Probabilistic Availability Quantification of PON and WiMAX Based FiWi Access Networks for Future...

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Probabilistic Availability Quantification of PON andWiMAX Based FiWi Access Networks for Future

Smart Grid ApplicationsMartin Levesque and Martin Maier, Optical Zeitgeist Laboratory, INRS

Abstract—Availability is one of the most important qualityattributes for smart grid communications, as qualitatively definedin the IEEE P2030 standard. However, the availability metricmust be quantified in order to validate given smart grid ap-plication requirements. In recent related work, availability hasbeen quantified for wireless and optical backhaul networks interms of communications reachability, while in some other workavailability was not formally defined in a fine-grained mannerand was assumed to be known. In this paper, we develop anovel multi-class probabilistic availability model for integratedpassive optical network (PON) and WiMAX networks in order toquantify this metric according to medium access control (MAC)protocol limits as well as fiber and base station failures. Theobtained numeric results show interesting availability behaviors,including the impact on availability depending on the numberof base stations. We also investigate optical traffic re-routingthrough WiMAX when fiber faults occur and show that thereexists a maximum amount of re-routed traffic for maximizingavailability. Furthermore, we investigate a scenario of real-worldsmart grid traffic configurations shared with regular traffic andfind the maximum sensor data rate to meet the availabilityrequirements.

Index Terms—Communications availability, fiber-wireless com-munications, PON and WiMAX networks, smart grid communi-cations.

I. INTRODUCTION

THE research on a new power system paradigm, the smartgrid (SG), has recently attracted significant attention

in the information technology (IT), telecommunications, andpower system communities [1]. The quantification of the com-munications requirements of several smart grid applicationshas been done in terms of security, reliability, bandwidth,and latency [2]. Quantifying the requirements of smart gridcommunications is crucial from a research perspective, sincethe predicted communications performance can be comparedto given requirements and new communications mechanismscan be developed to meet these requirements. Efforts havealso been made by standardization organizations. IEEE P2030is one of the first attempts to standardize smart grids [3]. IEEEP2030 presents generic design and implementation guidelinesamong systems in order to exchange data between smartgrid components. More specifically, the standard lists fourmain quality attributes: scalability, interoperability, informa-tion reliability, and security. In this work, we focus on thereliability quality attribute. Reliability is defined as the abilityto execute a function under given conditions for a given

This work was supported by FQRNT Doctoral Research Scholarship No.165516 and NSERC Strategic Project Grant No. 413427-2011.

Corresponding author: Martin Levesque, INRS, Montreal, QC, Canada H5A1K6 (email: [email protected]).

period of time. Information reliability has the following twomain data characteristics: (i) availability and (ii) level ofassurance. The IEEE P2030 standard qualitatively defines theavailability requirement to be either low, medium, or high.A critical information requiring high availability could, forinstance, cause catastrophic impacts if the requirement isnot met. Although qualitative requirements can be useful togive an idea about the critical importance of an operation,it is not sufficient from an engineering perspective though;instead qualitative requirements must be used in collaborationwith quantitative models. In this paper, we are interested inquantitatively defining the availability metric for a specificnext-generation communications architecture.

Availability is a well-known metric used by engineers toquantify the availability of a given piece of equipment [4]:

A =MTTF

MTTF +MTTR, (1)

where MTTF corresponds to the mean time to failure andMTTR is defined as the mean time to repair. In telecommu-nications, information can be exchanged and thus is availableif the network itself is available. In this paper, we quantify thenetwork availability, that is, the probability for a given packetto reach a given destination without failure. There exist twomain reasons that can cause a network failure if the networkis properly installed and not mobile:• One or several networking components physically fail.

From the equipment specifications, one can determineavailability A for certain components by using the givenvalues of MTTF and MTTR 1.

• One or several communications nodes are satu-rated/congested and thus packets are constantly dropped.

Note that it is not necessarily always trivial to find theexact values of MTTF and MTTR. For example, a fibercut could occur randomly and does not necessarily depend onthe product specifications. Therefore, in this work, we define Ain terms of probabilistic availability by combining parametersfor both failure types listed above.

A variety of communications technologies could be usedfor smart grid communications. According to Ovum Tele-coms, a leading analysis company in the telecommunicationsindustry, as Chinese utility companies such as State GridCorporation of China (SGCC)2 are migrating toward smart

1Examples are available for several N-Tron Ethernet switch series at: http://www.n-tron.com/pdf/network availability.pdf.

2Refer to: http://www.fiercetelecom.com/story/chinas-smart-grid-drive-creates-15-billion-opportunity-pon-vendors-says-ovu/2013-01-09.

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grids, backhaul networks based on passive optical networks(PONs) are planned to be widely deployed for two purposes:to enable emerging smart grid applications and to offer fiber-to-the-home (FTTH) services to consumers as a long-termand cost-effective communications solution for both smartgrid and triple-play services. China Telecom deployed morethan 70 millions FTTH ports from 2004 to 2012 [5]. Ac-cording to China Telecom, some of the major trends are:(i) increase of fiber penetration, whereby optical nodes getcloser to customers to improve bandwidth and quality ofnetwork services, (ii) higher volume and speed, from legacyEthernet/Gigabit PONs (EPONs/GPONs) to 10G-EPON to-wards 40G time division multiplexing (TDM) and wavelengthdivision multiplexing (WDM) PONs to provide at least 40Gbps in one PON system, and (iii) increased coverage bycombining fixed, wireless (e.g., wireless local area network(WLAN)), and mobile (e.g., 3G and Long Term Evolution(LTE)) nodes.

In this paper, we focus on an integrated fiber-wireless(FiWi) communications architecture based on PONs andWorldwide Interoperability for Microwave Access (WiMAX)[6]. Broadband access networks based on PONs have multi-ple significant merits: low operational expenditures (OPEXs),future-proofness, longevity, and high reliability as they usecompletely passive (e.g., unpowered) network components.Network operators (e.g., China Telecom) are seeking next-generation PONs (NG-PONs) that can transparently coexistwith legacy PONs such as IEEE 802.3ah/av 1/10 Gbps EPONarchitectures, enable gradual upgrades in order to avoid costlyand time-consuming network modifications, and stay flexiblefor future evolution paths. The final frontier of NG-PONs isthe integration with their wireless counterparts, giving riseto bimodal FiWi broadband access networks [7]. Wirelessnetworks such as WiMAX, ZigBee, and WLANs based onIEEE 802.11g/n/ac are highly available as they do not needto be physically connected and provide mobility, but they areless reliable due to channel interference. In this paper, forthe wireless domain, we focus on WiMAX since in Canada aseparate spectrum (30 MHz frequency spectrum in 1.8 GHz)is dedicated to the utilities and this technology provides long-range connectivity. Electricity companies in Australia havealso proposed WiMAX-based smart grid solutions [8]. Notethat LTE could also be used for smart grid communications. In[8], the authors have shown that LTE technologies can satisfythe latency and reliability requirements of distribution automa-tion (DA) networks. The considered traffic load and latencyrequirements were 1-480 bps and 1-4 seconds, respectively,which can be provided over LTE. However, for novel smartgrid applications requiring larger data rates, communicationsover LTE may not remain cost-effective. Furthermore, theminimum scheduling delays of LTE correspond to 65.5 ms and16.5 ms for the upstream and downstream directions, respec-tively [8]. Some smart grid applications require lower latencyin the order of 12-20 ms, such as real-time sensing/metering[9] and wide-area situational awareness (WASA). For compar-ison, the minimum upstream/downstream delay can be as lowas 3·2.5 ms with IEEE 802.16-2004 WiMAX [10]. Combiningboth fiber and wireless networks is a promising long-term

and cost-efficient solution for smart grid communications[11]. Therefore, we focus on this type of communicationsarchitecture in order to define network availability.

The performance of PON and WiMAX technologies havebeen widely studied in terms of different performance met-rics over the last few years. In [12], the authors developeda novel techno-economic analysis to compare EPON andWiMAX technologies with regard to equipment, installation,power consumption, and repairing costs. The medium accesscontrol (MAC) protocol performance in terms of delay andcapacity has also been modeled. The delay and throughputperformance of WiMAX technologies under non-saturated[13] and saturated [14] conditions has been quantified. A ca-pacity and delay analytical framework for TDM/WDM PONswas developed in [15], and the survivability performance ofFiWi access networks was studied in [16]. In our previouswork, we developed a novel capacity model for FiWi accessnetworks based on TDM/WDM PONs and Gigabit WLANs toevaluate the performance of FiWi network routing algorithms[17]. Although these analytical models allow to evaluate theperformance of different FiWi access network metrics, thenetwork availability quantification studies presented up to dateare limited and are briefly reviewed in the next section. Thispaper provides a probabilistic availability analytical frameworkfor WiMAX as well as legacy and NG-PONs. Note that themodel is applied to specific technologies, but several portionsof the model can be adapted to other technologies and couldbe extended, for instance, to include also the probabilisticreachability between nodes. The main novelties of this paperare as follows:• Availability is not only modeled considering the physical

equipment but also taking into account MAC protocollimits, which can affect availability depending on theamount of traffic routed in the network.

• The model considers both non-saturated and saturatedtraffic conditions as well as failure of optical fibers andWiMAX base stations (BSs).

• The traffic matrix takes into account multiple prioritizedclasses.

• In the presented numerical results, we investigate ascenario with traffic configurations based on smart gridtraffic patterns of smart grid applications we recorded.

• The quantified availability metrics can be used to com-pare them with given smart grid requirements, as avail-ability is one of the most important metrics of smart gridcommunications [3].

The remainder of the paper is structured as follows. Theconsidered communications architecture based on integratedTDM/WDM PON and WiMAX is described in Section II.The network availability analysis is developed in Section III.Section IV presents numerical results. Section V concludes thepaper.

II. INTEGRATED TDM/WDM PON AND WIMAXNETWORK ARCHITECTURES

There exist a variety of different ways to use/integrate PONand WiMAX technologies. In [18], the authors argue that

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Fig. 1: Conventional cascaded PON and WiMAX architecture.

both technologies are less likely to be integrated, as operatorschoose either WiMAX or PON since both share commoncharacteristics. Both technologies have a point-to-multipointtopology with a central node (optical line terminal (OLT)or BS) allocating bandwidth in the network. The end-users,connected to the optical network units (ONUs) for the PONcase or subscriber stations (SSs) for the WiMAX case, sendtheir bandwidth requirements to the central node, which inturn sends back the bandwidth allocations. The OLT in a PONsystem periodically polls ONUs, which subsequently sendback their current queue lengths in order to allocate bandwidth,whereas WiMAX uses a random access mechanism compliantwith IEEE 802.16 to send bandwidth request messages.

Clearly, such a non-cascaded architecture cannot take ad-vantage of both networks. In [19], different types of possibleEPON and WiMAX integrations were discussed, such asthe one depicted in Fig 1. A first type of integration is tocombine both networks independently, whereby ONUs andSSs remain unchanged and are interconnected to combineboth technologies. A different type of integration is the hybridarchitecture, whereby BSs and ONUs are merged, but theconnection mechanisms are not unified. The unification ofconnections in EPON and WiMAX networks represent anothertype of integration. Integrating EPON and WiMAX allowsto extend the EPON coverage at low cost, which can bevery beneficial for example in rural areas to avoid deployingfibers for sparsely distributed homes. In legacy wavelength-broadcasting (WB) TDM PONs (e.g., EPON and GPON) orWDM PONs, ONUs share the wavelength(s) for both upstreamand downstream traffic. Such an architecture typically cancover a distance of 20 kilometers.

As forecasted by major telecommunications providers suchas China Telecom, networking trends include a deeper fiberpenetration and growth of network speed. Wavelength-routing(WR) WDM PONs are expected to play a key role in responseto these trends. This type of PON system is characterizedby the replacement of one or several splitters/combiners withpassive arrayed-waveguide gratings (AWGs), which act as awavelength (de)multiplexer, whereby each wavelength may beshared by a subset of ONUs [11]. Fig. 2 shows an example ofsuch a WR WDM PON system composed of 5 stages with asplitting ratio of 6 in a pyramid-based topology (as proposedin [16]). A subset of ONUs have a WiMAX interface andcan therefore act as SS or BS. Note that this type of multi-stage PON system offers a higher coverage of up to 100 kms,thus also referred to as Long-Reach PON. This kind of hybrid

Fig. 2: Pyramid-based FiWi topology with WR WDM PONconsisting of 76 ONUs, whereby ONUs have WiMAX inter-faces and can act as SS or BS (splitting ratio: 6).

topology helps improve network availability as WiMAX canbe used temporarily in case of fiber failures and vice versa,which we model in the next section.

III. NETWORK AVAILABILITY ANALYSIS

Availability was the object of several recent research studiesin the context of optical and wireless networks. An availability-aware routing algorithm for hybrid wireless-optical broadbandaccess networks (WOBANs) was proposed in [20], wherepaths maximizing networking availability were selected. How-ever, the availability metric itself was not modelled and wasassumed to be known. In [21], availability was consideredin a novel provisioning path protection scheme for opticalWDM networks, whereby availability was computed based ona primary and a backup path according to the fiber length andoptical cross-connect (OXC) ports. Availability was also stud-ied for wireless networks according to the distance betweennodes and depending on their transmission and receptioncapabilities [22], [23]. In the context of smart grids, it isimportant to know the availability between wireless nodesduring installations. However, once installed, there is still aneed to quantify the availability depending on the failure ofthe components and considering the limitations of the MACprotocols in use. In the following, we quantify the networkavailability considering the MAC protocols of PONs andWiMAX as well as random fiber and BS failures under thefollowing assumptions:• For WiMAX, only the upstream direction is modelled

and thus no traffic can be routed in the downstreamdirection. Note that the main traffic generated from smartgrid sensors is typically sent upstream from sensors to thedistribution management system (DMS).

• We take into account two types of failure: (i) a fibercut at stage s occurring with probability pops , and (ii)a failure of a base station z occurring with probabilitypwiz . Both terms pops and pwiz represent the long-termfailure probabilities, that is, the probability that at anygiven time instant the component fails. These termscan be defined based on the MTTF and MTTR inEq. (1) obtained from the component specifications. Thedeveloped analysis below accommodates any type of fiberand BS failure.

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• The traffic is differentiated per class, whereby exist-ing/regular Cr and smart grid Csg traffic classes areconsidered. The overall traffic classes are given by C =Cr + Csg . All traffic classes C can be routed in the PONnetwork, but only the smart grid traffic (Csg) can useWiMAX dedicated to smart grids. Each traffic class hasan average payload length denoted by Lc,∀c ∈ C.

• As we take into account the MAC protocol performanceof both the PON and WiMAX networks, availabilityis defined by considering both frame drop and failureprobabilities. It relates to Eq. (1) as it represents thelong-term availability probability, but without explicitelydefining MTTF and MTTR.

• To find the MAC protocol limits, we adapt the analysisdeveloped in [17] for PON and [13] for WiMAX.

• We assume each FiWi node has a maximum queue lengthof Q frames and thus frames are dropped accordingto the following blocking probability equation undernonsaturated (0 ≤ ρ < 1, M/M/1/K model from [24,Eq. (3.43)]) and saturated conditions (ρ ≥ 1):

B(ρ) =

{(1−ρ)·ρQ+1

1−ρQ+1+1 , if 0 ≤ ρ < 1

1− 1ρ , if ρ ≥ 1

, (2)

where ρ represents the traffic intensity and B(ρ) theblocking/dropping probability.

• WiMAX frames are composed of mini-slots and duringa frame the mini-slots can be allocated to multiple SSs[25]. In this paper, similarly to [13], we assume a givenframe is fully allocated to a single SS.

To introduce our model, Fig. 3 illustrates our probabilisticmodel for a single-stage WB TDM/WDM PON. The availabil-ity of such a topology can be found by executing the followingsteps:• Calculate the blocking probability at the ouput ports

of SSs, ONUs, and OLT, corresponding to B(ρwiw ),B(ρop,u), and B(ρop,d) (derived thereafter in this sec-tion), respectively. Note that these three terms correspondto the generic blocking probability term defined in Eq. 2,which is generally applicable to any traffic intensity.

• Define the fiber failure probability at stages 0 and 1,corresponding to pop0 and pop1 and derived thereafter inthis section.

• Define the BS failure probability at each ONU/BS, cor-responding to pwiz and derived thereafter in this section.

• Calculate the probabilistic availability which is derivedthereafter in this section.

A. Network Model

1) Generic Definitions: In this paper, we focus on PONsconsisting of Λ bidirectional wavelength channels, indexed asλ = 1, 2, ...,Λ. We consider two distinct flavors:• Wavelength-broadcasting (WB) single-stage TDM/WDM

PON: A WB TDM/WDM PON consists of a legacysplitter/combiner as remote node (RN) and deploys Λwavelengths, which can be used by all ONUs. For legacyEPONs and GPONs we have Λ = 1.

Fig. 3: Probabilistic model for a single-stage WB TDM/WDMPON. B(ρwiw ), B(ρop,u), and B(ρop,d) correspond to theblocking probabilities at the output ports of the SSs, ONUs,and OLT, respectively; pop0/1 denotes the fiber failure probabil-ity at stage 0 or 1, and pwiz denotes the BS failure probability.

• Wavelength-routing (WR) multi-stage WDM PON: Theconventional splitter/combiner at the RN is replacedwith a wavelength (de)multiplexer. A typical wavelength(de)multiplexer is an arrayed-waveguide grating (AWG).A given wavelength λ can be used by a subset of theONUs forming a sector at each AWG output port.

The PON can be either single-stage or multi-stage. We letΞ denote the number of fiber stages, whereby we considerΞ = 2 for WB PONs and Ξ ≥ 2 for WR WDM PONs. Notethat we have Ξ = 2 for single-stage PONs since there is a fiberbetween the OLT and RN and another one between the RN andONUs. Note that a fiber stage corresponds to the maximumnumber of links between the OLT, intermediate RNs, andONUs, whereby multiple wavelengths can be shared in eachgiven link. A sector, using a given wavelength, corresponds tothe set of ONUs (end-users) sharing a given wavelength. LetSλ denote the set of ONUs in sector λ such that:

S1 ∪ ... ∪ SΛ = O, (3)

whereby O denotes the set of ONUs. Each wavelength channelhas a capacity of cop (in bps). Furthermore, the stage of ONUo is defined as s(o), ∀o ∈ O, s(o) ∈ {1, ...,Ξ}. Note that anONU cannot be installed at stage 0, which corresponds to thefiber between the OLT and first remote node.

As for the WiMAX network(s), let B denote the set of basestations:

∀z ∈ B, z ∈ {0, 1, ..., |O|}, (4)

where the index 0 corresponds to the OLT and z representsa zone aggregating traffic from subscriber stations (SSs),denoted asWz . SSs are either ONUs equipped with a WiMAXinterface or SSs with a WiMAX interface only. Thereby, theset of WiMAX SSs having only a wireless interface is givenby:

W =⋃z∈BWz −O. (5)

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This model is generic and can accommodate multiple dif-ferent FiWi access network topologies, which we illustrate inSection IV.

2) Pyramid-based Topology: The analysis accommodatesany type of topology. We describe below the pyramid-basedtopology since this type of topology presents interesting sym-metric properties. As depicted in Fig. 2, the number of ONUsdepends on the splitting ratio S and number of stages Ξ, andis given by:

2Ξ−2 · S +

Ξ−2∑i=1

2i−1 · (S − 2), (6)

where the first term corresponds to ONUs at the last stage(i = Ξ − 1) and the second term accounts for the ONUs atstages [1..Ξ− 2].

For a given set of ONUs attached to a RN, we let β denotesthe ratio of the number of BSs versus S. For example in Fig.2, we have β = 2

6 .

B. Traffic Matrix

We first define the set of FiWi nodes as follows:

N = {0} ∪ O ∪W. (7)

Next, we define the traffic matrix consisting of source-destination connections characterized by a traffic rate (inframes/sec.): Mi,j,c, whereby i, j ∈ N and c ∈ C. Mi,j,c = 0for i = j,∀c ∈ C. For the traffic being forwarded first inthe wireless domain, we define Mi,j,c following a matrix withthe same properties as Mi,j,c. Also, for all SSs w, w ∈ Wand i ∈ N , we have Mi,w,c = 0, that is, we do not modeldownstream WiMAX traffic.

C. Traffic Intensity

We next derive the traffic intensities for both PON andWiMAX systems. These traffic intensities are used to calculateblocking/dropping probabilities.

Each class c ∈ C has a priority αc such that∑c∈C αc = 1.

A given frame of class c is assigned the following probability,used thereafter in the analysis:

χc(R, ρ) =min(AdmS2c(R, ρ), Rc · Lc)

Rc · Lc, (8)

where R and ρ denote the set of traffic rates and trafficintensity at a given output port, respectively. The equationdefines the ratio of the per-class admitted data rate andper-class load, whereby AdmS2c(R, ρ) gives the bandwidthscheduled for class c, which is given by:

AdmS2c(R, ρ) = AdmS1c(R, ρ)−∑∀i6=c,αi=αc

min

(AdmS1c(R, ρ) ·

1

|[i ∈ C|αi = αc]|, Ri · Li

)(9)

with AdmS1c as follows:

AdmS1c(R, ρ) = (1−B(ρ)) ·∑i∈C

Ri · Li −∑

∀i,αi>αc

Ri · Li.

(10)1) Wireless Domain: The traffic rate of class c destined to

n from a given SS w with wireless support only is given by:

Rwiw,n,c = Mw,n,c, (11)

where n ∈ N . For a given ONU o in zone z having a WiMAXinterface sending to n (either a different ONU or the OLT),the traffic rate of class c takes into account the traffic forcedto be sent to the wireless domain and optical traffic destinedtowards faulty (due to frame drop or fiber failure) fibers:

Rwio,n,c = Mo,n,c +Rop,fo,n,c,

(12)

where the first term accounts for the traffic forced to be sentin the wireless domain. The second term corresponds to theoptical traffic from ONU o destined towards faulty fibers beingre-routed in the wireless domain and sent to a reachable basestation z (if any):

Rop,fo,n,c =

{%op,fo,n,c, if ∃z ∈ Wz ∧ o ∈ Wz

0, otherwise. (13)

%op,fo,n,c is given by:

%op,fo,n,c = δwio,z ·Mo,n,c ·(1− χc

([∑∀n′

Rop,uo,n′,∀c′

], ρop,uo

)·

0∏i=s(o)

(1− popi

)),

(14)

where δwio,z ∈ [0..1] corresponds to the ratio of the opticaltraffic destined to faulty fibers permitted to be re-routed in thewireless domain toward BS z. The traffic intensity for a givenwireless interface w ∈ Wz in zone z is defined as the trafficrate multiplied by the access delay:

ρwiw =∑c∈Csg

∑n∈{0}∪O

Rwiw,n,c · E[Tpkt, z], (15)

where E[Tpkt, z] corresponds to the WiMAX access delay ina given zone z, given in Eq. (17) in [13]. Note that in [13],a single wireless zone was used, thus in this work, since wehave several WiMAX zones, we have one access delay foreach given zone. Note that we adapt the traffic rate at each SSof [13, λ

N ] to:

1

|Wz|·∑c∈Csg

∑w∈Wz

∑n∈{0}∪O

LcTframe · cwi

·Rwiw,n,c,∀z ∈ B,

(16)where cwi denotes the WiMAX capacity.

Furthermore, we adapt the analysis in [13] for both non-

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saturated and saturated conditions (note that [13] applies onlyto non-saturated conditions). To do so, we bound the trafficintensity for a given SS, corresponding to [13, (16) and (17)],to be maximally 1/0.9 as follows:

P ∗0 =1− e− λ

N ·E[Tpkt]

min(1, λN · E[Tpkt])·(

1−min(

1,λ

N· E[Tpkt]

))(17)

and for T3:

T3 =1

2+

1

2 · (1−min(0.9, P0 · λ · Tframe)). (18)

Note that the variables in Eqs. (17, 18) refer to the variablesin the WiMAX analysis [13].

2) Fiber Domain: The upstream traffic rate of class c froma given ONU o (in zone z if it has a wireless interface) toONU/OLT n 6= o is given by:

Rop,uo,n,c = Mo,n +

δopo,0 · Mo,n,c ·

[1−

χc

([∑∀n′

Rwio,n′,∀c′

], ρwio

)·(

1− pwiz)]

+∑w∈Wo

Mw,n,c · (1− pwio ) ·

χc

([∑∀n′

Rwiw,n′,∀c′

], ρwiw

)+

∑o2∈Wo

Rop,eo2,n,c ·

[1−

χc

([∑∀n′

Rwio2,n′,∀c′

], ρwio2

)· (1− pwio )

],

(19)

where the first term accounts for the optical traffic, the secondterm corresponds to the traffic destined to a faulty WiMAXnode re-routed in the optical domain towards the OLT, the thirdterm represents the traffic coming from SSs with WiMAXsupport only, and the fourth term accounts for the trafficpreviously re-routed from the optical domain to the wirelessdomain and sent to o. δopo,0 corresponds to the ratio of thepermitted traffic re-routed from the wireless to the opticaldomain.

Similarly to [17, Eq. (6)] (but extended to multiple classes),the upstream traffic intensity for WB PONs is given by:

ρop,u =∑c∈C

LcΛ · cop

·∑o∈O

∑n∈{0}∪O−{o}

Rop,uo,n,c (20)

and similarly to [17, Eq. (4)] (extended to multiple classes)

ρop,uλ =∑c∈C

Lccop·∑o∈Sλ

∑n∈{0}∪O−{o}

Rop,uo,n,c (21)

for WR PONs.The downstream traffic rate of class c for WB PONs is given

by:

Rop,dc =∑o∈O

M0,o,c +

∑n∈O

Rop,un,o,c · χc

([ ∑∀n′∈O

Rop,uo,n′,∀c′

], ρop,u

)·

( 0∏i=1

(1− popi )

),

(22)

where the first term accounts for the traffic from the OLT toONUs and the second term for the traffic from ONUs to theOLT subsequently forwarded to another ONU. Similarly, thedownstream traffic rate of class c for WR PONs is given by:

Rop,dλ,c =∑o∈Sλ

M0,o,c +

Λ∑l=1

∑n∈Sl

Rop,un,o,c · χc([ ∑∀n′∈O

Rop,uo,n′,∀c′

], ρop,ul

)·

0∏i=s(n)

(1− popi ) +

∑w∈W0

Rwiw,o,c · χc([ ∑∀n′∈O

Rwiw,n′,∀c′

], ρop,uw

)·

(1− pwi0 ),

(23)

where the additional third term accounts for the wireless trafficsent to the OLT/BS and subsequently optically forwarded toa given ONU. The downstream traffic intensity for WB PONsis defined similarly to [17, Eq. (7)] (extended to multipleclasses):

ρop,d =∑c∈C

LcΛ · cop

·Rop,dc (24)

and similarly to [17, Eq. (5)]

ρop,dλ =∑c∈C

Lccop·Rop,dλ,c (25)

for WR PONs.In the remainder, for convenience, we use ρop,uΘ and ρop,dΘ to

denote the upstream and downstream intensities, respectively.Θ represents the PON flavor (empty for WB and λ for WR)and thus the proper traffic intensity equation must be used.

D. Probabilistic Availability

We first derive the probabilistic availability of a single-hopwireless link as follows:

Awi,wiw,z,c = χc

([∑∀n′

Rwiw,n′,∀c′

], ρwiw

)·(

1− pwiz), (26)

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representing the probability that no frame is dropped andthat no BS failure occurs. The probabilistic availability forthe downstream traffic is defined according to the droppingprobability at the OLT and fiber failure(s):

Aolt,onu0,o,c = χc

([Rop,dΘ,∀c′

], ρop,dΘ,c

)·s(o)∏i=0

(1− popi

). (27)

The probabilistic availability for the upstream ONU-OLTtraffic is derived from both PON and WiMAX availabilities:

Aonu,olto,0,c =

[1−[1− χc

([∑∀n′

Rop,uo,n′,∀c′

], ρop,uΘ,c

)·

(1− popz )s(z)+1

]·[

1− δwio,0 · Awi,wio,0,c

]·[

1− δwio,z · Awi,wio,z,c ·

χc

([∑∀n′

Rop,uz,n′,∀c′

], ρop,uΘ,c

)·

(1− popz )s(z)+1

]],

(28)

where o ∈ Wz . The equation defines the availability probabil-ity according to three unavailability probabilities: (i) the failureprobability of using the fiber between ONU o and the OLT,(ii) the failure probability of using the WiMAX link betweenONU o and the OLT, and (iii) the failure probability to firstroute wirelessly from ONU o to ONU o2 and then opticallyforward the frame from o2 to the OLT.

For improved readability, we derive some availability equa-tions for the traffic between SSs and OLT:

Awi,oltw,0,c = Awi,wiw,z,c · Aonu,oltz,0,c , w ∈ Wz (29)

and for the traffic from SSs to ONUs:

Awi,onuw,o,c = Awi,oltw,0,c · Aolt,onu0,o,c , w ∈W0. (30)

Finally, we derive the average probabilistic availability ofclass c in the FiWi access network as follows by taking intoaccount all possible paths:

Ac =1∑

n1∈N∑n2∈N ,n1 6=n2

Mn1,n2,c·(∑

z∈B

∑w∈Wz,w∈W

Mw,z,c · Awi,wiw,z,c +∑z∈B

∑w∈Wz,w∈W

Mw,0,c ·Awi,oltw,0,c +∑z∈B

∑w∈Wz,w∈W

∑o∈O−{z}

Mw,o,c · Awi,onuw,o,c +

∑o∈O

Mo,0,c · Aonu,olto,0,c +∑o1∈O

∑o2∈O,o2 6=o1

Mo1,o2,c · Aonu,olto1,0,c

· Aolt,onu0,o2,c+

∑o∈O

M0,o,c · Aolt,onu0,o,c

),

(31)

whereby the individual availabilities are as follows:• Single-hop wireless communications,• Wireless communications destined to the OLT,• Wireless communications destined to ONUs,• Optical traffic destined to the OLT,• Optical traffic from ONUs to ONUs, and• Optical traffic from the OLT to ONUs.Note that A represents the average network availability. We

derive a second availability metric representing the probabilis-tic availability on a per-node and class basis with confidenceinterval:

Apn,c = µ(ϑc)± z ·σ(ϑc)√|ϑc|

, (32)

where z is the z-score value (e.g., z = 1.96 for a 95 %confidence interval) and ϑc denotes a vector containing theprobabilistic availability for all nodes for a given class c,defined as follows:

ϑc =

[[Awi,wiw,z,c |z ∈ B, w ∈ Wz, w ∈ W, c ∈ C

],

[Aonu,olto,0,c |o ∈ O

],[Aolt,onu0,o,c |o ∈ O

]].

(33)

Note that the notation [[.], [.], ..., [.]] corresponds to a vectorbuilt by concatenating the elements contained in a set ofvectors.

Fig. 4 illustrates our probabilistic model for a WR WDMPON (Ξ = 3). As can be seen, the case of WR WDM PONis more complex compared with a WB TDM/PON (Fig. 3)as multiple stages are modeled and both WiMAX and PONsystems overlap.

E. Adaptation of Framework to Mesh-based Topologies

The structure of the FiWi network considered in this paperis tree-based. Note, however, that traffic can be routed either

7


Fig. 4: Probabilistic model for a WR WDM PON (Ξ = 3).

Fig. 5: Two-stage fixed-point iterations technique to solve thenonlinear equation system.

via WiMAX or PON. The framework can be also adapted tomesh-based structures. Toward this end, from a given node nto node d the availability equations would need to considermultiple alternate adjacent nodes an,d as follows:(

1−∏

i∈an,d

(1−Ai)). (34)

Furthermore, the traffic rates need to be adapted in a similarfashion as done in Eq. (19), whereby the traffic can be routedtowards either the PON or WiMAX interface.

F. Solving the Analytical Framework

In order to solve the above developed nonlinear equationsystem, we use a two-stage fixed-point iterations technique,as illustrated in Fig. 5. We first solve Eqs. (20, 21, 24, 25) forthe upstream and downstream PON traffic intensities. Next,we solve the set of WiMAX non-linear systems in order tofind the access delay for each zone. By means of fixed-pointiterations, we calculate the three unknown variables from [13,Eqs. (14-16)] followed by the recalculation of E[Tpkt, z] untilE[Tpkt, z] does not vary much (e.g., 10−8). Once stable, we go

0

50

100

150

200

250

300

200 400 600 800 1000

Com

puta

tion d

ura

tion (

in s

ec.)

Number of nodes

Measurement

Fig. 6: Computation duration (complexity) versus topologysize (number of nodes).

back to the first stage in order to calculate Eq. (15). These stepsare executed until the traffic intensities do not vary much. Wenote that we did not experience any computation problems asthe equation systems converge quickly within a few iterations.

To provide insights into the computation requirements,we measured the computation duration as a function of thetopology size (i.e., number of nodes), as depicted in Fig. 6.We observe from the figure that the complexity of the systemgrows linearly for medium to large number of nodes. For about1000 nodes, the availability performance can be computedwithin 4 minutes using a 800 MHz processor and 5.4 GBrandom access memory (RAM).

IV. NUMERICAL RESULTS

This section presents numerical results on the networkavailability equation A from Eq. (31) for WB and WR PONsintegrated with WiMAX.

A. Configurations

TABLE I: Configurations

Parameter ValueQ 100 frames

Tframe (in [13]) 0.005 sec.cop 109 bpscwi 75 · 106 bps

We set the WiMAX capacity per channel/zone to 75 Mbpsassuming 64-QAM modulation [26]. Furthermore, accordingto the WiMAX specifications, the frame duration varies be-tween 5 and 20 ms. For ease of calculation, we set the meanframe duration in the wireless networks to Tframe = 0.005and first set the average FiWi frame length to Tframe ·75 ·106

bits. We first assume a single traffic class. Table I presents themain parameters used in all scenarios described below. Forthe other WiMAX configurations, we use the same ones asconsidered in [13]. We use a traffic matrix with Mi,j,c = Φ,

8


0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10

Availa

bili

ty

Offered load (Gbps)

EPON (Λ = 1) WDM PON (Λ = 4)

WiMAX

Fig. 7: WiMAX, WB TDM, and WDM PON availabilityperformance vs. the offered load (pwi = pop = 10−5).

i ∈ N and j ∈ {0} ∪ O, whereby Φ represents a variabletraffic rate given in frames/sec.

B. WB TDM/WDM PONs

We first consider a single-stage WB TDM PON supporting 1Gbps per wavelength, which can be upgraded to a WDM PONsupporting multiple bidirectional wavelengths. We examinethe WiMAX and PON networks independently without anyinteraction between them. Fig. 7 shows the availability ofeach network versus offered load (OL). Note that the WiMAXcase corresponds to a single zone with 16 SSs. Expectedly,the WiMAX availability drops more quickly compared to thePONs due to its low capacity. The availability of PONs dropswhen the offered load gets close to their capacity.

Next, we consider a topology with a similar structure asdepicted in Fig. 1, whereby the number of ONU/BSs is setto 16 and each ONU/BS aggregates traffic from 3 SSs. Fig.8 compares the availability of an EPON and WDM PON.The availability drops at approximately 1 Gbps for the EPONdue to optical capacity saturation and the availability drops at1.3 Gbps in the WDM PON case due to WiMAX capacitysaturation.

Next, we investigate the fiber and BS failure probabilitiesin Figs. 9 and 10 for different traffic matrices. Overall, theimpact of fiber failures is significantly larger compared to BSfailures, which is explained by the fact that a larger number ofconnections uses the PON. The impact is especially apparentwhen the network is highly loaded.

C. Next-Generation FiWi Settings with Integrated WR WDMPONs

In the remainder, we focus on a topology with a WR WDMPON following the same structure as in Fig. 2 (a pyramid-based topology) with a splitting ratio of 16 (S = 16). Asopposed to the previous subsection, each ONU has a WiMAXinterface, either acting as a SS or BS, therefore all parts of theanalysis are used. The number of stages is set to 4 (Ξ = 4),thus a total of 14 + 2 · 14 + 4 · 16 = 106 ONUs are installed.

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10

Availa

bili

ty

Offered load (Gbps)

EPON and WiMAX WDM PON and WiMAX

Fig. 8: Integrated WB TDM/WDM PON availability perfor-mance vs. the offered load (pwi = pop = 10−5, |W| = 16 · 3).

0.1

1

1e-05 0.0001 0.001 0.01 0.1 1

Availa

bili

ty

pwi

Low load (OL = 0.4 Gbps) Mid load (OL = 1.2 Gbps)

High load (OL = 2.5 Gbps)

Fig. 9: Impact of WiMAX BS failure probability on overallnetwork availability for different network offered loads (pwi =pwiz ,∀z ∈ B).

0.001

0.01

0.1

1

1e-05 0.0001 0.001 0.01 0.1 1

Availa

bili

ty

pop

Low load (OL = 0.4 Gbps) Mid load (OL = 1.2 Gbps)

High load (OL = 2.5 Gbps)

Fig. 10: Impact of fiber failure probability on overall networkavailability for different offered loads (pop = pops ,∀s ∈{0, ...,Ξ− 1}).

9


0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 1 10

Availa

bili

ty

Offered Load (in Gbps)

Availability

Fig. 11: Availability performance vs. the offered load for apyramid-based WR WDM PON consisting of 106 ONUs witha splitting ratio of 16.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Availa

bili

ty

β

Low Load (OL = 0.7 Gbps) Medium Load (OL = 6.7 Gbps)

High Load (OL = 20.6 Gbps)

Fig. 12: Impact of the number of BSs per S (β) on availability.

Stages 0-2 use AWGs as a remote node and the last stageuses conventional splitters. Hence, a total of 14 + 2 · 14 + 4 =46 wavelengths are available, each dedicated to a differentsector, whereby the wavelengths at the last stage are sharedby 16 ONUs. Furthermore, a total of 64 SSs with WiMAXsupport only are attached to the last PON stage. Fig. 11 showsthe probabilistic availability versus the offered load. The firstavailability drop close to 1 Gbps is due to the saturation of theWiMAX channels for zones attached to ONUs at the last stage.The second drop close to 10 Gbps is due to the successivesaturation of PON channels.

We investigate the impact of the parameter β in Fig. 12.We observe that under low and high traffic loads, increasingthe number of BSs does not significantly improve the overallnetwork availability. However, at medium traffic loads, weobserve a significant network availability improvement byadding 4-7 BSs (0.3 ≤ β ≤ 0.5). A surprising behaviouris observed for β > 0.5, where an increasing number of BSsdoes not further improve network availability.

Another important parameter is the ratio of the trafficdestined towards faulty fibers permitted to be re-routed inthe wireless domain, denoted as δwi and illustrated in Fig.13 for different values of β. We observe that for each givenconfiguration depending on the number of BSs installed, thereexists a δwi that maximizes network availability, namely 0.17for β = 3

16 , 0.37 for β = 716 , and 0.7 for β = 11

16 .

0.32

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.4

0.41

0 0.2 0.4 0.6 0.8 1

Availa

bili

ty

δwi

β = 3/16 β = 7/16

β = 11/16

Fig. 13: Impact of the ratio of the traffic destined towardsfaulty fibers permitted be re-routed in the wireless domain(OL = 20.5 Gbps, δwi = δwii,j ,∀i, j ∈ N ).

1e-05

0.0001

0.001

0.01

0.1

1

pwi

1e-05 0.0001 0.001 0.01 0.1 1

pop

0

0.2

0.4

0.6

0.8

1

Availability

Fig. 14: Availability as a function of fiber and BS failures(OL = 686 Mbps).

D. Impact of Fiber and BS Failures

In this subsection, we study equipment failures using thesame topology as in the previous subsection. We calculatethe availability by varying the values of pwi and pop from10−6 to 1 for low (OL = 686 Mbps) and high (OL = 20.5Gbps) traffic loads, as depicted in Figs. 14 and 15. For bothtraffic loads, we denote a higher reduction of the availability aspop increases, as compared to pwi. Furthermore, the variationof the BS failure probability has a significant impact on theavailability only at low loads, as depicted in Fig. 14, whichwas already observed in the case of the WB WDM PON (SeeFig. 9).

E. Coexistence of Human-to-Human (H2H) and Smart GridTraffic

In the following, we consider three traffic classes: oneregular class for triple-play (video, voice, and data) traffic andtwo others for control and monitoring SG traffic. The controltraffic can be used, for instance, to turn ON/OFF controllable

10


1e-06

1e-05

0.0001

0.001

0.01

0.1

1

pwi

1e-06 1e-05 0.0001 0.001 0.01 0.1 1

pop

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Availability

Fig. 15: Availability as a funciton of fiber and BS failures(OL = 20.5 Gbps).

TABLE II: Experimental measurements of smart grid applica-tions based on the IEC 61850 standard.

Source node Average payload length Rate per sec.HVA/LV 500 bytes 1

30

Substation 5000 bytes 130

DER 224 bytes 2 · 160·10

Switch 100 bytes 2 · 160·10

power switches. Therefore, the data rate of this type of trafficis generally small and event-driven, but with high priority.As for the monitoring traffic, typical smart grid sensors aretime-based with configurable data rates. We captured thetraffic of experimental telecontrol smart grid applications in[27]. Table II shows the average payload length originatingfrom high-voltage/low-voltage (HV/LV) transformers, substa-tion, distributed energy resources (DERs), and controllableswitches. A single variable-value pair (following the formatof manufacturing message specification messages (MMSs) ofIEC 61850) accounts for 100 bytes. The measured payloadlength of 500 bytes for the HV/LV nodes corresponds toactive/reactive power, voltage, current, and location.

We assume a WB PON of 16 ONUs, with bidirectional1 Gbps link, and 10 SSs are aggregated per ONU. For thepayload length, we use Lr = 1500 · 8, Lm = 500 · 8, andLc = 100 · 8 for the regular, SG monitoring, and SG controltraffic, respectively, whereby the SG traffic payload lengthscorrespond to the ones listed in Table II. The traffic rate forthe regular traffic follows a uniform distribution in the WBPON, whereby ∀i, j ∈ O + {0},Mi,j,r = 250 (Total traffic:816 Mbps). As for the SG classes, we have one SS i perONU sending control frames to the OLT (Mi,0,c = 1

60·10 ) andSG monitoring frames are destined to the OLT, whereby thedata rate is varied, as depicted in Fig. 16. We set the classpriorities to αr = 0.2 and αm = αc = 0.4 for the regular andSG classes, respectively.

We assume that the availability requirements of the regular

1e-05

0.0001

0.001

0.01

0.1

1

0 1 2 3 4 5 6 7 8

Unavaila

bili

ty

Sensors monitoring datarate (Mbps)

Regular traffic limit

SG traffic limit

Regular traffic SG monitoring traffic

SG control traffic

Fig. 16: Unavailability as a function of sensor data rate (pop =pwi = 10−5). Regular traffic has an availability requirementof 99.99 % and 99.9999 % for SG traffic.

and SG traffic are different and are hence set to 99.99% and99.9999%, respectively. To meet these requirements under theconditions used in our scenario, the data rate should be set tomaximally 1.5 Mbps per sensor. Note that the frames of lowerpriorities are dropped first. Therefore, at a sensor monitoringdata rate of 1.5 Mbps, frames belonging to the regular class aredropped in order to be able to accommodate SG monitoringframes.

V. CONCLUSIONS

Backhaul networks based on PONs are expected to bewidely deployed for emerging smart grid applications andto offer FTTH services. There is a continuous trend of in-creasing communications coverage, by integrating FiWi-basedcommunications architectures. In this paper, we developed anovel multi-class probabilistic availability model to quantifythe availability metric defined in the IEEE P2030 standard.The model takes into account the MAC protocol limits ofintegrated PON and WiMAX networks as well as fiber andbase station failures. The obtained examples show severalinteresting availability behaviors. First, fiber backhaul failureshave a significant impact on availability especially at highloads, while WiMAX BS failures have a lower impact at lowand high loads. The model also enables to quantify availabilitydepending on the number of BSs available, whereby selectingthe right number of BSs helps maximize availability in acost-effective manner. We have also shown that there exists aunique ratio of traffic destined towards a faulty link permittedto be re-routed for maximizing availability. Each smart gridapplication has a certain given availability requirement. Weinvestigated a smart grid scenario with payload length and datarate configurations based on traffic measurements capturedfrom smart grid applications. By considering regular and smartgrid traffic classes with different priorities and availabilityrequirements, we have shown that under the investigatedconditions the sensors’ maximum monitoring data rate to meetthe availability requirements can be quantified. The developed

11


probabilistic availability model enables to verify wether or notrequirements are met based on predicted traffic and equipmentfailure probabilities.

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[7] N. Ghazisaidi, M. Maier, and C. M. Assi, “Fiber-Wireless (FiWi) AccessNetworks: A Survey,” IEEE Communications Magazine, vol. 47, no. 2,pp. 160–167, Feb. 2009.

[8] P. Cheng, L. Wang, B. Zhen, and S. Wang, “Feasibility Study ofApplying LTE to Smart Grid,” in Proc., IEEE First InternationalWorkshop on Smart Grid Modeling and Simulation (SGMS), Brussels,Belgium, Oct. 2011, pp. 108–113.

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PLACEPHOTOHERE

Martin Levesque received his M.Sc. degree inComputer Science from UQAM, Quebec, Canada in2010. During 2010-2011 he worked as a SoftwareEngineer where he developed and maintained severalhigh traffic websites and file sharing systems. Duringsummer 2013, he was a visiting researcher at EDFR&D in Clamart, France, and researched on ad-vanced smart grid multi-simulations. Since 2011, heis pursuing the PhD degree in Telecommunicationsat INRS, Optical Zeitgeist Laboratory, Canada. Mar-tin Levesque was a recipient of master and doctoral

scholarships while pursuing his graduate studies. He served as a reviewerof numerous major journals and is a member of the IEEE IES TechnicalCommittee on Smart Grids.

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PLACEPHOTOHERE

Martin Maier is a Full Professor with the Insti-tut National de la Recherche Scientifique (INRS),Montreal. He has joined INRS as an AssociateProfessor in May 2005. He received the MSc andPhD degrees both with distinctions (summa cumlaude) in electrical engineering from the TechnicalUniversity Berlin, Berlin, Germany, in 1998 and2003, respectively. He was a Visiting Researcher atthe University of Southern California (USC), LosAngeles, CA, in spring 1998 and Arizona StateUniversity (ASU), Tempe, AZ, in winter 2001. In

summer 2003, he was a Postdoc Fellow at the Massachusetts Institute ofTechnology (MIT), Cambridge, MA. Before joining INRS, Dr. Maier wasa Research Associate at CTTC, Barcelona, Spain, November 2003 throughMarch 2005. He was a Visiting Professor at Stanford University, Stanford,CA, October 2006 through March 2007. Dr. Maier was a recipient of thetwo-year Deutsche Telekom doctoral scholarship from June 1999 throughMay 2001. He is also a co-recipient of the 2009 IEEE CommunicationsSociety Best Tutorial Paper Award and the Best Paper Award presented atThe International Society of Optical Engineers (SPIE) Photonics East 2000-Terabit Optical Networking Conference. He served on the Technical ProgramCommittees of IEEE INFOCOM, IEEE GLOBECOM, and IEEE ICC, andis an Editorial Board member of the IEEE Communications Surveys andTutorials as well as ELSEVIER Computer Communications. He is a SeniorMember of IEEE.

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