(1.College of Physics and Information Science,Hunan Normal University,Changsha 410081,China;

2.Network Convergence Laboratory,University of Essex,Colchester CO4 3SQ,United Kingdom)

Virtualized Wireless SDNs:Modelling Delay Throughh the Use of Stochastic Network Calculus

Lianming Zhang1,Jia Liu1,and Kun Yang2

(1.College of Physics and Information Science,Hunan Normal University,Changsha 410081,China;

2.Network Convergence Laboratory,University of Essex,Colchester CO4 3SQ,United Kingdom)

Software-defined networks(SDN)have attracted much attention recently because of their flexibility in terms of network manage⁃ment.Increasingly,SDN is being introduced into wireless networks to form wireless SDN.One enabling technology for wireless SDN is network virtualization,which logically divides one wireless network element,such as a base station,into multiple slices, and each slice serving as a standalone virtual BS.In this way,one physical mobile wireless network can be partitioned into multi⁃ple virtual networks in a software-defined manner.Wireless virtual networks comprising virtual base stations also need to provide QoS to mobile end-user services in the same context as their physical hosting networks.One key QoS parameter is delay.This pa⁃per presents a delay model for software-defined wireless virtual networks.Network calculus is used in the modelling.In particu⁃lar,stochastic network calculus,which describes more realistic models than deterministic network calculus,is used.The model en⁃ables theoretical investigation of wireless SDN,which is largely dominated by either algorithms or prototype implementations.

wireless software defined networks(SDN);wireless network virtualization;QoS modelling;upper bound delay;stochastic network calculus

1 Introduction

Software-defined networks(SDN)have attracted much attention recently because they enable flexible net⁃work management[1],[2].The bulk of research on SDN has focused on wired networks and OpenFlow [3],[4],but there is an increasing tendency to introduce SDN into wireless networks[5],[6].SDN brings to wireless networks the same benefits it brings to wired networks,e.g.,separation of control and forwarding planes,but it also creates some ra⁃dio-specific issues[5].

One of the key enabling technologies of SDN is network vir⁃tualization.Wireless mobile network virtualization enables physical mobile network operators(PMNO)to partition their network resources into smaller slices and assign each slice to an individual virtual mobile network operator(VMNO).These virtual networks are the managed in a more dynamic,cost⁃ef⁃fective way.We call these virtualized individual networks vir⁃tual wireless networks(VWNs).VMNOs pay the PMNO using a pay⁃as⁃you⁃use model.Wireless network virtualization has its real⁃world bearings in mobile cellular networks.Wen et al. summarise some current trends and perspectives in wireless virtualization[7].

The purpose of network virtualization is to provide services to end users.To satisfy user requirements and abide by the ser⁃vice⁃level agreement with the customer,virtual network opera⁃tors need to provide quality of service(QoS)in their networks. One important QoS metric is network delay,which is critical for real⁃time services such as voice.In this paper,we address the delay requirements of different services(flows).A key is⁃sue is how a PMNO allocates resources to an individual VM⁃NO in order to satisfy service delay requirements within the VMNO.Guarantee that this delay requirement will be met in a VWN is a challenge to mobile network operators[14],[15].

Before allocating or scheduling resources,it is essential to understand the behaviours of virtual networks,especially in terms delay bounds.Although much work has been done on modelling physical wireless networks themselves,little has been done in the way of modelling virtual networks.Some ini⁃tial work in this area can be found in[8],but the network being considered is a mesh network.Furthermore,this model does not differentiate physical networks from virtual networks.

As in[9],we partition a physical network node,such as a base station,into multiple slices.This partitioning can be car⁃ried out in a dynamic manner using software,i.e.,supporting software⁃defined radio networks.Each slice represents a virtu⁃al network node.

The predominant theoretical bases for network modelling

are probability theory and queue theory.In this paper,we use a new modelling tool called network calculus,which is a set of recent developments that enable the derivation of performance bounds in networking[10],[11].Applications of network calcu⁃lus are wide⁃ranging and include QoS control,resource alloca⁃tion and scheduling,and buffer and delay dimensioning[10]. We have previously researched the use of network calculus in wireless sensor networks[12].Our recent work[13]extends on this,moving into the new area of wireless network virtualiza⁃tion but using deterministic network calculus.Deterministic network calculus cannot describe service flow distribution or characteristics and thus cannot realistically model real⁃world scenarios.In this paper,we go one step further and use sto⁃chastic network calculus,which enriches the expressiveness of the service flow and is thus a more realistic modelling tool.

The technical aim of this paper is to propose a delay model for VWN under a more realistic service flow model using sto⁃chastic network calculus.This paper makes the following main contributions:

· It mathematically describes the different roles of a typical VWN system using stochastic network calculus.

·It describes a delay model for the above virtual wireless net⁃work by expressing network delay in its upper bound and in a closed⁃form manner.The proposed model can help ana⁃lyze delay guarantee for per⁃flow granularity.

We do not consider a particular networking technology,such as Wi⁃Fi or LTE⁃A.The proposed model is generic enough to be applicable in any network.

2 Related Work

SDN,represented by OpenFlow,has been successful for in⁃novating on network operations and service provisioning.It al⁃so reduces complexity in terms of network configuration and management.Costanzo et al.identify the benefits of SDN for wireless and mobile communications,although their exemplar is a wireless personal area network[6].

Network virtualization is a strong enabler of wireless SDN because it provides a flexible,efficient way of deploying cus⁃tomized services on a shared infrastructure[14],[15].Recent⁃ly,wireless virtualization has attracted attention because of it benefits in several scenarios[9],[16],[17].

A lot of research has been done on wireless network virtual⁃ization[9],but there is a lack of formal modelling of wireless virtual networks.System modelling can be a useful means of studying the fundamental features of a system.In this paper, we aim to fill this gap by providing a model for virtualized wire⁃less networks.In particular,we focus on one important feature of virtual networks:network delay.

There are various approaches to delay⁃aware resource con⁃trol in wireless networks.Tao et al.investigate the resource⁃al⁃location problem in a multiuser OFDM system with both delay⁃constrained and non⁃delay⁃constrained traffic[18].However, they do not discuss the affect of the delay mechanism on perfor⁃mance.Another approach is to convert average delay con⁃straints into equivalent average rate constraints using queuing theory[19],[20].These approaches are linked to a particular resource⁃allocation or packet⁃scheduling algorithm and are thus specific to the corresponding algorithms.We provide a more generic model of wireless virtual networks that is agnos⁃tic to resource⁃allocation algorithms and specific network tech⁃nology.In our recent work[13],we describe a more expressive network⁃modelling tool,called stochastic network calculus.

Stochastic network calculus is used to analyze performance guarantee in information systems[21],[22].It has its founda⁃tions in the min⁃plus convolution and max⁃plus convolution queuing principles,and it has tremendous potential in dealing with queuing⁃type problems.It complements classical queuing theory[21].In[22],Ciucu et al.discuss sharp bounds in sto⁃chastic network calculus.Stochastic network calculus has can be used to compute per⁃flow queuing system metrics in a uni⁃fied manner for a large class of scheduling algorithms.Further⁃more,the per⁃flow results can be extended in a straightforward manner,from a single queue to a large class of queuing net⁃works that are amenable to convolution⁃form representation in an appropriate algebra.

Here,we summarize representative work in which network calculus is used to model QoS parameters,in particular,delay. In[23],network calculus is used to compute the delay of indi⁃vidual traffic flows in feed⁃forward networks under arbitrary multiplexing.In[24],the maximum end⁃to⁃end delay is calcu⁃lated,again for feed⁃forward type networks.In[25],Schmitt et al.propose an analytical framework for analyzing worst⁃case performance and to dimension resources in a sensor network. In[26]-[28],the authors present research on the deterministic performance bound on end⁃to⁃end delay for self⁃similar traffic regulated by a fractal leaky bucket regulator in an ad hoc net⁃work[26],wireless sensor network[27],and wireless mesh net⁃work[28].Working with the concept of flows and micro⁃flows, Zhang et al.[12]use arrival curves and service curves in net⁃work calculus to propose a two⁃layer scheduling model for sen⁃sor nodes.The authors develop a guaranteed QoS model that includes upper bounds on buffer queue length,network delay, and effective bandwidth.In[29],Azodolmolky et al.,describe the functionality of the SDN switch and controller and present an analytical model,based on network calculus theory,for de⁃lay and queue length boundaries of the SDN switch and buffer length of the SDN controller and SDN switch.In[6],Costanzo et al.present a complete SDN for wireless personal area net⁃works and call it software⁃defined wireless network(SDWN).

3 Description of System Model

Fig.1shows a virtual wireless network with virtual queue, the benefit of which is described in[13].Each slice is allocat⁃ed a virtual queue in the hosting physical network or network

node.All these virtual queues share the data rate capacity of the physical network node,i.e.,the physical BS under the con⁃trol of a scheduler.The scheduler takes into account the QoS requirements of the slices when scheduling resources.Each slice,denoted S1,S2,...,Sn,represents a virtual base station.

Fig.1 highlights the following two key elements in a virtual⁃ized network:physical BS and virtual BS(i.e.,slice).Each slice represents a virtual mobile network(VMN)and has a slice ID.A slice is used by many end users,i.e.,u1,u2,u3…un. A user is physically represented by a mobile node in the net⁃work and may have multiple flows,i.e.,F1,1,F1,c1,Fn,1,Fn,cn.For example,the smart phone may be used to check emails while listening to music online.Here,email and music each repre⁃sents a flow Fn,cn.The biggest differentiator between flow types is the delay requirement.Voice flow has more stringent delay requirements than non⁃real time emails.A flow represents a session,and each flow has an ID.

Fig.1 shows the relationship between the four key elements in a WVN:physical BS,virtual BS,users,and flows.Packets from different users and of the same type(e.g.,real⁃time)are denoted Uiand are put into the same queue in a slice.A leaky bucket source model is used to regulate the flows of each slice queue because this model is simple and practical.A leaky bucket regulator is applied to each slice queue to both regulate the flows so that non⁃rule flows can be controlled in certain conditions.A flow regulated by the leaky bucket regulator is given by envelopeα(t)[12]:

where b is the burst parameter,r is the average arrival rate, and t is time.

4 Proposed Upper Bound Delay Modelling Using Stochastic Network Calculus

In this section,we describe the above wireless SDN system using stochastic network calculus.Then we deduce the delay upper bound for this wireless SDN model.Detailed information about stochastic network calculus can be found in[21],[22], and network calculus in general is described in[10].Notations used in this paper are listed in Table 1.

4.1 System Description Using Stochastic Network Calculus

We model the wireless SDN presented in section 3 using sto⁃chastic network calculus.The process of the model is as fol⁃lows:First,a flow enters a virtual BS and is regulated by a leaky bucket regulator(1).The arrival curve is denoted α(t). Second,we assume a first come,first served(FCFS)strategy for a queue.This is reasonable because the packets in the same queue are of the same service type.Other more compre⁃hensive queuing strategies may be applied here as well.Final⁃ly,the aggregated flows from a slice are scheduled in the same way that a generalized processor sharing(GPS)server would schedule them in a physical BS[30].The system is further ex⁃plained as follows:

▼Table 1.Notations

▲Figure 1.Virtual wireless network and its roles.

The parameters in(2)to(5)are shown in Table 1.The flows of the slice i obtain the bandwidth weight.The sum bandwidth of the slices is,at most,the total bandwidth of the physical BS. Each slice entering the physical BS has a certain service curve that is not only decided by the total service curve of the physi⁃cal BS scheduler but also the arrival curve of the slice.

4.2 Proposed Delay Model

Proposition 1:In an interval[0,t],the least stochastic upper delay bound of physical BSi can be computed using

The symbol Pr[Di(t)≥di]represents the probability that the delay of the flows passing through the slice i is greater than di. When the value of the right side of(14)is at the minimum,we obtain parameter δ.The other parameters are shown in Table 1.Service rate and latency are the two key parameters of the physical BS;the former is equivalent to the network band⁃width,and the latter is the maximum service delay of the physi⁃cal BS.

Proof:We can derive(7)from(2)and(3):

From(7),we have

From(10)and(17)in[31],we obtain

Substituting(5)into(9)gives

Then,using Chernoff’s Bound Theorem gives

Using network calculus[10]gives

From(12),we get the following:

Substituting(4)and(8)into(13),we get

We can derive(6)from(9),(10),(11)and(14).

5 Numerical Results and Analysis

5.1 Network/Flow Parameter Setup

The two⁃level model shown in Fig.1 and described section 3 is used for all physical base stations.The service curvesβ(t)of the physical base stations are given in(8).Slices 1 to 3 are de⁃noted A1(t),A2(t)and A3(t),respectively,and are used to show the evaluation results.We assume that A1(t)contains three flows:A1,1(t),A1,2(t),A1,3(t);we assume that A2(t)contains two flows:A2,1(t)and A2,2(t);and we assume that A3(t)contains one flow:A3,1(t).Here we assume that every flow is regulated by the leaky bucket regulator α(t)(1).Without loss of generality,we specify the reservation of bandwidth weight for each slice ac⁃cording to the size of the flows.The bandwidth weight μiof the slices,the average arrival rate ri,k,and the burst parameter bi,kof the six flows are shown in Table 2.

In terms of evaluation,we investigate network delay as a function of both flow arrival rate and service rate of the physi⁃cal BS.

5.2 Network Delay

Figs.2,3,4 and 5 show the impact of the same set of vari⁃ables,i.e.,d,R,T and ρ,on Pr[D≥d]of a virtual slice.Figs. 6,7,8 and 9 show,respectively,the curved surface of the up⁃per bounds on the delay probability as a function of a)the de⁃lay and service rate,b)the service rate and latency,c)the ser⁃vice rate and network bandwidth utilization,and d)the latency and network bandwidth utilization for slice 1.

Fig.2 shows the delay probability curves as a function of de⁃

lay.Pr[D≥d]of A1(t),A2(t)and A3(t)decreases slightly and lin⁃early in relation to d and is almost insensitive to d when traffic for each flow is relatively small.In Fig.2,if d=0.52,Pr[D≥d]of A1(t),A2(t)and A3(t)is 0.1414,0.0645 and 0.0134,respec⁃tively.If d=0.56,Pr[D≥d]of A1(t),A2(t)and A3(t)is 0.1224, 0.0527 and 0.0098,respectively.Fig.3 shows how Pr[D≥d] decreases as R increases and how Pr[D≥d]approaches 0 for all slices.This exponential trend suggests network bandwidth is critical to Pr[D≥d].In Fig.3,if R=20,Pr[D≥d]of A1(t), A2(t)and A3(t)is 0.6628,0.5744 and 0.4176 respectively.If R=100,Pr[D≥d]of A1(t),A2(t)and A3(t)is 0.1269,0.0555and 0.0106,respectively.

▼Table 2.Parameters of the Three Slices and Their Flows

▲Figure 2.Pr[D≥d]vs.delay forR=100,T=0.001,t=0.025,ρ=1.

▲Figure 3.Pr[D≥d]vs.service rate ford=0.55,T=0.001,t=0.025,ρ=1.

Fig.6 shows the joint impact of d and R on Pr[D≥d]of A1(t).Pr[D≥d]increases as both R(network bandwidth)and d decrease.The impact of R on Pr[D≥d]is more obvious than that of d.Fig.4 shows how Pr[D≥d]increases in relation to T (maximum service delay).Pr[D≥d]starts slow but increases much more significantly as T increases and the network be⁃comes more loaded or even congested.This trend applies to all slices,and the heavier the slice load,the more obvious the trend is.In Fig.4,if T=0.1,Pr[D≥d]of A1(t),A2(t)and A3(t) is 0.1812,0.0918 and 0.0234,respectively.If T=0.5,Pr[D≥d]of A1(t),A2(t)and A3(t)is 0.7649,0.7034 and 0.5741,respec⁃tively.

Fig.7 shows the curved surface of Pr[D≥d]as a function of R and T.Pr[D≥d]decreases as R increases and T decreases. Fig.5 shows how the effect ofρon Pr[D≥d]is roughly the same as that of R on Pr[D≥d](Fig.3).

Fig 8 shows the joint effect ofρand R on Pr[D≥d],and Fig.9 shows the joint effect ofρand T on Pr[D≥d].The joint effect ofρand R is complex:Pr[D≥d]increases as R decreas⁃es(Fig.3)and decreases asρincreases(Fig.5).Pr[D≥d]in⁃

creases as T increases and decreases asρincreases(Fig.9).

▲Figure 4.Pr[D≥d]vs.latency forR=100,d=0.55,t=0.025,ρ=1.

▲Figure 5.Pr[D≥d]vs.bandwidth utilization for R=100,T=0.001,d=0.55,t=0.025.

▲Figure 6.Pr[D≥d]ofA1(t)vs.delay and service rate for T=0.001,t=0.025,ρ=1.

▲Figure 7.Pr[D≥d]ofA1(t)vs.service rate and latency for t=0.025,d=0.55,ρ=1.

In summary,the parameters of the flow regulators and ser⁃vice curves in the physical BS and virtual BS play an impor⁃tant role in modelling guaranteed delay.In particular,Pr[D≥d]decreases as d decreases;Pr[D≥d]decreases as R increas⁃es;Pr[D≥d]decreases asρincreases;and Pr[D≥d]decreas⁃es as T decreases.To improve network performance and guar⁃anteed delay in an SDN,certain mechanisms can be used to re⁃duce Pr[D≥d].These mechanisms may involve rational sched⁃uler parameters,such as network bandwidth and maximum ser⁃vice delay.

6 Conclusion and Future Work

In this paper,we have proposed a simple but realistic model for describing the upper bound delay of a wireless virtual net⁃work in the context of SDN.The model takes into account ser⁃vice flows,which represent service types,and virtualized net⁃works,as presented by slices.In particular,we have used a fin⁃er system modelling and performance analysis tool,called sto⁃ chastic network calculus,to describe the proposed model.We also deduced closed⁃formed formulas for the upper bound de⁃lay.In future work,we will propose a scheduling algorithm based on the above QoS model.Another area of future work is to extend the model to include other network parameters,such as throughput.

▲Figure 8.Pr[D≥d]ofA1(t)vs service rate and bandwidth utilization forT=0.001,t=0.025,d=0.55.

▲Figure 9.Pr[D≥d]ofA1(t)vs.latency and bandwidth utilization forR=100,t=0.025,d=0.55.

Acknowledgements

This research was supported in part by the grant from the National Natural Science Foundation of China(60973129). The authors would like to thank there viewers for their valu⁃able comments.

[1]H.Kim and N.Feamster,“Improving network management with software defined networking,”IEEE Communication Magazine,vol.51,no.2,pp.114-119, 2013.doi:10.1109/MCOM.2013.6461195.

[2]S.Sezer,S.Scott⁃Hayward,P.K.Chouhan,B.Fraser,D.Lake,J.Finnegan,N. Viljoen,M.Miller,and N.Rao,“Are we ready for SDN?Implementation chal⁃lenges for software⁃defined networks,”IEEE Communications Magazine,vol.51, no.7,pp.36-43,2013.doi:10.1109/MCOM.2013.6553676.

[3]N.McKeown,T.Anderson,H.Balakrishnan,G.Parulkar,L.Peterson,J.Rex⁃ford,S.Shenker,and J.Turner,“Openflow:enabling innovation in campus net⁃works,”ACM SIGCOMM Computer Communication Review,vol.38,no.2,pp. 69-74,2008.doi:10.1145/1355734.1355746.

[4]K.Choumas,N.Makris,T.Korakis,L.Tassiulas,and M.Ott,“Exploiting Open⁃Flow resources towards a content⁃centric LAN,”in Proceedings Of Second Euro⁃pean Workshop on Software Defined Networks(EWSDN2003),Berlin,2013,pp. 93-98.doi:10.1109/EWSDN.2013.22.

[5]C.Chaudet and Y.Haddad,“Wireless software defined networks:challenges and opportunities,”in Proceedings of IEEE InternationalConference on Micro⁃waves,Communications,Antennas and Electronics Systems(COMCAS2013),Tel Aviv,Israel,2013,pp.1-5.

[6]S.Costanzo,L.Galluccio,G.Morabito,and S.Palazzo.“Software defined wire⁃less networks:unbridling SDNs,”in Proceedings of European Workshop on Soft⁃ware Defined Networking(EWSDN2012),Darmstadt,Germany,2012,pp.1-6.

[7]H.Wen,P.K.Tiwary,and T.Le⁃Ngoc,“Current trends and perspectives in wire⁃less virtualization,”in Proceeding of 2013 InternationalConference on Selected Topics in Mobile and Wireless Networking,Montreal,QC,Canada,2013,pp.62-67.doi:10.1109/MoWNet.2013.6613798.

[8]R.Matos,C.Marques,S.Sargento,K.A.Hummel,and H.Meyer,“Analytical modeling of context⁃based multi⁃virtual Wireless Mesh Networks,”Elsevier Inter⁃national Journal on Ad hoc Networks,vol.13,pp.191-209,2014.doi:10.1016/j. adhoc.2011.05.004.

[9]X.Lu,K.Yang,Y.Liu,D.Zhou,and S.Liu.An elastic resource allocation algo⁃rithm enabling wireless network virtualization,Wiley International Journal on Wireless Communications and Mobile Computing[Online].Available:http://on⁃linelibrary.wiley.com/doi/10.1002/wcm.2342/full

[10]J.⁃Y.Le Boudec,and P.Thiran,Network calculus.Springer Verlag,2004.

[11]M.Fidler,“A survey of deterministic and stochastic service curve models in the network calculus,”IEEE Communications surveys&tutorials,vol.12,no. 1,pp.59-86,2010.doi:10.1109/SURV.2010.020110.00019.

[12]L.Zhang,J.Wu,and X.Deng,“Modelling the guaranteed QoS for wireless sen⁃sor networks:a network calculus approach,”EURASIP Journal Wireless Com⁃munication and Networking,vol.82,2011.doi:10.1186/1687⁃1499⁃2011⁃82.

[13]J.Liu,L.Zhang,and K.Yang,“Modeling guaranteed delay of virtualized wire⁃less networks using network calculus,”in Proceedings of the 10th International Conference on Mobile and Ubiquitous Systems:Computing,Networking and Ser⁃vices(MOBIQUITOUS 2013),Tokyo,Japan,December 2-4,2013.

[14]N.M.M.K.Chowdhury,and R.Boutaba,“Network virtualization:state of the art and research challenges,”IEEE Communications Magazine,vol.47,no.7, pp.20-26,2009.doi:10.1109/MCOM.2009.5183468.

[15]G.Schaffrath,C.Werle,P.Papadimitriou,A.Feldmann,R.Bless,A.Greenhal⁃gh,A.Wundsam,M.Kind,O.Maennel,and L.Mathy,“Network virtualization architecture:Proposal and initial prototype,”in Proceedings of ACM VISA,NY, USA,2009,pp.63-72.doi:10.1145/1592648.1592659.

[16]R.Kokku,R.Mahindra,H.Zhang,and S.Rangarajan,"NVS:A substrate for virtualizing wireless resources in cellular networks,"IEEE/ACM Transactions on Networking,vol.20,no.5,pp.1333-1346,2012.doi: 10.1109/ TNET.2011.2179063.

[17]S.Ahn and C.Yoo,"Network interface virtualization in wireless communica⁃tion for multi⁃streaming service,"in Proceedings of International Symposium on Consumer Electronics(ISCE),Singapore,2011,pp.67-70.doi:10.1109/ ISCE.2011.5973785.

[18]M.Tao,Y.C.Liang,and F.Zhang,“Resource allocation for delay differentiat⁃ed traffic in multiuser OFDM systems,”IEEE Transactions on Wireless Commu⁃nications,vol.7,no.6,pp.2190-2201,2008.doi:10.1109/TWC.2008.060882.

[19]D.S.W.Hui,V.K.N.Lau,and H.L.Wong,“Cross⁃layer design for OFDMA wireless systems with heterogeneous delay requirements,”IEEE Transactions on Wireless Communications,vol.6,no.8,pp.2872-2880,2007.doi:10.1109/ TWC.2007.05716.

[20]C.C.Zarakovitis,Q.Ni,D.E.Skordoulis,and M.G.Hadjinicolaou,"Power⁃ef⁃ficient cross⁃layer design for OFDMA systems with heterogeneous QoS,imper⁃fect CSI,and outage considerations,"IEEE Transactions on Vehicular Technolo⁃gy,vol.61,no.2,pp.781-798,2012.doi:10.1109/TVT.2011.2179817.

[21]Y.Jiang,“Stochastic network calculus for performance analysis of Internet net⁃works-An overview and outlook,”in Proceedings of International Conference on Computing,Networking and Communications(ICNC),Maui,HI,2012.doi: 10.1109/ICCNC.2012.6167501.

[22]F.Ciucu,F.Poloczek,and J.Schmitt,“Sharp bounds in stochastic network cal⁃culus,”ACM SIGMETRICS Performance Evaluation Review,vol.41,no.1,pp. 367-368,2013.doi:10.1145/2494232.2465746.

[23]J.B.Schmitt,F.A.Zdarsky,and M.Fidler,“Delay bounds under arbitrary mul⁃ tiplexing:when network calculus leaves you in the lurch....,”in Proceedings of 27th IEEE International Conference on Computer Communications(INFOCOM’08),Phoenix,USA,April 2008.doi:10.1109/INFOCOM.2008.228.

[24]A.Bouillard,L Jouhet,and E.Thierry,“Tight performance bounds in the worst⁃case analysis of feed⁃forward networks,”in Proceedings of the 29th IEEE Inter⁃national Conference on Computer Communications(INFOCOM’10),San Diego, USA,2010,pp.1316-1324.doi:10.1109/INFCOM.2010.5461912.

[25]J.B.Schmitt,F.A.Zdarsky,and L.Thiele,“A comprehensive worst⁃case calcu⁃lus for wireless sensor networks with in⁃network processing,”in Proceedings of the 28th IEEE International Real⁃Time Systems Symposium(RTSS’07),Tuc⁃son,USA,December 2007,pp.193-202.doi:10.1109/RTSS.2007.17.

[26]L.Zhang,“Bounds on end⁃to⁃end delay jitter with self⁃similar input traffic in ad hoc wireless network,”in Proceedings of 2008 ISECS International Colloqui⁃um on Computing,Communication,Control,and Management(CCCM’08), Guangzhou,China,August 2008,pp.538-541.doi:10.1109/CCCM.2008.23.

[27]L.Zhang,S.Liu,and H.Xu,“End⁃to⁃end delay in wireless sensor network by network calculus,”in Proceedings of 2008 International Workshop on Informa⁃tion Technology and Security(WIST’08),Shanghai,China,December 2008, pp.179-183.

[28]H.Qi,Z.Chen,and L.Zhang,“Towards end⁃to⁃end delay on WMNs based on statistical network calculus,”in Proceedings of the 9th International Conference for Young Computer Scientists(ICYCS’08),Zhang JiaJie,Hunan,China,No⁃vember 2008,pp.493-497.doi:10.1109/ICYCS.2008.347.

[29]Azodolmolky S,Nejabati R,Pazouki M,Wieder P,Yahyapour R,and Simeoni⁃dou D,“An analytical model for Software Defined Networking:A network cal⁃culus⁃basedapproach,”inGlobalCommunicationsConference(GLOBECOM2013), Atlanta,USA,December 09-13,2013.doi:10.1109/NOCS.2009.5071447.

[30]A.K.Parekh and R.G.Gallager,“A generalized processor sharing approach to flow control in integrated services networks:the single node case,”IEEE/ACM Transactions on Networking,vol.1,no.3,pp.344-357,1993.doi:10.1109/ 90.234856.

[31]R.R.Boorstyn,A.Burchard,J.Liebeherr,and C.Oottamakorn,”Statistical ser⁃vice assurances for traffic scheduling algorithms,”IEEE Journal on Selected Ar⁃eas in Communications Special Issue on Internet QoS,vol.18,no.10,pp.2651-2664,2000.doi:10.1109/49.898747.

Manuscript received:2014⁃02⁃24

Biograpphhiieess

Lianming Zhang(zlm@hunnu.edu.cn)received his PhD from Central South Univer⁃sity,China.He received his BS degree and MS degree from Hunan Normal Universi⁃ty,China.He is currently a professor in the school of Physics and Information Sci⁃ence,Hunan Normal University.His current research interests include software⁃de⁃fined networking,complex networks,and network calculus.He completed a two⁃year postdoctoral fellowship in complex networking at South China University of Technology.He has published more than 90 papers.

Jia Liuis currently an MS student in the College of Physics and Information Sci⁃ence,Hunan Normal University.She also received her BEng.degree from Hunan Normal University.Her research interests include wireless communications,wire⁃less networks,and network calculus.She has published one journal paper in Spring⁃er/ACM Mobile Networks and Applications(MONET)and one conference paper in IEEE MOBIQUITOUS 2013.

Kun Yang(kunyang@essex.ac.uk)received his PhD from University College Lon⁃don.He received his BSc degree and MSc degree from Jilin University,China.He is currently a chair professor in the School of Computer Science and Electronic Engi⁃neering,University of Essex,and leads the Network Convergence Laboratory(NCL) there.Before joining the University of Essex in 2003,he worked for several years at UCL on several EU research projects.His main research interests include heteroge⁃neous wireless networks,fixed mobile convergence,future Internet technology and network virtualization,cloud computing and networking.He manages research proj⁃ects funded by various sources such as UK EPSRC,EU FP7 and industries.He has published 60+journal papers.He serves on the editorial boards of both IEEE and non⁃IEEE journals.He is a Senior Member of IEEE and a Fellow of IET.