwork embedding processrdquo Computer Communicationsvol 129 pp 80ndash88 2018

[39] A Jahani LM Khanli M T Hagh andM A BadamchizadehldquoEE-CTA energy efficient concurrent and topology-awarevirtual network embedding as a multi-objective optimizationproblemrdquo Computer Standards amp Interfaces vol 66 Article ID103351 2019

[40] SWang J Bi J Wu A V Vasilakos and Q Fan ldquoVNE-TD avirtual network embedding algorithm based on temporal-difference learningrdquo Computer Networks vol 161 pp 251ndash263 2019

[41] H Yao X Chen M Li P Zhang and L Wang ldquoA novelreinforcement learning algorithm for virtual network em-beddingrdquo Neurocomputing vol 284 pp 1ndash9 2018

[42] C K Dehury and P K Sahoo ldquoDYVINE fitness-based dy-namic virtual network embedding in cloud computingrdquo IEEEJournal on Selected Areas in Communications vol 37 no 5pp 1029ndash1045 2019

[43] Z Li Z Lu S Deng and X Gao ldquoA self-adaptive virtualnetwork embedding algorithm based on software-definednetworksrdquo IEEE Transactions on Network and Service Man-agement vol 16 no 1 pp 362ndash373 2019


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Submit your manuscripts atwwwhindawicom

provider (ISP) into two separate entities infrastructureprovider (InP) and service provider (SP) [8 9] In such anetwork environment an infrastructure provider manages asubstrate network (SN) which is composed of substratenodes and physical communication links Infrastructureproviders use virtualization technology to abstract the un-derlying substrate resources into functional entities providea unified programming interface shield the underlyingphysical network resources from the upper layers and splitthe substrate resources into multiple virtual slices for use bydifferent service providers In addition infrastructure pro-viders can communicate and collaborate with each other tocreate a complete interdomain infrastructure for serviceproviders Service provider rents slices of substrate resources(eg nodes and links) and then creates his own customizedvirtual network (VN) to provide value-added services (egcontent distribution) to end users Service provider deployscustomized protocols in virtual networks and is responsiblefor running managing and maintaining virtual networksService provider generates a virtual network requestaccording to the needs of the service (eg CPU resourcesmemory resources bandwidth resources latency re-quirements and geographic location) and forwards therequest to the infrastructure provider to establish a virtualnetwork Regardless of whether the virtual network spansmultiple infrastructure providersrsquo domains the serviceprovider has a unified view of the entire virtual network

)e problem of allocating infrastructure network re-sources according to virtual network requests with node andlink resource constraints is called virtual network embed-ding (VNE) problem [10] )e virtual network embedding isan important challenge in the realization process of networkvirtualization technology and has become a hotspot formany scholars )e VNE problem is an NP-hard problem[11] Even after all virtual nodes have been mapped map-ping virtual links with bandwidth resource constraints is stillNP-hard )e virtual network embedding problem is one ofthe main challenges faced by network virtualization tech-nology It has become a hot issue in this research field andhas received extensive attention from researchers Manyresearches on VNE have been reported [5] However there isvery few focus on parallelizable virtual networks embedding[12] In the parallelizable virtual network embedding(PVNE) environment a virtual node could be mapped intomultiple substrate nodes With parallelization severalsubstrate nodes can parallely accomplish the computation towhich the virtual node is dedicated With parallelization asubstrate network can carry more virtual networks Forexample in cloud computing MapReduce processes large-scale data in parallel [13 14] Another important advantageof parallelization is its robustness that the computation canquickly migrate to other substrate nodes in the event of asubstrate node crash [15] With the development of cyber-physical system and cloud computing [16] the need forparallelization will increase Parallelization also providesnew ideas for the implementation of cyber-physical systemand cloud computing [17]

Consider the example in Figure 1 where Figure 1(a)shows a virtual network request and Figure 1(b) shows a

substrate network )e corresponding number near eachvertex (or edge) is the CPU (or bandwidth) capacitycon-straint )is virtual network request contains four virtualmachines (VMs) linked by four links If parallelization is notsupported this request would be rejected as there is nosubstrate node that has more than 50 units of available CPUresources which the virtual node d requires In PVNE afeasible mapping scheme is shown in Figure 1(c))emastermapping is a⟶B b⟶H c⟶C d⟶D and theslave mapping is a⟶Φ b⟶Φ c⟶Φ d⟶ F )elink mapping is ab⟶ BH Ac⟶ BC bed⟶HD CD⟶ CD In the mapping scheme ofFigure 1(c) it can be seen that the virtual nodes a b and c aremapped to one substrate node respectively However nophysical node can satisfy the resource requirements of thevirtual node d In this case a parallelized mapping scheme isrequired that is a virtual node is mapped to multiplesubstrate nodes Finally the virtual node d is mapped to thephysical nodes D and F

In this paper we study parallelizable virtual networkembedding (PVNE) and propose an efficient parallelizablevirtual network embedding (EPVNE) algorithm We sum-marize the main contributions here as follows

(i) Based on optimization theory and resource in-tegration technology an optimized mathematicalmodel for parallel virtual network embedding isproposed

(ii) A new heuristic algorithm named EPVNE is pro-posed to reduce the cost of embedding the VNrequest and increase the VN request acceptanceratio Firstly the substrate nodes and virtual nodesneed to be ordered by a metric proposed in thispaper which is simple and can be calculated effi-ciently Secondly when mapping each virtual nodenv our algorithm will search for a substrate nodewhose CPU value is just larger than nv Substratenodes with more CPU resources will be retained forsubsequent virtual network mapping requests

(iii) Extensive simulations have been performed toevaluate the performance of our new algorithm)eresults demonstrate that our algorithm performsbetter than the existing heuristic algorithms

)is paper is organized as follows the related work isintroduced in Section 2 In Section 3 we describe systemmodel and problem formulation We then present theproposed algorithm in Section 4 Section 5 introduces theexperimental conditions and related parameter settings andfocuses on the experimental results Finally Section 6concludes the paper

2 Related Work

Researchers have conducted in-depth research on virtualnetwork embedding algorithms from different perspectivesand a series of research results have emerged Several papers[3ndash5] summarize the virtual network mapping algorithmFischer A [5] summarized the research content of VNE








a

c d

1

b

8

3

7

1010

20 50

(a)

A

B H

C D

9

34

13 17 F

35

77

20 40

4830

10 20

(b)

A

B H

C D

9

34

13 17 F

35

77

a b

dc

20 40

4830

10 20

dprime

(c)



















s Als) where Ns






























Objective


1113944lijisinLs




⎛⎝ ⎞⎠



i


(1)


Constraints




i


forallu isin Nv

1113944iisinNs


iisinNslceil xu

i

CPU(u)rceil 1

1113944iisinNs


iisinNslceil xu

i

CPU(u)rceil gt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)



i





Remarks





ij

and xui xi

u respectively






CPU(m)

(8)


















ALGORITHM 1 EPVNE












01

02

03

04

05

06

07

08

09

Acc

epta

nce r

atio

EPNVELazyP





6 Conclusion




Data Availability




References










1

105

11

115

12

125

CPU

ratio


EPNVELazyP


Figure 3 CPU ratio

0

2

4

6

8

10

12

14

Z ra

tio


EPNVELazyP


Figure 4 Z ratio







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








a

c d

1

b

8

3

7

1010

20 50

(a)

A

B H

C D

9

34

13 17 F

35

77

20 40

4830

10 20

(b)

A

B H

C D

9

34

13 17 F

35

77

a b

dc

20 40

4830

10 20

dprime

(c)



















s Als) where Ns






























Objective


1113944lijisinLs




⎛⎝ ⎞⎠



i


(1)


Constraints




i


forallu isin Nv

1113944iisinNs


iisinNslceil xu

i

CPU(u)rceil 1

1113944iisinNs


iisinNslceil xu

i

CPU(u)rceil gt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)



i





Remarks





ij

and xui xi

u respectively






CPU(m)

(8)


















ALGORITHM 1 EPVNE












01

02

03

04

05

06

07

08

09

Acc

epta

nce r

atio

EPNVELazyP





6 Conclusion




Data Availability




References










1

105

11

115

12

125

CPU

ratio


EPNVELazyP


Figure 3 CPU ratio

0

2

4

6

8

10

12

14

Z ra

tio


EPNVELazyP


Figure 4 Z ratio







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


















s Als) where Ns






























Objective


1113944lijisinLs




⎛⎝ ⎞⎠



i


(1)


Constraints




i


forallu isin Nv

1113944iisinNs


iisinNslceil xu

i

CPU(u)rceil 1

1113944iisinNs


iisinNslceil xu

i

CPU(u)rceil gt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)



i





Remarks





ij

and xui xi

u respectively






CPU(m)

(8)


















ALGORITHM 1 EPVNE












01

02

03

04

05

06

07

08

09

Acc

epta

nce r

atio

EPNVELazyP





6 Conclusion




Data Availability




References










1

105

11

115

12

125

CPU

ratio


EPNVELazyP


Figure 3 CPU ratio

0

2

4

6

8

10

12

14

Z ra

tio


EPNVELazyP


Figure 4 Z ratio







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










Objective


1113944lijisinLs




⎛⎝ ⎞⎠



i


(1)


Constraints




i


forallu isin Nv

1113944iisinNs


iisinNslceil xu

i

CPU(u)rceil 1

1113944iisinNs


iisinNslceil xu

i

CPU(u)rceil gt 1

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)



i





Remarks





ij

and xui xi

u respectively






CPU(m)

(8)


















ALGORITHM 1 EPVNE












01

02

03

04

05

06

07

08

09

Acc

epta

nce r

atio

EPNVELazyP





6 Conclusion




Data Availability




References










1

105

11

115

12

125

CPU

ratio


EPNVELazyP


Figure 3 CPU ratio

0

2

4

6

8

10

12

14

Z ra

tio


EPNVELazyP


Figure 4 Z ratio







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia















ALGORITHM 1 EPVNE












01

02

03

04

05

06

07

08

09

Acc

epta

nce r

atio

EPNVELazyP





6 Conclusion




Data Availability




References










1

105

11

115

12

125

CPU

ratio


EPNVELazyP


Figure 3 CPU ratio

0

2

4

6

8

10

12

14

Z ra

tio


EPNVELazyP


Figure 4 Z ratio







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












01

02

03

04

05

06

07

08

09

Acc

epta

nce r

atio

EPNVELazyP





6 Conclusion




Data Availability




References










1

105

11

115

12

125

CPU

ratio


EPNVELazyP


Figure 3 CPU ratio

0

2

4

6

8

10

12

14

Z ra

tio


EPNVELazyP


Figure 4 Z ratio







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



6 Conclusion




Data Availability




References










1

105

11

115

12

125

CPU

ratio


EPNVELazyP


Figure 3 CPU ratio

0

2

4

6

8

10

12

14

Z ra

tio


EPNVELazyP


Figure 4 Z ratio







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


research articledownloads.hindawi.com/journals/wcmc/2019/8416592.pdfmapping algorithm can be divided...

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