arxiv:2103.16981v2 [cs.ni] 2 apr 2021

MULTI-CORE FIBER AND POWER-LIMITED OPTICAL NETWORKTOPOLOGY OPTIMIZATION WITH MILP

A PREPRINT

Bjoern AnnighoeferInstitute of Aircraft Systems

University of StuttgartStuttgart, Germany

0000-0002-1268-0862

Adrian ZeyherInstitute of Aircraft Systems


0000-0003-2888-4509

Johannes ReinhartInstitute of Aircraft Systems


0000-0002-3512-5220

April 6, 2021

ABSTRACT

Optical networks with multi-core fibers can replace several electronics networks with a single topology.Each electronic link is replaced by a single fiber, which can save space, weight, and cost, whilehaving better segregation and EMI resistance. This is, for instance, of high interest in safety-criticalcyber-physical systems, such as aircraft avionics networks. Finding the optimal topology requiresfinding the optimal number of components, component locations, inter-meshing, and signal routing,while assuring the appropriate optical power level at each participating device. A Mixed-integerLinear Programming (MILP) representation is presented for the optimization of the topology ofoptical multi-core fiber networks. The optimization approach retrieves a globally optimal topologywith respect to weight or cost, i.e. it builds the optimal network topology from a set of switch andcable types, which differ in the number of fibers, attenuation, connectors, and other properties. Thenovelty of our approach is the consideration of translucent and opaque optical switches as well as therepresentation of cable and device attenuation directly in the MILP constraints. Moreover, arbitraryinstallation and routing resource restrictions are considered. The application of the approach to fivededicated scenarios yields in each case an optimal solution and validates our method. The applicationto an excerpt of an aircraft cabin network shows the retrieval of the global optimum in less 30 min fora topology with 48 signals and 23 components.

Keywords attenuation · damping · connectors · power level · translucent · opaque · safety-critical ·cyber-physical ·MILP · avionics · multi-core fiber · resources · resource limited · model-based

1 Introduction

Optical networks differ from electrical networks by usinglight as an information carrier. Instead of voltage changesin metallic conductors, modulated light is passed from onenode to another through fine optical fibers, also knownas fiber optics or fiber. This results in much higher band-widths compared to electrical networks, while the weightand volume of the components is smaller. Furthermore,optical networks do not generate electromagnetic interfer-ence during transmission and do not themselves react tosuch interference. In addition, less electrical power is re-quired to transmit signals, which has an additional positiveimpact on weight and costs. (cf. [1])

No electromagnetic interference as well as weight, power,and space saving are highly desirable properties in thefield of aviation. Modern large transport airplanes arebased on Distributed Integrated Modular Avionics (DIMA).DIMA is a concept for a safety-critical sharing of com-puting and network resources. Throughout the aircraftfuselage, generic computer modules are installed. Locally,I/O modules read sensors and control actuators. Centrallyplaced processing modules calculate control commands.All DIMA modules are connected with a common high-bandwidth Aircraft Data and Communication Network(ADCN). The modules and network are configurable. Theplanning and development of a DIMA avionics systemarchitecture is time and resource intensive, due to a highnumber of software components, signals, modules, pe-

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A PREPRINT - APRIL 6, 2021

ripherals as well as a complex interaction. It is assumedthat potential is wasted and a (partial) automation of de-sign steps results in better reliability, weight, cost, etc.[2, p. 2]. Special attention has to be paid to the networkconnecting the modules. Many implemented functionsare safety-relevant and have mandatory reliability and la-tency requirements. A prominent example is the flightcontrol system. Since the number of functions increasesexponentially over time [3], the ADCNs used today willreach their limits in terms of data rate and latency in aforeseeable time. Optical networks are being consideredfor future deployments as an alternative to the commonlyused ARINC 629 and ARINC 664 (AFDX) networks [4].

This work presents a Mixed-Integer Linear Programming(MILP) representation, which generates an optimal net-work topology for a given installation space from a givenset of signals and optical components. The algorithm con-siders necessary signal flows, cable properties such asattenuation and the number of fibers, and generic resourceconstraints of hardware and locations.

The remainder of this work is organized as follows. Chap-ter 2 defines our topology optimization problem and in-volved components. In Chapter 3 we review the state-of-the-art and compare it to our approach. Chapter 4 derivesthe MILP representation of the topology optimization prob-lem. In chapter 5 the algorithm is validate using five ded-icated scenarios. In chapter 6 the algorithm is applied tooptimize the network of an In-flight Entertainment Sys-tem (IFE). Chapter 7 discusses the results and gives anoutlook.

2 Topology Optimization Problem andLimitations

The general scope of topology optimization in this workis depicted in fig. 1. We assume a given installation spacefor the optical network consisting of possible switch andend system installation locations as well as cable routes.Within this installation space a number of end systems existthat have communications needs in terms of signals, whichhave to be transmitted and received. Given the installationlocations and possible cable routes, an optimal optical net-work is desired that connects all end systems and fulfillstheir communication needs. The communication needs arefulfilled if a path from sender to receiver for each signalis determined. The optical network has to be built frompredefined switch and cable types. Switches can either beopaque or translucent. Opaque switches convert opticalinformation to electrical and back and, therefore, decou-ple the optical power of connected cables. Translucentswitches forward the original optical signal, i.e. input andoutput power levels are dependent. Moreover, switchescan differ in the number of ports, costs as well as genericresources required for their installation (e.g. space) andinternal resources required for signal routing (e.g. band-width or number of signals). Cables differ in cost, weight,and resources like switches. Moreover, multi-core fiber

cables are assumed. Different cable types can offer a dif-ferent number of cores. It is assumed that each signalrequires a single core. This allows for a physical segre-gation of signals. A valid topology must not exceed theresources of any installation space or cable route. A validrouting must not exceed the resources of any switch orcable and the optical power level of signals has to be in themin-max-range acceptable for the individual components.Attenuation shall be considered in cables, connectors, andtranslucent devices. Multiple optimization objectives exist,e.g. the cost or weight of the network components, theinstallation effort, and the number of components in total.A pre-definition of single aspects shall be possible, e.g.fixed switches or cables.

Signal ASignal BSignal C

Signal ASignal BSignal C

Cables ?

Switches ?

- Attenuation?- Direction?

- Number of ports?- Space restriction?

Opaqueortranslucent?

Type 1

Type 2...

Type 1

Type 2...

2or4 coresor ...?

Maximumtopology

End systemwith communicationneeds

Signal routing ?

FIGURE 1: SCOPE OF TOPOLOGY OPTIMIZATION

2.1 Multi-core Fiber Transmission

By using multiple cores in a single optical cable, the trans-mission speed can be increased with the help of paralleltransmission. We refers to cores instead of fibers to preventthe ambiguity of fiber as a synonym for optical cables. Theuse of a single cable with multiple cores is common, whichare called Multi-Core Fibers (MCF).

Fig. 2 shows an example of a signal path in a switch.Each of the connected cables has three cores. Each signalis assigned to its own core. The routing of which signalpasses through which core can be done dynamically orstatically. In the latter case, the paths are calculated inadvance as suggested in this work. The physical routingmethod depends on the switch type, i.e. how the signal istransferred from one cable to the other.

Due to physical coupling effects between the individualcores, the range of multi-core fiber technology is degraded,but is still in the range of kilometers [5, p. 355]. Mechan-ical limitations such as the minimal bend radius and themaximal force loads are stricter than for single-core fibers.

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Multi-core fiber Multi-core fiberSwitch

Cable/fiber Signal pathCores

FIGURE 2: SIGNAL PATH WITHIN AN OPTICAL NETWORKTHAT USES MULTI-CORE FIBER TRANSMISSION.

2.2 Network Devices

This section explains the network devices in an opticalnetwork that are relevant in this work. The componentsinclude devices that can generate, forward (route), andreceive signals.

2.2.1 Transceiver

Transceivers combine transmitter and receiver in one de-vice. They provide a bidirectional interface between thenetwork and a computer module. Their task is to generatethe optical signals and convert them back into electronicinformation.

Besides the form factor and the connectors, especiallythe optical and electrical properties are relevant for theselection. The latter are irrelevant for this work and are,therefore, omitted.

2.2.2 Transmitter

The wave characteristics of the transmission are deter-mined by the transmitter. Important key points are thewavelength, spectral width, and the transmission power.Further properties of the transmitter are wave modes, re-flections, etc. [6, p. 4].

2.2.3 Receiver

For an optical signal to be received, the receiver has tobe tuned to the appropriate wavelength. In addition, thepolarization and power of the signal have to be compati-ble. Light-sensitive semiconductors, such as photo-diodes,convert optical signals into electronic signals, which canbe measured for data extraction. The received power hasto be within the allowed range of the detector. If the poweris too low, it is no not possible to distinguish between sig-nal and thermal noise, i.e. the signal-to-noise ratio is toolow [7]. If the received power is too large, the detector isoverloaded. Modulations of the luminous flux cannot bedetected. Overloading can cause permanent damage to thedetector and should, therefore, be avoided.

2.2.4 Connectors

Connectors attach optical cables to devices. Diverse con-nector types have been developed for a wide range of cabletypes and transmission technologies. The transmission inthe connector is lossy due to fiber interruption. Lenses,polishing of the ends, and special geometries are used toreduce the attenuation.

2.3 Switches

Switches form the heart of a network. Switches controlsignal routes between nodes. In classic copper networks,this routing is based on identifiers in the data packets. Incontrast, routing in optical networks can also be basedon the physical signal characteristics. This can be, forexample, the wavelength in Wave-Division Multiplexing(WDM) or the core used in multi-core fiber transmission.

2.3.1 Opaque Switches

Opaque switches convert the incoming optical signals toelectrical signals for routing. After signal conversion, thesignals are routed through electrical conductors to a trans-mitter located at the output. The transmitter convertsthe signal back into an optical signal. The advantagesof opaque switches are digital signal processing and aflexibility of the routing. Incoming attenuated signals areamplified. In addition, electrical signals can be controlledwithout mechanical components, which simplifies routing.Dynamic route changes are easy to establish. However, apower supply is mandatory and the transmission speed inthe optical network is limited by the electrical components.Opaque switches are easy to configure and cheaper thanso-called translucent switches [8, p. 5].

2.3.2 Translucent Switches

In translucent switches no conversion of the optical signaltakes place. An incoming signal is routed through fixedoptical fibers or miniature mirrors, depending on the tech-nology. Advantages over opaque switches result from thefact that no conversion is required. Translucent switchesare independent of the data rates [9, p. 1]. The signal trans-mission is delay-free over the entire wavelength range ofthe fibers used [10].

A disadvantage is the lack of signal conditioning. Signalsare not amplified, which requires care in signal attenuationduring path planning in network design.

Routing in translucent switches based on Micro-Electro-Mechanical Systems (MEMS) is accomplished via small,electromechanically adjustable micromirrors [8, p. 13f.].Dynamic routing can be established, but electrical poweris required for the actuation.

An alternative are translucent switches with fixed opticalfibers. In this case, the routing is directly specified by thephysical connection from the input to the output, whichlimits this technology to static routing. Since this type does

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not contain any electrical or moving mechanical compo-nents, no power supply is required.

2.4 Attenuation

For a successful optical signal transmission, the attenua-tion has to be considered comprehensively. The possibletransmission power of transceivers is limited as well as theinput power range of receivers.

2.4.1 Reasons for Attenuation

An optical signal is attenuated by various disturbances andlosses along the signal path. Physical effects such as scat-tering and absorption cause a majority of the attenuationwithin the fiber [11].

All disturbances and impurities in a fiber lead to additionallosses [7]. These losses are proportional to the signalpower. Therefore, when considering attenuation losses,they are calculated in decibels. Therefore, attenuationlosses are quantified in decibels and absolute quantitiesare specified in dBm (decibel milliwatts). All calculationsin this work are carried out in decibels, which leads toan additive expression compatible to MILP. Additionalattenuation is caused by connectors and splices. If the fiberends are not exactly aligned, portions of the signal willget lost. This loss is called insertion loss [7]. In addition,reflections occur at the fiber ends, which also attenuatesthe signal.

2.4.2 Design Calculation

For the design purposes, all the attenuation that occur isaccumulated and compared with the transmit and receivecharacteristics of the transceivers. By means of fig. 3 thecalculation is explained. In comparison to the presentationof the calculation in [7], the connector losses at the trans-mitter and receiver are explicitly presented, since these arelater assigned to the cables. Starting from the transmitter,the signal is sent out with the transmit power (1). Withinthe connector, first losses occur in the form of an inser-tion loss and losses due to reflection. Along the cable, thetransmit power reduces linearly with cable length due todispersion and absorption effects. At the splice (2), fiberinterference causes signal attenuation. The transparentswitch used in this example has an intrinsic attenuation,which is evident from the signal attenuation between input(3) and output (4). The switch is connected via connec-tors, which cause the abrupt signal degradation at points(3) and (4). The connector (5) connects the cable to thereceiver. Since the signal is received after the connector,the connector attenuation also has to be considered. Thesignal power at the receiver (5) has to be within the dy-namic range, which is determined by the sensitivity and theoverload limit. The signal in this example is receivable. If(5) is below the sensitivity, thermal noise hides the signal.If it is above the overload limit, then the detector cannotprocess the signal.

Distance to sender

Sign

alst

reng

thin

dBm

Sender Switch Receiver

Transmit power

OverloadSensitivity

(1)

(2)(3)

(4)

(5)

ConnectorSplice

FIGURE 3: EXEMPLARY SIGNAL STRENGTH CURVE (CF. [7]).THE SWITCH IS TRANSLUCENT IN THIS EXAMPLE.

3 Related Work

Optimization is a common issue for (optical) networks.Typical subjects of optimization are routing [12–19],scheduling [14, 15, 20], robustification [19, 21], and fre-quency sharing [17, 22]. The mentioned works are ex-emplary and the list is not complete. Some of the worksaddress multiple issues at once. MILP or ILP are com-mon methods to solve the problems. Whereas the existingsolutions for routing match our required routing from de-fined senders to defined receives with respect to resourceconstraints, the underlying network topology needs to bepredefined. That is not the case for our problem. Topol-ogy optimization is a less frequent research topic, but ithas been addressed for electrical [23–26] and optical net-works [27–33]. The approaches for electrical networksomit optical properties such as attenuation and multi-corecables. The approaches for optical networks are verydomain-specific. They deal with special applications (e.g.telecommunication, dynamic routing) or certain opticalnetwork technologies (e.g. WDM), that do not match oursetup, i.e. static routing in a restricted space while choos-ing optimal cables, switches, and topology. In no existingapproach the placement and choice of components is sub-ject to optimization in parallel. The approach most similarto our needs is [34]. It suggests a Binary Program rep-resentation for a topology optimization of Ethernet-likenetworks. The approach considers switch types, port, andresource restrictions as well as signal routing and signalsegregation constraints. We extend that approach to opticalnetworks by adding multi-core cable types and attenuation.

4 A Mixed-integer Linear ProgrammingRepresentation for TopologyOptimization

The topology optimization problem is expressed as a MILP,whose objective function and constraints are linear andcontain both binary, integer, and continuous variables:

maximize

4


∑j∈B

cjxj +∑j∈I

cjxj +∑j∈C

cjxj (1)

subject to∑j∈B

aijxj +∑j∈I

aijxj +∑j∈C

aijxj

≤=≥ bi,∀i ∈M

(2)lj ≤ xj ≤ uj ∀j ∈ N = B ∪ I ∪ C (3)xj ∈ {0, 1} ,∀j ∈ B (4)xj ∈ Z,∀j ∈ I (5)xj ∈ R,∀j ∈ C (6)

The solution x is a set of variables xj , j ∈ N that satisfiesthe conditions (2)-(6). The aim is to maximize an objectivefunction (1). This is a weighted sum with cost factorscj , j ∈ N [35, p. 2]. The equalities and inequalities (2)are constraints [36, p. 84]. Each comprises coefficients aij .The indices associated with the constraints are included inthe setM . The setsB, I , andC are the index sets of binary,integer, and continuous variables [35, p. 2]. Formally,the variable data type is specified by the equations (4),(5), and (6). (3) allows to set additional upper (uj) andlower (lj) bounds for xj , j ∈ N . The cost function (1) ismaximized. Problems that require the objective function tobe minimized are addressed by inverting the const function:

min{f(x)

}≡ max

{−f(x)

}(7)

Essential for topology optimization is the encoding of theexpected topology in the solution vector

x = (xDk

Tl , . . . ∈ {0, 1} (8)

, xFj

Tt , . . . ∈ {0, 1} (9)

, xFj

useAB, xFj

useBA ∈ Z (10)

, xFj

allowAB, xFj

allowBA, . . . ∈ {0, 1} (11)

, xSi

Fj ,AB, xSi

Fj ,BA, . . . ∈ {0, 1} (12)

, xSi

Dk,doesRx, xSi

Dk,doesTx, xSi

Dk,Rx, xSi

Dk,transm. ∈ {0, 1} (13)

, xSi

Dk,TxAvail, xSi

Dk,Tx, xSi

Fj ,powerAB, xSi

Fj ,powerAB ∈ R (14)

, xSi

Dk,opaqueRx ∈ {0, 1}). (15)

x is based on the fact that the maximum topology is cal-culated a priori from the given installation space, i.e. themaximum number of switch devices and links is known.Solving the MILP extracts the optimum from the maxi-mum topology.

Variables (8) and (9) encode the existence of all possibledevices and cables of their chosen type. The set D denotesall possible devices (switches) in the network. The elementDk ∈ D is the ith device. The total number of devices pos-sible is denoted by |D|. Further, F represents the set of allpotential cables. This includes those cables automatically

generated to form a fully connected network. A singlecable of this set is referred to as Fj ∈ F . |F| is the totalnumber of possible cables. The notation Fj = (DA, DB)expresses which two devicesDA andDB are connected bycable Fj , with DA 6= DB . Two cables Fm and Fn linkingthe same two devices DA and DB are legal. Furthermore,the direction is considered, i.e. (DA, DB) 6= (DB , DA).

Signals are represented by the set S, the signal count is|S|. Source and target DS and DT of a signal Si ∈ S areindicated by the notation Si = (DS , DT ). Multiple signalsare allowed between the same source and target device, butsignal aggregation is recommended as each signal requiresits own core assignment. Bandwidth limitations have to beconsidered in signal aggregation.

4.1 Device, Cable types, and Properties

Devices and cables are assigned with property vectors. Avector contains all the quantities of the type. Class E is arepresentative of device set D or cable set F .

4.1.1 Type declaration

A type t belonging to the element class E is represented bythe constant property vector

τEt =(τE1t

· · · τEit · · · τE|τE |t

)T, τEt ∈ R|τ

E |.(16)∣∣τE ∣∣ denotes the number of properties that are identical for

all representatives of the element class. In general, eachproperty is a real value. Sometimes a restriction to discretevalues is useful. Across all types, τ̂Ei gives the maximumvalue of a property τEi . τ̌Ei is the minimum value.

All representatives of the element class can be combinedinto a constant matrix TE . The number of types of theelement class E is given by

∣∣TE ∣∣.TE =

[τE1 · · · τE|TE |

],TE ∈ R|τ

E |×|TE | (17)

4.1.2 Type Assignment

For the assignment of a type to an element e ∈ E , this hasa type vector xeT which encodes the actual type. A one inthe vector entry xeTt means the element e is of type t. Azero corresponds to saying that the element is not of theassociated type t. Other values are not allowed in

xeT =

(xeT1· · ·xeTt · · ·x

eT|TE |

)T

, xeT ∈ {0, 1}|TE | .

(18)

For unambiguous type assignment, the choice of the typevector xeT is restricted to have at most a single entry equalto one. A limiting SOS constraint [36, 147 ff] is used. Asuitable choice of the (in)equalities determines whether

5


the element e has to exist by definition or can be removedby optimization.

If an element has to exist this means that a type is assignedto it. The SOS equation 19 specifies the type assignmentobligation.

|TE |∑t=1

xeTt = 1 (19)

The elements that exist according to definition are com-pared with the optional elements. An element to which notype is assigned has all assignment entries set to zero. Forthis the following SOS equation ensures a valid assignment:

|TE |∑t=1

xeTt ≤ 1 (20)

In addition to defining whether an element must exist inthe final topology, the ability to restrict the allowed types isrequired. This is done by specifying forbidden mappings t:

xeTt = 0 (21)

In the following it is necessary to know whether an elementexists in the network or not. This is given by sum in thetwo conditions (19) and (20). This notation is used as anabbreviation in the following.

‖xeT ‖1 :=

|TE |∑t=1

xeTt (22)

4.1.3 Element properties

From the type matrix TE and the type vector xeT of theelement e, the current properties of the element are deter-mined with (23). These are stored in an element propertyvector pe.

pe = TE ·xeT =(pe1 · · · pei · · · pe|τE |

)T, pe ∈ R|τ

E | (23)

For a property i the equation is:

pei = τEi1 · xeT1

+ · · ·+ τEi|TE |· xeT|TE |

(24)

If the element does not exist, i.e. xeT = 0, then all elementproperties pei = 0.

4.1.4 Device Properties

The device properties listed in table 1 are essential toconsider the requirements on attenuation and routing con-straints.

The port count pDkports limits the number of available cable

ports. It is the main resource constraint. Whereas, thefive properties pDk

∆ , pDkRx , pDk

Rx, pDk

Tx and pDk

Txare used to

TABLE 1: OVERVIEW OF THE DEVICE PROPERTIES OF A DE-VICE Dk ∈ D.

Property Symbol Type Unit

Port count pDkports Z+ —

Internal signal attenuation pDk∆ R dB

Minimum received power pDkRx R dBm

Maximum received power pDk

RxR dBm

Minimum transmit power pDkTx R dBm

Maximum transmission power pDk

TxR dBm

Translucent transmission pDktrans {0, 1} —

Cost value pDkcost R+ —

Type vector xDkT {0, 1}|T

D| —

account for attenuation. For these properties, the follow-ing conditions are imposed depending on the translucentproperty pDk

trans:

pDktrans = 0 ⇒ pDk

∆ = 0 (25)

pDktrans = 1 ⇒ pDk

Rx = pDk

Rx= pDk

Tx = pDk

Tx= 0 (26)

Devices Dk for which pDktrans = 1 are translucent devices.

If pDktrans = 0, the device is opaque.

Further, it is required that the maximum and minimumvalues of the transmit and receive power are chosen assuch:

pDkRx ≤ p

Dk

Rx(27)

pDkTx ≤ p

Dk

Tx(28)

The conditions (25) to (28) are applied to the propertyvalues of the device types. The conditions can trivially besatisfied by mindful device type definition.

The last property pDkcost is solely for the cost function.

4.1.5 Cable Properties

Variables related to cable type and thus cable propertiesare listed in table 2.

The core count pFjcores and the signal attenuation pFj

∆ are themost important properties of an optical cable. The costvalue pFj

cost is in use exclusively in the cost function. Thecable type vector xFj

T is included for completeness.

The transmission direction of a cable is controlled by theproperties pFj

uni, pFj

AB and pFj

BA.

The index AB indicates the direction of transmission fromdevices DA to DB . The index BA denotes the oppo-site direction. This convention is used below for otherdirection-related variables.

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TABLE 2: OVERVIEW OF CABLE PROPERTIES OF A CABLEFj ∈ F .

Property Symbol Type Unit

Number of cores pFjcores Z+ —

Attenuation pFj

∆ R dB

Cost value pFjcost R+ —

transmission constraints

Unidirectional pFj

uni {0, 1} —

Direction A→ B allowed pFj

AB {0, 1} —

Direction B → A allowed pFj

BA {0, 1} —

Type vector xFj

T {0, 1}|TF | —

TABLE 3: ALLOWED AND DISALLOWED VALUES FOR THEDIRECTIONALITY PROPERTIES OF A CABLE Fj .

pFj

uni pFj

AB pFj

BA Explanation

0 1 1 bidirectional1 1 1 Unidirectional, no direction specified1 1 0 Unidirectional, direction A→ B1 0 1 Unidirectional, direction B → A

• 0 0 Not allowed, no direction allowed0 0 1 Not allowed, inconsistent0 1 0 Not allowed, inconsistent

The property pFj

uni describes whether the cable Fj can trans-mit signals bidirectionally (value zero) or unidirectionally(value one). The two properties pFj

AB and pFj

BA define theallowed directions. If the value is one, a transmission inthe associated direction is allowed. Consistency betweenunidirectionality and allowed directions has to be ensured.Table 3 lists all valid values.

Although a cable may only transmit unidirectionally, thedirection is not restricted (pFj

uni = pFj

AB = pFj

BA = 1). Thiscan be used to model unidirectional cables where the trans-mission direction is not specified in advance.

4.2 Port Count Limits

The number of ports of a deviceDk ∈ D limits the numberof cables. No more cables can be connected than there areports.

Cables Fj whose definition contains the device Dk asstart or end points are relevant for the constraint. This istrue for the subset

{F | Fj = (Dk, ∗) ∨ Fj = (∗, Dk)

}⊆

F . Since an existing cable have to be connected to thedevice, the sum of all existing cables is the number ofconnected cables. A sum norm of the type vector equal toone describes the existence of a cable Fj . The inequality29 relates the number of connected cables to the number

of ports of the device Dk.∑Fj∈{F|Fj=(Dk,∗)∨Fj=(∗,Dk)}

∥∥∥xFj

T

∥∥∥1≤ pDk

ports (29)

This constraint also prohibits the connection of cables ofdevices that do not exist. In this case pDk

ports = 0, whichprohibits all potential cables. Conversely, this requires thatdevices connected by a fixed cable always have to haveports and thus have to be existing device instances.

4.3 Directionality and Core Count Limits

Another resource constraint is the directionality of thecables as well as the maximum number of transmissionspossible. The variables listed in table 4 are necessary forthese constraints. The total number of transmissions is

TABLE 4: LIST OF VARIABLES OF EACH CABLE Fj ∈ F .THESE ARE REQUIRED TO SET UP THE CONSTRAINTS

Designation Symbol Type

Transfer count A→ B xFj

useAB Z+

Transfer count B → A xFj

useBA Z+

Transfer allowance A→ B xFj

allowAB {0, 1}Transfer permit B → A x

Fj

allowBA {0, 1}

limited by the core count pFjcores of a cable Fj ∈ F . Each

core may only transmit a single signal. The number oftransmissions is counted for each cable by the variablesxFj

useAB and xFj

useBA. The sum of the count variables gives thetotal count, which is limited by the core count as follows:

xFj

useAB + xFj

useBA ≤ pFjcores . (30)

Directionality is modeled by limiting the count variablesdepending on the directionality properties. The two auxil-iary binary variables xFj

allowAB and xFj

allowBA are introducedto limit the direction of unidirectional cables. The twoinequalities 31 and 32 limit the direction depending on thetype definition for unidirectional cables. If a direction isforbidden for the selected type, the associated property iszero, which also forbids the corresponding auxiliary vari-able to be used. For bidirectional types it is required thatboth directions are allowed.

xFj

allowAB ≤ pFj

AB (31)

xFj

allowBA ≤ pFj

BA (32)

Equation 33 controls direction depending on whether thecable is allowed to be used bidirectionally or unidirection-ally.

xFj

allowAB + xFj

allowBA = 2 ·∥∥∥xFj

T

∥∥∥1− pFj

uni (33)

To explain the equation, three cases are considered:

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1. No type is assigned to the cable, it is∥∥∥xFj

T

∥∥∥1=

pFj

uni = 0. No transfers are allowed at all. This isconsistent with the assumption of a non-existentcable.

2. The type associated with the cable allows bidirec-tional transmission. Consequently,

∥∥∥xFj

T

∥∥∥1= 1

and pFj

uni = 0 holds, i.e.

xFj

allowAB + xFj

allowBA = 2,

so that both directions are allowed.3. In this case, the cable is assigned a type with

unidirectional transmission∥∥∥xFj

T

∥∥∥1= p

Fj

uni = 1.Substituting (33) gives:

xFj

allowAB + xFj

allowBA = 1.

Unless there is a restriction on the direction via(31) or (32), the MILP solver is free to determinethe direction of the unidirectional cable.

The final limit for the count variables is set using the in-equalities 34 and 35. If the cable is forbidden, the countvariable is forced to zero, i.e., no transfers. If transfers areallowed in that direction, the bound is raised to the largestallowed transfer count. This corresponds to the maximumcore count across all cable types, described by the constantτ̂Fcores.

xFj

useAB ≤ xFj

allowAB · τ̂Fcores (34)

xFj

useBA ≤ xFj

allowBA · τ̂Fcores (35)

4.4 Routing

A valid routing of a signal Si ∈ S, Si = (DS , DT ) con-sists of a continuous, i.e., uninterrupted, path from thesource device DS to the target device DT .

The variables listed in table 5 encode the path of signalSi. The path encodes the devices Dk ∈ D that transmitor receive the signal as well as the allocations to directedcables Fj ∈ F . If the value is one, the signal is transmittedover the device or cable. If it is zero, the signal does notpass over it.

TABLE 5: EACH SIGNAL Si ∈ S HAS THE LISTED VARI-ABLES, WHICH CONTAIN INFORMATION FOR THE ROUTINGCONSTRAINTS. EACH LISTED VARIABLE IS REQUIRED MULTI-PLE TIMES; ONCE FOR EACH ELEMENT LISTED IN THE LASTCOLUMN.

Designation Symbol Type Exists for

Fj in directionA→ B xSiFj,AB {0, 1} ∀Si ∈ S, ∀Fj ∈ F

Fj in directionB → A xSiFj,BA {0, 1} ∀Si ∈ S, ∀Fj ∈ F

Dk receives signal xSiDk,doesRx {0, 1} ∀Si ∈ S, ∀Dk ∈ D

Dk transmits signal xSiDk,doesTx {0, 1} ∀Si ∈ S, ∀Dk ∈ D

4.4.1 Start and End of a Signal Path

Translucent devices cannot be the source or destination ofa signal. Therefore, the source and target devices DS andDT of a signal Si = (DS , DT ) have to be opaque. This isrepresented by two constraints:

pDStrans = 0 (36)

pDTtrans = 0 (37)

For the source device DS it is required that the signal isemitted by it. In addition, it is specified that the sourcedevice does not receive the signal. On the other hand, ithas to be specified that the target device DT receives thesignal and does not transmit it:

xSi

DA,doesTx = 1 (38)

xSi

DA,doesRx = 0 (39)

xSi

DB ,doesTx = 0 (40)

xSi

DB ,doesRx = 1 (41)

4.4.2 Intermediate Devices

Regarding other devices, a device Dk ∈ D \ {DS , DT }can transmit the signal only if it was received before:

xSi

Dk,doesTx = xSi

Dk,doesRx (42)

Either the device transmits and receives or does nothing.The coupling of devices is encoded with the variablesxSi

Fj ,AB and xSi

Fj ,BA from table 5. They encode whether thesignal Si is transmitted over the cable Fj and thus the twodevices are connected by that cable.

4.4.3 Device-cable-coupling

The coupling between the individual devices is expressedby the cable usage. Since cables have a direction, this mustbe considered when setting up the constraints. Accord-ing to table 5 each signal Si for each cable Fj has twodirectional variables that encode the transmission direc-tion of the signal. If a device Dk emits a signal Si, then

xSiDk,doesTx

DeviceDk DeviceDu

xSiDu,doesRx

xSiFj,AB

xSiFj,BA

Fj = (Dk, Du)

Cable

xSiDk,doesRx x

SiDu,doesTx

.

FIGURE 4: RELATION OF THE SIGNAL PATH ENCODING VARI-ABLES OF A SIGNAL Si WITH THE ELEMENTS OF A NETWORK

a cable must transmit it away from the device. This isexpressed with eq. 43. A cable Fj = (Dk, ∗) that starts

8


at an emitting device Dk carries the signal in the directionA → B. A cable Fj = (∗, Dk) carries signals to deviceDk (direction B → A). This relationship of the variablesto the individual elements is outlined by fig. 4.

A signal Si is always routed over a cable towards thereceiving device Dk. The directional relationships forreceiving is revers compared to sending. Eq. 44 sums upthe number of incoming path segments of the signal Si forthe device Dk.

xSi

Dk,doesTx =∑

Fj∈{F|Fj=(Dk,∗)}xSi

Fj ,AB +∑

Fj∈{F|Fj=(∗,Dk)}xSi

Fj ,BA (43)

xSi

Dk,doesRx =∑


Fj ,BA +∑


Fj ,AB (44)

Both equations restrict, in addition, that at most one out-going and one incoming transmission of the signal exist.Multicasting is prohibited. The equations also prevent asignal from passing the same device twice.

4.4.4 Signal-core-coupling

The type-dependent core-count limits the routing. The twocount variables xFj

useAB and xFj

useBA for each cable Fj ∈ Fcount the number of transmitted signals in each direction.Signal routing is directly coupled with the cable selectionand indirectly with the device selection.

xFj

useAB =∑Si∈S

xSi

Fj ,AB (45)

xFj

useBA =∑Si∈S

xSi

Fj ,BA (46)

The total number of transmitted signals for each cable Fj ∈F is calculated by adding up the individual transmissionpath variables xSi

Fj ,AB and xSi

Fj ,BA of each signal Si whiletaking into account the transmission direction.

4.5 Attenuation

This work puts a special focus on including signal attenua-tion in the topology optimization problem. The additionalvariables listed in table 6 are added for attenuation compu-tation. These represent the power levels in all devices andcables for each signal Si ∈ S . An exception is the variablexSi

Dk,opaqueRx.

4.5.1 Transmit Power Limits

A fully connected network with |D| devices is assumed.The longest possible signal path in it passes all |D| devices.Only translucent intermediate devices are of interest forthe power considerations. Opaque devices can repowerthe signal. Between each pair of devices, the signal is

TABLE 6: ADDITIONAL VARIABLES REQUIRED FOR ATTENU-ATION

Designation Symbol Type Exists for

device-related variablesReceived input power x

SiDk,Rx R ∀Si ∈ S, ∀Dk ∈ D

Opaque output power xSiDk,transmit R ∀Si ∈ S, ∀Dk ∈ D

Translucent output power xSiDk,TxAvail R ∀Si ∈ S, ∀Dk ∈ D

Used output power xSiDk,Tx R ∀Si ∈ S, ∀Dk ∈ D

Is opaque signal receiver xSiDk,opaqueRx {0, 1} ∀Si ∈ S, ∀Dk ∈ D

cable-related variablesPower at cable endAB x

SiFj,powerAB R ∀Si ∈ S, ∀Fj ∈ F

Power at cable endBA xSiFj,powerBA R ∀Si ∈ S, ∀Fj ∈ F

transmitted via a cable. In total, this results in |D| − 1cables in the signal path assuming that each device is notpassed more than once. The signal power at the receivingdevice is calculated by adding the transmit power to all at-tenuations occurring along the signal path. Conversely, thetransmit power can be calculated starting from the receivepower. Inserting maximum and minimum attenuation val-ues yields the extreme conditions. A positive attenuationvalue is signal amplification. Maximum attenuation cor-relates with the smallest attenuation value. The minimumattenuation or gain correlates with the largest attenuationvalue.

With eq. 47 a general limit is derived. All power valuesoccurring in the network have to be within in the interval[−Plim, Plim].

Plim = Plim,Tx +(|D| − 2

)· Plim,∆D

+(|D| − 1

)· Plim,∆F + Plim,Rx ≥ 0 (47)

The four terms necessary for attenuation are calculatedfrom the maximum and minimum values of the cable anddevice types.

First, Plim,Tx denotes the largest absolute transmit power.For the optimization only the largest maximum as wellas the smallest minimum transmit power τ̂D

Txand τ̌DTx are

required. Since τ̌DTx ≤ τ̌DTx≤ τ̂D

Txand τ̌DTx ≤ τ̂DTx ≤ τ̂D

Tx,

τ̂DTx and τ̌DTx

are eliminated, i.e. the largest minimum andsmallest maximum transmit power are removed:

Plim,Tx :=max

(∣∣∣τ̌DTx

∣∣∣ , ∣∣∣τ̂DTx

∣∣∣) (48)

In the same way, Plim,Rx represents the largest receivedpower. Again, the calculation requires only the largest max-imum and smallest minimum power τ̂D

Rxand τ̌DRx. Similar to

the receive power τ̌DRx ≤ τ̌DRx≤ τ̂D

Rxand τ̌DRx ≤ τ̂DRx ≤ τ̂D

Rx,

the smallest maximum as well as largest minimum receivedpower τ̌D

Rxand τ̂DRx do not need to be considered.

Plim,Rx :=max

(∣∣∣τ̌DRx

∣∣∣ , ∣∣∣τ̂DRx

∣∣∣) (49)

9


The remaining two terms are determined similarly.Plim,∆D is determined by the largest absolute value ofthe internal device attenuation. The maximum absolutevalue of the cable-side transmission loss is used for thePlim,∆F .

Plim,∆D :=max(∣∣τ̌D∆ ∣∣ , ∣∣τ̂D∆ ∣∣) (50)

Plim,∆F :=max(∣∣τ̌F∆ ∣∣ , ∣∣τ̂F∆ ∣∣) (51)

4.5.2 Cable attenuation

The power calculation is carried out according to anadapted calculation rule derived from [11] and [7]. Alongthe signal path, starting with the transmitted power, allattenuation influences are accumulated. The largest attenu-ations are scattering and absorption effects in cables anddevices as well as mechanical inaccuracies in the align-ment (s. sec. 2.4). The latter is referred to as insertionloss at connectors. The insertion loss is not assigned to thedevices, but is considered as a part of the cable design.

All attenuations along a cable Fj ∈ F are assumed tobe summable to a single attenuation value pFj

∆ . This ispossible since the length of each cable is known from theinstallation space dimensions. Furthermore, it is assumedthat the plug-in or splice connections are included in thisattenuation value.

The basic idea of the power calculation is to add up thesegment’s individual attenuations. However, since no sig-nal path information is available when setting up the MILPconstraints, all possible paths have to be considered. Thisleads to the model shown in fig. 5.

xSiDA,Tx

DeviceDA DeviceDB

xSiFj,powerAB

Cable Fj = (DA, DB)

= 0

= 1+p

Fj∆

0dBm

xSiFj,AB

0dBmxSiFj,BA

0 =

1 = +pFj∆

xSiDB,Tx

xSiDB,Rx

xSiDA,Rx x

SiFj,powerBA

FIGURE 5: POWER FLOW ALONG A CABLE Fj =(DA, DB

)FOR A SIGNAL Si . THE UPPER HALF OF THE FIGURE SHOWSHOW THE POWER INFORMATION PROPAGATES IN THE CABLEDIRECTION. THE POWER FLOW AGAINST THE DIRECTION ISSHOWN IN THE LOWER HALF OF THE FIGURE.

For each signal Si ∈ S and each cable Fj ∈ F a set ofconstraints is defined, which calculates the power changealong cable Fj depending on the signal path. The path aswell as the powers depend on the direction.

If the signal Si is transmitted as depicted in fig. 5 fromdevice DA to device DB , then the signal power at thecable’s end, xSi

Fj ,powerAB, is obtained by adding the cable

attenuation pFj

∆ to the transmit power xSi

DA,Tx of DA tobe determined. The attenuation value is negative if signalattenuation occurs. Therefore, the attenuation has to beadded.

If the signal is not transmitted over that link, thenxSi

Fj ,powerAB = 0dBm is applied to the power at the ca-ble’s end. This choice is reasonable for the attenuationconsideration in the devices. For the opposite directionB → A the conditions apply analogously. Formally, theseconditions can be described mathematically as follows.xSi

Fj ,AB is the transmission state.

xSi

Fj ,powerAB =

{xSi

DA,Tx + pFj

∆ if xSi

Fj ,AB = 1

0 if xSi

Fj ,AB = 0(52)

xSi

Fj ,powerBA =

{xSi

DB ,Tx + pFj

∆ if xSi

Fj ,BA = 1

0 if xSi

Fj ,BA = 0(53)

These distinctions are incompatible with the MILP con-straints. Eight inequalities are needed for a MILP com-patible notation using the Big-M method [37, p. 2]. Ineach case, the transmission state of the respective directionxSi

Fj ,AB or xSi

Fj ,BA serves as the decision variable. Sincepower is accounted, the power limit Plim serves as theBig-M constant.

−Plim ·(1− xSi

Fj ,AB

)≤ xSi

Fj ,powerAB − xSi

DA,Tx + pFj

∆

≤ Plim ·(1− xSi

Fj ,AB

)(54)

−Plim · xSi

Fj ,AB ≤ xSi

Fj ,powerAB ≤ Plim · xSi

Fj ,AB

(55)

−Plim ·(1− xSi

Fj ,BA

)≤ xSi

Fj ,powerBA − xSi

DB ,Tx + pFj

∆


Fj ,BA

)(56)

−Plim · xSi

Fj ,BA ≤ xSi

Fj ,powerBA ≤ Plim · xSi

Fj ,BA

(57)

4.6 Attenuation Consideration in Devices

Attenuation behaves different for devices. Power calcula-tions there are more complex as for cables. The challengeis to model opaque and translucent devices by a commonset of constraints. Separation is not possible due to thecommon type model.

4.6.1 Received Power

Fig. 6 shows how power flows through a device Dk ∈ D.Assuming that at most one cable Fj carries the signalSi ∈ S to the device, the incoming signal strength iscalculated by summation. For this purpose the power at thecable’s end is set to 0 dBm when no signal is transmitted.This incoming signal power xSi

Dk,Rx corresponds to thepower at the end of the cable transmitting to the device:

xSi

Dk,Rx =∑


Fj ,powerAB +∑


Fj ,powerBA

(58)

10


Receiver

xSiDk,Tx

Sender

0dBm

xSiDk,TxAvail

xSiDk,transmit

xSiFj,powerAB

Check powerpDkRx ≤ x

SiDk,Rx ≤ p

DkRx

xSiDk,Rx

pDktrans = 0

pDktrans = 1

xSiDk,opaqueRx = 1

xSiDk,doesTx = 1

xSiDk,doesTx = 0

+pDk∆

DeviceDk

xSiFj,powerBA

FIGURE 6: POWER FLOW WITHIN A DEVICE Dk FOR A SIG-NAL Si . THE MIXTURE OF OPAQUE AND TRANSLUCENT DE-VICES CREATES COMPLEXITY WITHIN THE ATTENUATIONCONSIDERATION.

The auxiliary variable xSi

Dk,opaqueRx encodes whether anopaque type is assigned to the associated device Dk andthe signal Si is received by the device. It is determinedfrom combination of the receive state xSi

Dk,doesRx and thecomplementary value of the device property pDk

trans.

According to the general encoding of logical AND condi-tions [38, 4 ff] this results in MILP :

xSi

Dk,opaqueRx ≤ xSi

Dk,doesRx (59)

xSi

Dk,opaqueRx ≤(1− pDk

trans

)(60)

xSi

Dk,opaqueRx ≥(1− pDk

trans

)+ xSi

Dk,doesRx − 1 (61)

If the device Dk is opaque, then the received power shouldneither overpower the sensor (pDk

Rx) nor be below its sensi-

tivity pDkRx . Formally, however, it is considered whether the

signal Si is received at all. Indeed, if there is no reception,then in the received power xSi

Dk,Rx = 0 is meaninglessand the sensitivity is not considered. All incoming signalpowers are zero, thus their sum is also zero. A check isperformed if the device Dk is opaque and the signal Si isreceived. This condition is just represented by the auxiliaryvariable xSi

Dk,opaqueRx. Formally, the implication below isobtained with the decision variable xSi

Dk,opaqueRx:

xSi

Dk,opaqueRx = 1 =⇒ pDkRx ≤ x

Si

Dk,Rx ≤ pDk

Rx(62)

As before, this is represented with the Big-M notation [37,p. 2]. Again, the power limit Plim is used as the Big-Mconstant.

xSi

Dk,Rx − pDk

Rx≤ Plim ·

(1− xSi

Dk,opaqueRx

)(63)

−xSi

Dk,Rx + pDkRx ≤ Plim ·

(1− xSi

Dk,opaqueRx

)(64)

It should be noted that attenuations occurring during recep-tion in the device are not modeled. If an optical path to the

sensor inside the opaque device degrades the signal, thisattenuation is added to the sensitivity range in advance dur-ing the type declaration. Further possibly desired marginsare to be added in advance to the declaration.

4.6.2 Signal Dispatch and Internal Attenuation

The power calculation of the signal dispatch requires sev-eral steps. First, the theoretically available transmit powerxSi

Dk,TxAvail is determined. In the case of opaque devices,this corresponds directly to the actual transmitter powerxSi

Dk,transmit. For translucent devices, the available transmitpower is calculated from the received power xSi

Dk,Rx andthe internal attenuation pDk

∆ of the device:

xSi

Dk,TxAvail =

{xSi

Dk,Rx + pDk

∆ if pDktrans = 1

xSi

Dk,transmit if pDktrans = 0

(65)

The decision variable pDktrans = 1 encodes whether an

opaque or translucent device is present. At this point,the device attenuation of translucent devices is included.For opaque devices there is no consideration of attenua-tion values, these (if present) are accounted for with thetransmitter power as part of the device type declaration.The inclusion of the case distinction (65) is based again onBig-M:

− Plim ·(1− pDk

trans

)≤ xSi

Dk,TxAvail − xSi

Dk,Rx − pDk

∆

≤ Plim ·(1− pDk

trans

)(66)

− Plim · pDktrans ≤ xSi

Dk,TxAvail − xSi

Dk,transmit ≤ Plim · pDktrans

(67)

The transmitter power xSi

Dk,transmit is freely selectable

within a power range[pDk

Tx , pDk

Tx

]. Consequently, trans-

mitters with adjustable power can be considered in theoptimization. A distinction between opaque and translu-cent devices is not made. For the latter, the range is set to{0}. Moreover, an additional constraint is required:

pDkTx ≤ x

Si

Dk,transmit ≤ pDk

Tx(68)

The available transmit power xSi

Dk,TxAvail is calculated inde-pendently of the fact whether the device Dk transmits thesignal Si at all. The latter is stored in the status variablexSi

Dk,doesTx. If Si is not transmitted, a transmit power of0 dBm is desired. A case differentiation is introduced forcalculating the actual transmitted signal power xSi

Dk,Tx:

xSi

Dk,Tx =

{xSi

Dk,TxAvail if xSi

Dk,doesTx = 1

0 if xSi

Dk,doesTx = 0(69)

When the signal Si is transmitted, the actual transmit powerxSi

Dk,Tx is equal to the available power xSi

Dk,TxAvail. If no

11


TABLE 7: THE ACTUAL TRANSMIT POWER CONSIDERED FORTHE POSSIBLE STATES OF THE TRANSLUCENT PROPERTY ANDTRANSMIT STATUS OF A SIGNAL Si IN DEVICE Dk

State pDktrans xSi

Dk,doesTx xSiDk,Tx

Opaque, not transmitting 0 0 0 dBmTranslucent, no signal 1 0 0 dBm

Opaque, transmitting 0 1 xSiDk,transmit

Translucent, with signal 1 1 xSiDk,Rx + p

Dk∆

transmission takes place, the transmit power is zero. Foran overview of what actual transmit powers xSi

Dk,Tx areassumed for the different cases see table 7. The case dis-crimination (69) is again represented by four constraintsusing the Big-M method. The Big-M constant must not beexceeded or undercut by the absolute power bound Plimused as a limit.

− Plim ·(1− xSi

Dk,doesTx

)≤ xSi

Dk,Tx − xSi

Dk,TxAvail


Dk,doesTx

)(70)

− Plim · xSi

Dk,doesTx ≤ xSi

Dk,Tx

≤ Plim · xSi

Dk,doesTx (71)

5 Validation

Validation is performed with small scenarios each dedi-cated to a certain aspect of the optimization approach.

The optimization is implemented in Python. The input andoutput data are modeled with the Open Avionics Architec-ture Model1 (OAAM) [39]. OAAM is a domain-specificmodel developed to represent the logical and physical ar-chitecture of cyber-physical aircraft systems and the un-derlying avionics architecture. It includes elements tomodel the possible installation space, signal needs, andthe component types with their properties used in the opti-mization. The required information is read from OAAMwith Python and transformed to the matrix representationof the MILP. The result of the optimization is convertedback into model elements and stored in OAAM. The MILPproblem is solved using Gurobi 8 on an Intel Core i3-2100with 8 GB RAM.

Two basic models are used for validation experiments:

Model A as depicted in fig. 7 defines three switches aswell as a fixed cable and type between devices 0 and 1.All other cables are subject to the optimization. For theoptimization, opaque and translucent switches (s. tab. 8)are available. Switch 2 is fixed as an opaque device. Twocable types (s. tab. 9) are defined: The unidirectional two-core cable type as well as the bidirectional three-core cabletype. The predefined cable instance between switch 0 andswitch 1 is unidirectional.

1http://www.oaam.de

Model B is a network consisting of five switches 0 to 4.Five possible cables are declared as depicted in fig. 8.The switch types are again opaque and translucent. Threebidirectional cable types are defined: A single-core cabletype with high attenuation as well as two- and three-corebidirectional types (s. tab. 9). There is no restriction onthe type assignment for switches and cables.

All scenarios use cost minimization as an objective func-tion, i.e.

minimize∑Dk∈D

pDkcost +

∑Fj∈F

pFj

cost. (72)

0

12

Possible switchopaquetransparent or opaqueall switches

Possible cablepredefinedopen

FIGURE 7: VALIDATION MODEL A

0

1

23

4

Possible switchPossible cable

FIGURE 8: VALIDATION MODEL B

TABLE 8: PROPERTIES OF THE SWITCH TYPES FOR VALIDA-TION. ENTRIES WITH — ARE MODELED WITH THE VALUEZERO. RX AND TX VALUES ARE IN dBm.

PropertiesTypes τDportst

τD∆tτDRxt

τDRxt

τDTxtτD

TxtτDtranst

τDcosttModel

opaque 4 — −14 0,5 −5 0 0 300 A,Btranslucent 2 −0,5dB — — — — 1 100 A,B

5.1 Scenario 1: Unidirectional Cables

The first scenario is based on model A with three definedsignals identified by letters A, B and C. Fig. 9 depictsoptimized network topology. The colors of the devices andcables indicate the associated types, while the letters at theconnection illustrate the respective signal path.

All three switches are assigned to the opaque switch type.Each transmits and receives at least one signal. All con-nections are optimized to use unidirectional cables. Their

12

http://www.oaam.de


TABLE 9: PROPERTIES OF CABLE TYPES FOR VALIDATION.ATTENUATION INCLUDES CONNECTORS AND INTERNAL CA-BLE LOSSES

PropertiesTypes τF

corestτF∆t

τFcostt

Direction Model

Bidirectional, 1 core 1 −15,0 dB 1 0 (Bidirectional) BBidirectional, 2 cores 2 −2,0 dB 30 0 (Bidirectional) BUnidirectional, 2 cores 2 −2,0 dB 30 3 (Unidirectional) ABidirectional, 3 cores 3 −2,0 dB 50 0 (Bidirectional) A,B

A,CB,C

B

0 Target: BSource: A

1 Target: A,CSource: B2 Source: C

Switch(#0) opaque(#1) transparent

Cable(#0) Optical Wire Bidirectional(#1) Optical Wire Unidirectional

FIGURE 9: OPTIMIZED TOPOLOGY OF SCENARIO 1

defined transmission directions are indicated by arrows.The cable connection between devices 0 and 1 is prede-fined in the model, but its direction was undefined. A cycleformed by the cables can be observed. This correspondsto the signals to be transmitted. Only one of the two sig-nals A and B can be transmitted via the cable betweendevices 0 and 1 due to directionality. The signal that isnot transmitted via this cable, in this case B, has to take adetour via switch 2. Signal C fits within the second avail-able core of the unidirectional cables in each case. Thetwo generated cables (0–2 and 1–2) must exist because ofthe required paths for signal B and C, but the type is notspecified. Since there is no need to use the more expensivebidirectional cable (cost: 50), the less expensive unidirec-tional cable (cost: 30) is chosen. Therefore, this topologyhas the lowest possible cost under the given constraints.

5.1.1 Scenario 2: Attenuation

As a second scenario, the topology of model B with twosignals A and B is optimized. The network cost servesas the objective function. Two opaque and translucentswitch types per position are possible. The translucentswitch can connect at most two cables in the validation.The three bidirectional cable types are selectable. Thesingle-core cable type has a high attenuation. Sourcesand destinations of signals A and B can be taken from theoptimized topology in fig. 10.

A

AB

B

0 Target: ASource: B

1

2 Source: A3

4 Target: B


Cable(#0) 1-Core High Attenuation(#1) 2-Core(#2) 3-Core


As expected, all source and destination switches are opaqueswitches in the optimized topology. Instead of the (inten-tionally) inexpensive single-core cable type the signifi-cantly costlier dual-core cable type is used, even thoughit is not needed due to the number of signals. The reasonis the attenuation over the signal paths. This is verifiedusing signal A as an example. This results in the followingcalculation for the cable type used:

pD2

Tx+ p

(D1,D2)∆ + pD1

∆ + p(D0,D1)∆ =

0dBm− 2 dB− 0,5 dB− 2 dB =

−4,5 dBm

pD2Tx + p

(D1,D2)∆ + pD1

∆ + p(D0,D1)∆ =

−5 dBm− 2 dB− 0,5 dB− 2 dB =

−9,5 dBm

These values are within the permissible range of−14 dBmto 0,5 dBm of the opaque receiver. If only one cable isexchanged for the one with poorer attenuation, the receivercannot receive the signal:

pD2

Tx+ p

(D1,D2)∆ + pD1

∆ + p(D0,D1)∆ =

0dBm− 15 dB− 0,5 dB− 2 dB =

−17,5 dBm

pD2Tx + p

(D1,D2)∆ + pD1

∆ + p(D0,D1)∆ =

−5 dBm− 15 dB− 0,5 dB− 2 dB =

−22,5 dBm

Consequently, the cable type with poor attenuation is notpossible. In addition, a common signal routing via switch

13


3 is not possible, because the translucent switch 3 allowsat most two connections. It would have to be replacedby an opaque switch for a more compact architecture, butthis comes with overall higher cost. Instead of the twotranslucent switches and one cable (cost: 230), one opaqueswitch would be needed (cost: 300). The result is, thus,minimized in cost and all signal power levels are legal.

5.1.2 Scenario 3: Core Count

Scenario 3 extends scenario 2. In addition, two moresignals C and D are defined. Both start at switch 0 andhave switch 2 as their destination. This requires two morecores. The result of the optimization is shown in fig. 11. Itcan be seen that the path of signal B is unchanged. Also,the switch and cable types along its path are the same.However, a change of the cable types for the other signalpaths is necessary. Between switches 0, 1, and 2, the threesignals A, C and D are transmitted. This makes the thirdcable type with three cores mandatory.

A,C,D

A,C,DB

B

0 Target: ASource: B,C,D

1

2 Target: C,DSource: A3

4 Target:B


Cable(#0) 1-core high attenuation(#1) 2-core(#2) 3-core

FIGURE 11: OPTIMIZED TOPOLOGY OF SCENARIO 3. COM-PARED TO SCENARIO 2, TWO CABLES HAVE A HIGHER CORECOUNT.

5.1.3 Scenario 4: Core count vs. Switch Ports

Scenario 4 extends scenario 3 with the additional signal E.E has the same target and destination as the signals C andD. This time, there is a significant change in the topology,which can be seen in fig. 12. The four signals A, C, D,and E require at least two connections to switch 2. Theavailable cable types are limited to a maximum of threecores. Compared to scenario 3, one additional core fromswitch 0 to 2 is necessary. This results in the change ofswitch 3 that is mandatory. Instead of the previous translu-cent switch type, the opaque type is assigned, because ofthe required connection capacities. The other switch types

remain optimal according to the requirements. Consider-ing the cables, the two path options (0-3-2 and 0-1-2) haveboth two segments. Instead of introducing one upgradedand one additional 3-core cable to the solution of scenario2 (cost +70), only one is necessary (cost -20). Therefore,the result is the cheapest possible topology.

C,D

C,D

A,E

B

A,B,E

0 Target: ASource: B,C,D,E

1

2 Target: C,D,EQuelle: A3

4 Target: B



FIGURE 12: OPTIMIZED TOPOLOGY OF SCENARIO 4 ADDINGA FIFTH SIGNAL.

5.1.4 Scenario 5: Free interconnection

As a last validation scenario, a further variation of thethird scenario is optimized. The network is now automati-cally completed by the algorithm, i.e. no cable routes arepredefined. All cables can be created.

The resulting topology shown in fig. 13 corresponds tothe expectation. A smaller topology is not achievablewith the given specifications, because the start and targetswitch are fixed by the signals. The result correspondsto the minimum required number of cables with the mostreasonable cable types in terms of cost and attenuation.

6 Optimization of an In-flightEntertainment Network

An In-flight Entertainment system (IFE) scenario is usedto demonstrate our approach on a larger scale.

The scenario comprises three rows of seats of an aircraftwith a center aisle. Each row of seats consists of twogroups of seats, one to the left and one to the right ofthe aisle. Each of these seating groups contains a centralcomputer, called a Seat Unit, which controls the screensand controls. There is a panel above each seating groupwith call buttons, indicator, and reading lights. Each panel

14


A,C,D

B

0 Target: ASource: B,C,D

1

2 Target: C,DSource: A3

4 Target: B




has a dedicated control unit, i.e. Call and Light Panel. TheIFE is able to display live TV and movies at each seatingarea. Two streaming computers (Stream Server) providethe data to be transmitted over the optical network. BothStream Servers can provide each seating group with a videostream. The Stream Servers also control the lamps of theCall and Light Panels. All inputs to the control elementsof the seating groups and the call buttons are processedby two Control Servers. These command also the StreamServers. The Control Servers and Stream Servers are fullybidirectionally networked for communication with eachother. In a real aircraft, multiple of these aisle segmentswould be placed sequentially.

The optimization space for the IFE scenario is shown infig.14. The points outlined in black represent possiblepositions for switches. There are two types to choose from,namely an opaque switch and a translucent switch. Allother devices have their type already assigned. The ID ofdevices are listed in table 10.

TABLE 10: DEVICE AND IDS USED IN THE IFE SCENARIO

ID Type

1, 2, 3 Call and Light Panel (A)6, 7, 8 Call and Light Panel (B)11, 12, 13 Seat Unit (B)16, 17, 18 Seat Unit (A)20, 21 Stream Server22, 23 Control Server0, 4, 5, 9, 10, 14, 15, 19 possible Switches

Table 11 lists the available device types with their prop-erties. With the exception of one translucent switch, alldevices are opaque.

1516

17

18

19

20

21

22

23

0

1

2

34

1011

12

13

14

5

6

7

89

Devices(#0) Call and light panel(#1) Seat unit(#2) Switch opaque(#3) Switch translucent Possible cable

Possible switch

(#4) Stream server(#5) Control server

FIGURE 14: DEFAULT NETWORK OF THE SCENARIO IFE.

TABLE 11: DEVICE TYPES AND PROPERTIES OF IFE SCE-NARIO. RX AND TX ENTRIES ARE IN dBm. ENTRIESMARKED WITH — ARE MODELED WITH THE VALUE ZERO.

PropertyName τD

portstτD∆t

τDRxt

τDRxt

τDTxt

τDTxt

τDtranst

τDcostt

Call and Light P. 2 — −14 0,5 −5 0 0 —Seat Unit 2 — −14 0,5 −5 0 0 —Switch Opaque 6 — −14 0,5 −5 0 0 6000Switch Transl. 6 −0,5 dB — — — — 1 5600Stream Server 3 — −14 0,5 −5 0 0 —Control Server 3 — −14 0,5 −5 0 0 —

Due to the fixed assignment of the device types #0, #1,#4 and #5, no cost values are used. Costs are assignedto the two switch types subject to the optimization. It isassumed that a translucent device is less expensive thanan opaque device that requires active components. Theavailable cable types are listed in table 12. They all haveidentical attenuation characteristics, but differ based ontheir number of cores. The chosen numbers are inspiredby the [40]. The optimization objective is the total cost forcables and switches.

With two exceptions, the cables are free of choice. Neithera type is assigned, nor is the selection restricted. Only theconnection between the two Stream Servers 20 and 21, aswell as the connection between the two Control Servers 22and 23 are limited to the two-core cable type. In total, 48signals are necessary for this scenario as listed in table 13.

The IFE scenario resulted in a MILP with 49 316 Con-straints. 8454 binary, 162 integer and 9260 real variablesare used. Finding a first solution requires 12 s. After threeminutes, the solution within a limit of one percent devia-

15


TABLE 12: PROPERTIES OF CABLE TYPES IN IFE SCENARIO.ALL CABLE TYPES DECLARED ARE BIDIRECTIONAL. AT-TENUATION INCLUDES CONNECTORS AND INTERNAL CABLELOSSES.

PropertyNb. Name τFcorest τF∆t

τFcostt

#0 Optical Wire 2 Core 2 −2 dB 10#1 Optical Wire 4 Core 4 −2 dB 40#2 Optical Wire 8 Core 8 −2 dB 80#3 Optical Wire 12 Core 12 −2 dB 90

TABLE 13: SIGNAL DEFINITIONS AND OPTIMIZED PATHS FORTHE IFE SCENARIO. (S = SOURCE, T = TARGET)

ID S. Path T.

A 20 -15- 16B 20 -15- 17C 20 -15- 18D 20 -15-10- 11E 20 -15-10- 12F 20 -15-10- 13G 21 -10-15- 16H 21 -10-15- 17I 21 -10-15- 18J 21 -10-14- 11K 21 -10- 12L 21 -10- 13M 20 -0- 1N 20 -0- 2O 20 -0- 3P 20 -0-5- 6Q 20 -0-5- 7R 20 -0-5- 8S 21 -5-0- 1T 21 -5-0- 2U 21 -5-0- 3V 21 -5- 6W 21 -5- 7X 21 -5- 8Y 16 -15-20-0-4- 22Z 17 -15-20-0-4- 22

ID S. Path T.

BA 18 -15-20-0-4- 22BB 11 -14- 23BC 12 -14- 23BD 13 -14- 23BE 1 -4- 22BF 2 -4- 22BG 3 -4- 22BH 6 -5-21-10-14- 23BI 7 -5-21-10-14- 23BJ 8 -5-21-10-14- 23BK 22 - 23BL 23 - 22BM 20 - 21BN 21 - 20BO 22 -4-0- 20BP 22 -4-0-5- 21BQ 23 -14-10-15- 20BR 23 -14-10- 21BS 20 -0-4- 22BT 20 -15-10-14- 23BU 21 -5-0-4- 22BV 21 -10-14- 23

tion from the global optimum is obtained. It takes another9min to confirm an optimal solution. The entire programexecution requires 23min, of which a total of 6min arespent on data input, output, and internal model building.Another 6min are attributable to Gurobi and subsequentprocessing.

The optimized topology is shown in fig. 15. Six of theeight possible switches are instantiated in the topology astranslucent devices (yellow). The remaining two possibleswitch positions are unused. The structure of the cablesis interesting. The cable assignment is symmetrical alongone axis. Although there is a symmetry of the definedsignals in in the top strands (0-4 and 5-9) as well as in thebottom strands (10-14 and 15-19), this symmetry is notreflected in the optimized topology. Instead, a symmetry

of left and right is observed, resulting from the control andstream device positions.

5

6

7

89

1011

12

13

14

1516

17

18

19

20

21

22

23

1

2

34

0

Devices(#0) Call and light panel(#1) Seat unit(#2) Switch opaque(#3) Switch translucent(#4) Stream server(#5) Control server

Cables(#0) Optical wire 2 core(#1) Optical wire 4 core(#2) Optical wire 8 core(#3) Optical wire 12 core

FIGURE 15: SPATIAL REPRESENTATION OF THE OPTIMALSOLUTION OF THE IFE SCENARIO. THE COLORS REPRESENTTHE ASSIGNED DEVICE AND CABLE TYPES. GRAY DEVICESAND CABLES WERE REMOVED BY THE OPTIMIZER.

A careful inspection of the optimization result uncoversa shortcoming in our cost function: While two signals tothe Seat Unit with number 11 follow short paths from startto destination, one signal is not optimal in terms of hopcount. Instead of using the available core of the connection10-11, it runs via Switch 14. However, since the alternativepath with fewer hops does not result in a better objectivefunction value, it is not picked by the optimizer.

7 Conclusion

A topology optimization for an optimal network consid-ering attenuation, multi-core fibers as well as translucentand opaque switches can be represented and solved as aMILP problem. The validation scenarios indicate a correctfunctioning of the core-count, direction, and attenuationconstraints. The demonstration with the small IFE systemshows that the methodology is appropriate for envisionedtopology design. 30 min are an acceptable optimizationtime. However, further routing constraints (e.g. hop countand segregation) are needed to limit the signals routedthrough a device. The used domain-specific model OAAMincludes ways to specify such constraints, which were notconsidered so far. A critical consideration is the problemsize. For the demonstrated system section, already a five-digit number of variables and constraints are necessary.

16


Surprisingly, the demonstration model was solved com-paratively quickly. Comparing the presented approach foroptical networks with the optimization of classical net-works, runtimes in the order of several hours to days canbe expected for larger problems, e.g. a complete avionicssystem. In future, the limitations imposed will be revised:The fact that only one single signal is transmitted in onecore is a limitation in practice. Future implementationsshould, therefore, look at relaxing this limitation. In ad-dition, for future systems, it makes sense to extend theapproach to other optical transmission techniques, primar-ily the commonly used Wavelength-Division-Multiplexingtechnique. Each technology has its own advantages anddisadvantages, which is why there is also a potential needfor optimization in the selection of a suitable technology.In conclusion, it can be stated that topology optimizationof optical networks is feasible within an acceptable com-putation time and provides added value to the design ofaircraft systems.

Acknowledgements

This paper is based on research work carried out in theDELIA project (contract code: 16KIS0940) funded bythe German Federal Ministry of Education and Research(BMWI) in the IKT 2020 program.

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