A theoretical approach to a safety-based predictive adaptation of wireless communication channel parameters in harsh environments

  • L. GausEmail author
  • M. Schwarz
  • J. Boercsoek
Original Research Article


This paper presents an approach to a real-time optimization of safety parameters in wireless communication systems. When considering the GEC–model (Generalized Erasure Channel) and the black channel design of a communication channel, then the PFH (Probability of Failure per Hour) value can be estimated using the parameters ε – BER (bit-error-rate), φ – BLR (bit-loss-rate), v – number of safety related messages per second, n – message length and dmin – minimum distance of a linear code. The number of safety related messages per second v and the message length n, including the information block k and the checksum block r, can be varying between the permissible bounds. Accordingly, the variable parameters can be adjusted at run-time with additional assimilation of the used cyclic code. It allows the real-time prediction and optimization of the safety parameters. In this paper, the concept of the parameter estimation is discussed and based on it the optimization problem is defined.


Safety parameter Probability prediction Communication Black channel GEC CRC 


During the last years, the trend towards the usage of wireless communication rises in technical facilities (Yoshigoe 2010; Kadri 2012; Zhu et al. 2018). Taking processes in harsh environments into account, it is crucial to avoid any process accidents and not least to maintain the safe operation. Generally, the communication system is an important part of the entire safety related application. Therefore, besides the safety requirements for the hardware and the software of a technical system the safety requirements for the data communication have to be considered.

The safety requirements are described in several standards such as the IEC 61508–2 “Functional safety for electrical/electronic/programmable electronic safety related systems, Part 2” (IEC 61508 2000), the IEC 61784–3 “Industrial Process Measurement and Control, Part 3” (IEC 61784-3 2016) and the DIN EN 50159 “Railway applications – Communication, signaling and processing systems - Safety-related communication in transmission systems” (DIN EN 50159 2011).

To guarantee the safety of a plant or other technical system, the so-called Safety Functions (SF) are integrated within the system. SFs are “functions to be implemented by an E/E/PE1 safety-related system or other risk reduction measures that is intended to achieve or maintain a safe state for the EUC” (IEC 61508 2000).

In a communication process the transmission errors, repetitions, deletion, insertion, re-sequencing, corruption, delay and masquerade can cause a faulty communication. Those failures are to be considered while ascertaining the failure measure of the communication process (IEC 61508 2000).

The motivation of the present work is the definition of an optimization problem for an adaptation procedure allowing to react on transmission deviations and therefore, to keep the transmission at the required reliability.

The benefits of executing the optimization routine at run–time can be a higher efficiency of the communication part, maintaining of safe operation and less down time of the technical process.

The remaining of the paper is organized as follows: Section 2 discusses the use of wireless vs. wired communication in harsh environments. In section 3 some necessary definitions are outlined. Also, this section explains all-important assumptions, which has taken place in the described work. Then, in section 4 the theoretical approach to the estimation methods of the required parameters is introduced followed by the definition of the optimization problem. Finally, section 5 summarizes the approach and indicates further investigations.

Wireless vs. wired comunication in harsh environments

A communication system plays a weight-bearing role in a technical system. A failure or an incorrect functionality of the communication system or its parts can paralyze the entire system and cause a dangerous situation. By using wired technologies, particularly in harsh environments, the environmental conditions can comprise the reliability of the cable, signal integrity and life performance (GORE 2013). Such harsh conditions include low pressure, low to high temperature, temperature shock, contamination by fluids, solar radiation, (freezing) rain, humidity, fungus, salt fog, sand and dust, leakage, acceleration, vibration, acoustic noise, (ballistic) shock, gunfire vibration icing, acidic/vibro-acoustic/temperature and explosive atmosphere (Pirich 2011). For instance, the potential destructive effects on an underwater fiber optic cable are corrosion, moisture, high pressure, high forces in the axial and lateral directions and high temperatures (Pirich 2011).

Generally, a cable system can be defeated by the following factors (GORE 2013):
  • Electrical stress can compromise signal integrity due to electromagnetic interference, crosstalk, attenuation and conductor resistance (GORE 2013).

  • Mechanical stress can take place when a cable system is induced by random, rolling and torsion types of motion. Especially in harsh environments, cables can be impacted on sharp surfaces. This can cause strong abrasion and cable-cut (GORE 2013).

  • Environmental stress arises from the physical characteristics of the operational area and can drastically reduce the life-time of the cable system. As an example, raised fragility as a result of low temperatures or destroying of cable materials by gases and liquids can be mentioned (GORE 2013).

  • Application-specific stress can be caused by the specific design characteristics of the operational application, such as the choice and the utilization of the inappropriate technologies or materials. Also, the insufficient safety measures can contribute to increase of the application-specific stress (GORE 2013).

A variety of these factors can be refused by applying wireless solutions. Primarily, the mechanical stress and the environmental stress can be eliminated in this manner. There are also many other advantages of using wireless technologies opposed to the wired solution in industrial automation. Especially in terms of harsh environments, the use of wired networks yields some restrictiveness. In (Pereira da Cunha 2013), the following limitations of direct-wired sensors in hostile environments are named: (i) reliability problems due to degradation and breakage of physical connections over time; (ii) extra weight due to all the wires and connections; (iii) complicated and costly sensor installation and maintenance; (iv) difficult to be placed in rotating parts and (v) limited overall number of sensors that can be monitored due to complexity of the wiring.

Wireless communication delivers the possibility to handle such obstacles and bring several benefits in general, such us lower costs of installing, maintaining, troubleshooting and fast commissioning (Kadri 2012). And there are also advantages for special applications, e.g. harsh environments, such as freedom to place network nodes in more versatile independent locations, the capability to request information from multiple sensor devices with the same interrogator and a reduction in sensor system and weight (Pereira da Cunha 2013).

Currently, there are several wireless technologies, which can be deployed at different levels in industrial facilities (YOKOGAWA 2015):
  • Radio-frequency identification (RFID): RFID is an automatic identification technology, which makes it possible to obtain real-time information about the physical objects comprised in a technological process (Li et al. 2010). The possible application areas for RFID are: identification and access control (e.g. LPG tank), certification and anti-counterfeiting, logistic (e.g. train car, container and tobacco pallets), ticketing.

  • Wireless sensor networks (WSN): WSNs can be characterized as the limited power, memory, processing and communication capacity of small and low-cost sensor nodes (Kung et al. 2008). Their field of application includes many areas such as wireless measurements, condition monitoring and disaster prevention in commercial, industrial and medical domains.

  • Wireless LANs: industrial WLANs are primarily implemented with the IEEE 802.11 family of standard (Willig 2008). They are mainly suited for mobile operator terminals, data logging, security and maintenance.

  • Wireless WANs: wireless WANS are conceived for data transmission about large geographical areas and include long-distance broadband backhaul and high-bandwidth video applications.

The wireless technologies outlined above can be used in an arrangement side by side within a plant. The approach described in this work is applicable by RFID, WSN and WLAN technologies.

Caused by Industry 4.0 and IoT industrial wireless networks are a research-intensive domain. Some approaches can be found in (Zolfaghari et al. 2017; Hassan et al. 2016; Krishna et al. 2018). To judge by the literature investigation, it seems that the WSNs are actually the most researching part of wireless communication domain, e.g. in harsh environments (Xoshigoe 2017; Pereira da Cunha et al. 2016; Aqueveque et al. 2018; Verma et al. 2018; Saffari et al. 2018).

The issues discussed at the beginning, as well as a considerable academic interest in wireless communications in general and in particular in the context of harsh environmental operations, all those clearly show the relevance of this subject.

Definitions and assumptions

Safety related communication process

Basically, the safety related communication can be performed using a safety application protocol. There are several safety-related protocols for industry available: PROFIsafe, FF-SIF, INTERBUS-Safety, CIP-Safety et al.. Besides the message transmission, such protocols have to detect and if applicable to correct the possible errors.

In the IEC 61508, one of the possibilities to ensure the required safety communication level is the implementation of the communication channel as a “black channel” model, where parts of the communication channel are not designed in accordance with the standard. The principle of black channel is illustrated in Fig. 1.
Fig. 1

The principle of black channel

In case of the black channel approach, the transfer of the safety related messages can be performed across the particular safety layer parallel to safety irrelevant data.

In the safety layer additional measures for the design of the safe communication such as cyclic exchange of messages, protection by a linear code or the multiple transmission with the following comparison have to be implemented (IEC 61784-3 2016). Beyond these measures, the watchdog timer controls the timing performance of the transmission.

The linear code, which is used in most safety protocols, is the Cyclic Redundancy Check (CRC) (Hannen 2012).


The CRC was firstly described by W.W. Peterson and D.T. Brown (Peterson and Brown 1961). This is the polynomial code for error detection over the Binary Galois Field GF(2) = {0, 1}. The polynomials in GF(2) have the form:
$$ {a}_{n-1}{x}^{n-1}+{a}_{n-2}{x}^{n-2}+...+{a}_1{x}^1+{a}_0 $$
with coefficients ai ∈ {0, 1}. In this manner, each string of binary symbols can be expressed as an polynomial in GF(2). For example, the string 10,101 can be represented by the polynomial
$$ 1\cdot {x}^4+0\cdot {x}^3+1\cdot {x}^2+0\cdot {x}^1+1={x}^4+{x}^2+1 $$

The CRC computation is based on the polynomial division in GF(2), which is the same as the decimal division with the subtraction in modulo 2. It means, instead of subtraction the exclusive OR (XOR) operation can be used in the intermediate steps of the division.

According to desired performance, the transmitter and the receiver predetermine the generator polynomial G(x) with the degree r. For the sending process, the transmitter multiplies the message polynomial M(x) by xr. This means, the message is extended by r zero bits at the end. Then, the extended message xr ⋅ M(x) is divided by the generator polynomial G(x). In the next step, the remainder R(x) = xrM(x)modG(x) of the division is concatenated with the original message M(x). Hence, the message to transmit has a form Mt(x) = xr ⋅ M(x) + R(x) and is divisible by G(x).

After receiving a message Mt(x), the receiver divides this message by G(x) and checks the reminder Rt(x) = (xr ⋅ M(x) + R(x))modG(x). The reminder will be nonzero if the transmission was faulty (Peterson and Brown 1961).

There are several generator polynomials used in different applications. Some of them are presented in Table 1.
Table 1

Some examples of applied CRC polynomials



Used in:


x8 + x2 + x + 1



x16 + x12 + x5 + 1



x16 + x15 + x2 + 1

USB, Modbus


x32 + x26 + x23 + x22 + x16 + x12 + x11 + x10 + x8 + x7 + x5 + x4 + x2 + x + 1

IEEE 802.3: Ethernet

In a simplified sheme, a message with a CRC checksum is shown in Table 2. Here, the original message M(x), is called payload, contains k bits and the appended CRC checksum consists of r bits. The generated message to be send is n bit long with n = k + r.
Table 2

The message with the CRC checksum

Payload M(x)

CRC checksum R(x)

k bits

r bits

Related to the generator polynomial, the minimum distance dmin is an important metrics for channel coding. dmin is based on the Hamming distance dij, which represents the measure of the difference of two code words Ci and Cj in a block code. More precisely, Hamming distance dij is the number of diverse corresponding elements of Ci and Cj. The minimum distance dmin denotes the smallest value of set {dij} for the M = 2k binary code words (Proakis 2000).

In present literature, listings of generator polynomials exist with accompanying minimum distance as stated in (Koopman and Chakravarty 2004).

Safety parameters

Theoretically, in terms of safety the main desired requirement on the wireless communication system is the errorless and lossless transmission of the data. On the practical point of view, it is impossible to eliminate all transmission errors and erasures because of noise, interference, fading effects, jamming as well as deliberate corruption (Pendli 2014). But there are measures for reducing or detecting them. Such measures help to decrease the residual risk. Residual risk is defined in IEC 61508 as “risk remaining after protective measures have been taken” (IEC 61508 2000). Since the residual risk cannot be completely eliminated, the primary aim is to minimize the residual risk up to a tolerable bound while operating of a safety-related system (IEC 61508 2000).

In the IEC 61508, such bounds are defined by the value of Probability of Failure per Hour (PFH) and are classified in Safety Integrity Levels (SIL). As required in this standard, the PFH value has to be calculated during the design phase. The calculation is based on the hazard and risk analysis of the whole system (IEC 61508 2000).

As recommended in the IEC 61784, the maximum residual risk of the safety communication channel should not exceed 1% of the maximum permitted PFH of the achieving SIL. In the table below, the respective upper bounds of PFH are shown in relation to SIL Table 3.
Table 3

SIL respective PFH upper bounds


PFH of safety function

PFH of safety communication channel


< 10−8

< 10−10


< 10−7

< 10−9


< 10−6

< 10−8


< 10−5

< 10−7

Generalized Erasure Channel

From the safety layer point of view, the model of a channel for a digital transmission can be described by the Generalized Erasure Channel (GEC) stated in (Wacker and Boercsoek 2007). In this model the channel is assumed with the set of two possible input symbols {0, 1} and the set of three possible output symbols {0, 1, e}. The output symbol e indicates the erasure symbol which occurs if the received symbol can be identified neither as a “0” nor as a “1”. Under these assumptions, the transmission behavior of the channel can be specified by the transition probabilities:
$$ P\left(e|0\right)=P\left(e|1\right)=\zeta $$
$$ P\left(0|1\right)=P\left(1|0\right)=\eta $$
$$ P\left(0|0\right)=P\left(1|1\right)=\theta $$
with ζ ≥ 0, η ≥ 0, θ ≥ 0 and ζ + η + θ = 1.
The graphical representation of the probabilistic transmission behavior of the GEC is shown in Fig. 2.
Fig. 2

Transition probabilities of GEC (Wacker and Boercsoek 2007)

In fact, because of physical effects, it is rarely possible to assume a symmetric cannel. Normally, the symmetry can be achieved by additional measures, e.g. by transmitting each block twice (Pendli 2014).


To quantify the error-proneness of a communication channel, the Bit Error Rate (BER) is used. The BER indicates the ratio of the number of bits falsified during the transmission process to the total number of transmitted bits.

To quantify the susceptibility of the communication channel to the loss of bits, the Bit Loss Rate (BLR) is defined. The BLR denotes the ratio of number of bits, which cannot be identified, to the total number of transmitted bits.

Probability of undetected error and PFH

The GEC model of the communication channel is assumed as described in 3.4. For transmissions protected by a linear code C the probability of undetected error Pue is given by (Wacker and Boercsoek 2007):
$$ {P}_{ue}\left(\zeta, \eta, \theta, C\right)=\sum \limits_{l=1}^n{A}_l\cdot {\eta}^l\cdot {\theta}^{n-l} $$
where Al is a number of code words of weight l (given by the number of nonzero bits). ζ, η and θ are the transition probabilities according to section 2.4, n is the block length. In this paper, the complete message is considered as a block. Therefore, hereinafter the parameter n referred to as a length of the message.
Now, let the parameter ε and φ denote the bit-error rate (BER) and the bit-loss rate (BLR), respectively. Then, the transitions probabilities ζ, η and θ can be given as detailed in (Sköllermo and Skoglund 2003):
$$ \zeta =\varphi $$
$$ \eta =\varepsilon \cdot \left(1-\varphi \right) $$
$$ \theta =\left(1-\varepsilon \right)\cdot \left(1-\varphi \right) $$
Under consideration of these equations the probability of undetected error is given in dependence to Eq. (6) and after some transformations by (Pendli 2014):
$$ {P}_{ue}\left(\varepsilon, \varphi, C\right)={\left(1-\varphi \right)}^n\cdot {P}_{ue}\left(\varepsilon, C\right) $$
$$ {P}_{ue}\left(\varepsilon, C\right)=\sum \limits_{l=1}^n{A}_l\cdot {\varepsilon}^l\cdot {\left(1-\varepsilon \right)}^{n-l} $$
The PFH for a GEC can be calculated as follows (Pendli 2014):
$$ PFH=3600\cdot {P}_{ue}\left(\varepsilon, \varphi, C\right)\cdot v\cdot 100\cdot \left(m-1\right) $$

number of safety related messages per second and


number of communicating devices.

Finally, the formula for the calculation of PFH for one channel with two communicating devices can be derived by substituting Eq. (10) in Eq. (12):
$$ PFH=36\cdot {10}^4\cdot v\cdot {\left(1-\varphi \right)}^n\cdot {P}_{ue}\left(\varepsilon, C\right) $$

Mathematical approach for online SIL estimation

For the proposed approach, a safety communication with one channel is assumed. It consists of a transmitter, a receiver and one channel in a black channel format. Since the communication is cyclic in such a channel, it is therefore possible to observe the communication and to adapt the relevant safety parameters in such a way that the required SIL can be achieved. Equation (13) is suitable to estimate the PFH value for the allocated time and the related SIL. Due to the “black channel” approach, only channel parameters of the safety layer can be optimized.

In the following section the mathematical approach is discussed that estimates the non-changeable factors and the optimization of the variable parameters.

Estimation of BER

In communication systems, the BER can be determined by the SNR (Signal-to-Noise Ratio) of the communication path, which can be measured at the receiver. Let γb denote the signal-to-noise ratio per bit:

$$ {\gamma}_b=\frac{E_b}{N_0} $$
where Eb is the mean energy per bit and N0 is the noise spectral density. Then, the BER for different modulation types can be determined as given in (Proakis 2000). In Table 4 a few expressions are presented.
Table 4

BER expressions for some modulation techniques

Modulation Technique


BPSKa(antipodal signals)

\( Q\left(\sqrt{2{\gamma}_b}\right) \)

BPSK (orthogonal signals)

\( Q\left(\sqrt{\gamma_b}\right) \)

DBPSKb and BFSKc for non-coherent detection

\( \frac{1}{2}{e}^{-{\gamma}_b} \)

aBPSK: binary phase-shift keying

bDBPSK: differential BPSK

cBFSK: binary frequency-shift keying

In this table Q(x) for x ≥ 0 is the Gaussian probability density function (PDF):
$$ Q(x)=\frac{1}{2}-\frac{1}{\pi}\underset{0}{\overset{x}{\int }}{e}^{-{t}^2} dt $$

According to (Wacker and Boercsoek 2008), it is required from the Technical Control Board of Germany to assume the worst-case value of BER with ε = 10−2 in case of the black channel application, and if no other information about the BER is available.

Estimation of BLR

The estimation of the BLR could occur using the value of the channel capacity. This value can be calculated using the parameters of the transmission process.

Considering the transmission between a transmitter and a receiver, every communication channel offers a certain channel capacity. The channel capacity is the maximum bitrate which allows a reliable transmission over a communication channel. From the view of the transmission participants the channel capacity owns a tight value at a certain time and is independent of the used channel model. Based on this fact, the expressions for the calculation of the channel capacity from different channel models can be used for estimation of BLR.

For instance, the normalized capacity as a function of SNR for band-limited AWGN (Additive White Gaussian Noise) channel (Pirich 2011) is given by:
$$ \frac{C_{W,{\gamma}_b}}{W}={\log}_2\left(1+\frac{C_{W,{\gamma}_b}}{W}\ {\gamma}_b\right) $$

Here, \( {\mathrm{C}}_{\mathrm{w},{\upgamma}_{\mathrm{b}}} \) stands for the channel capacity and W denotes the bandwidth.

After some transformations of Eq. (16) that leads to the following expression as shown below:
$$ {2}^{C_{w,{\gamma}_b}}={\left(1+\frac{C_{W,{\gamma}_b}}{W}\ {\gamma}_b\right)}^W $$
From the view of the GEC model, the channel capacity is given by (Wacker and Boercsoek 2007):
$$ {C}_{\varsigma, \kern0.5em \eta, \kern0.5em \theta }=\eta {\log}_2\eta +\theta {\log}_2\theta -\left(\eta +\theta \right){\log}_2\left(\eta +\theta \right)+\eta +\theta $$
In terms of Eqs. (7)–(9) the Cς, η, θ can be represented as Cε, φ:
$$ {C}_{\varsigma, \eta, \theta }={C}_{\varepsilon, \varphi } $$
$$ {C}_{\varepsilon, \varphi }=\left(1-\varphi \right)\left(\varepsilon {\log}_2\varepsilon +\left(1-\varepsilon \right){\log}_2\left(1-\varepsilon \right)+1\right) $$
Now, with the considerations described at the beginning of this section \( {C}_{W,{\gamma}_b}={C}_{\varepsilon, \varphi } \) can be set. Thus follows:
$$ {2}^{C_{\varepsilon, \varphi }}={\left(1+\frac{C_{\varepsilon, \varphi }}{W}\ {\gamma}_b\right)}^W $$
Inserting expression (20) for Cε, φ in Eq. (21) results in:
$$ 2\frac{\left(1-\varphi \right)}{w}{\cdot}^{\left(\varepsilon {\log}_2\varepsilon +\left(1-\varepsilon \right){\log}_2\left(1-\varepsilon \right)+1\right)}=1+\frac{\left(1-\varphi \right)\left(\varepsilon {\log}_2\varepsilon +\left(1-\varepsilon \right){\log}_2\left(1-\varepsilon \right)+1\right)}{W}\ {\gamma}_b $$
In the next step:
$$ {\left(2\cdot {\varepsilon}^{\varepsilon}\cdot {\left(1-\varepsilon \right)}^{\left(1-\varepsilon \right)}\right)}^{\frac{\left(1-\varphi \right)}{W}}=1+\frac{\left(1-\varphi \right)\left(\varepsilon {\log}_2\varepsilon +\left(1-\varepsilon \right){\log}_2\left(1-\varepsilon \right)+1\right)}{W}\ {\gamma}_b $$
By the substitution of:
$$ x=\frac{\left(1-\varphi \right)}{W} $$
$$ a=2\cdot {\varepsilon}^{\varepsilon}\cdot {\left(1-\varepsilon \right)}^{\left(1-\varepsilon \right)} $$
It follows:
$$ {a}^x=1+x\ {\gamma}_b\left(\varepsilon {\log}_2\varepsilon +\left(1-\varepsilon \right){\log}_2\left(1-\varepsilon \right)+1\right) $$
$$ {\displaystyle \begin{array}{l}{2}^{a^x}={2}^{1+x\cdot {\gamma}_b\cdot \left(\varepsilon {\log}_2\varepsilon +\left(1-\varepsilon \right){\log}_2\left(1-\varepsilon \right)+1\right)}\\ {}=2\cdot {\left({2}^{\left(\varepsilon {\log}_2\varepsilon +\left(1-\varepsilon \right){\log}_2\left(1-\varepsilon \right)+1\right)}\right)}^{x\cdot {\gamma}_b}\\ {}=2\cdot {\left(2\cdot {\left({2}^{\log_2\varepsilon}\right)}^{\varepsilon}\cdot {\left({2}^{\log_2\left(1-\varepsilon \right)}\right)}^{\left(1-\varepsilon \right)}\right)}^{x\cdot {\gamma}_b}\\ {}=2\cdot {\left(2\cdot {\varepsilon}^{\varepsilon}\cdot {\left(1-\varepsilon \right)}^{\left(1-\varepsilon \right)}\right)}^{x\cdot {\gamma}_b}\end{array}} $$
And finally, with Eq. (25):
$$ {2}^{a^x}=2\cdot {a}^{x\cdot {\gamma}_b} $$

In the next step the value of BLR φ can be determined by the solution of the Eq. (28). Here, x is the variable and a is assumed as known, because it can be calculated from ε by Eq. (25). The solution could be proceeded numerically.

Upper bound for Pue

As aforementioned, the estimation of the probability of undetected error Pue can be carried out with the expression of Eq. (11). In this expression, the message length n, the value ε of BER and the weight spectrum [A1, A2, …, An] are needed for the calculation. While the message length in the communication application is known and the value of BER can be calculated based on other known parameters, e.g. as explained in section 4.1, the estimation of the weight distribution turns out to be an unresolved problem for most of the codes (Afanassiev and Davydov 2017). Therefore, for the application the “worst-case” estimation of Pue were selected. Worst case estimation is a common approach in the field of functional safety.

The worst-case value of Pue with \( {P}_{ue}^{\ast } \) is selected. The inequality of upper bound for Pue(ε, C) was derived by (Wacker and Boercsoek 2008) for all 0 ≤ ε ≤ ½:
$$ {P}_{ue}\left(\varepsilon, C\right)\le \frac{72}{121}\cdot \frac{\sqrt{2\pi n}}{2^r\cdot d!}\cdot {n}^d{\varepsilon}^d+{2}^n{\left(\sqrt{\varepsilon}\right)}^j $$

j = n if n ≥ 3 and even


j = n − 1 if n ≥ 4 and odd.

Here is d = dmin the minimum distance of the code C, ε is the value of BER, n is the length of the message and r the length of the checksum.

It can be seen, that the right expression in (29) can have one of two values depending on the length of the message. In the present approach, the even number of bits in a message can be assumed, because the frames of protocols in the safety layer are organized in byte blocks. Therefore, the value j = n can be used.

When summarizing this consideration with the inequality (29), the upper bound \( {P}_{ue}^{\ast } \) can be calculated with the following expression:
$$ {P}_{ue}^{\ast }=\frac{72}{121}\cdot \frac{\sqrt{2\pi n}}{2^r\cdot d!}\cdot {n}^d{\varepsilon}^d+{\left(2\sqrt{\varepsilon}\right)}^n $$

Optimization problem

After defining the expression for calculation of the PFH in a communication channel and identifying the deployable estimation of necessary parameters, the optimization problem can be formulated.

Focusing on (13) and (30) the upper bound of PFH for a communication channel can be calculated by
$$ PF{H}^{\ast }=36\cdot {10}^4\cdot v\cdot {\left(1-\varphi \right)}^n\cdot \left(\frac{72}{121}\cdot \frac{\sqrt{2\pi n}}{2^r\cdot d!}\cdot {n}^d{\varepsilon}^d+{\left(2\sqrt{\varepsilon}\right)}^n\right) $$

The PFH* is a function which is generally dependent on six variables: v, k, r, d, φ und ε. Two of these variables, the BLR φ und the BER ε, are conditioned by the parameters of the underlying channel and cannot be adjusted in the safety layer. Contrarily, the other four parameters, the message length n, the length of the CRC checksum r, the minimum distance d of CRC and the number of safety related messages per second v, are changeable within the safety layer at run-time.

Therefore, in the provided approach, the PFH is assumed as a function f(v, n, r, d) with four variable parameters and the remaining parameters are assumed as predetermined at the time of consideration.

The general aim of the optimization proposal examined in this paper is the minimizing of the communication effort. In terms of the safety layer, it means the minimization of the message length n and the number of safety related messages v. With this target, the optimization problem can be defined as follows:
$$ {\left[{v}_{opt},{n}_{opt},{r}_{opt},{d}_{opt}\right]}^T=\arg \underset{\begin{array}{c}v\in R\\ {}n\in R\end{array}}{\min }f\left(v,n,r,d\right) $$
$$ \mathrm{s}.\mathrm{t}.{n}_{min}\le k+r\le {n}_{max} $$
$$ {v}_{min}\le v\le {v}_{max} $$
$$ f\left(v,n,r,d\right)\le PF{H}_{SILx} $$
$$ f\left(v,n,r,d\right)=36\cdot {10}^4\cdot v\cdot {\left(1-\varphi \right)}^n\cdot \left(\frac{72}{121}\cdot \frac{\sqrt{2\pi n}}{2^r\cdot d!}\cdot {n}^d{\varepsilon}^d+{\left(2\sqrt{\varepsilon}\right)}^n\right) $$

Here, the PFHSILx denotes the upper bound of the required SIL in the application.


This paper presented a mathematical approach to optimize safety parameters of a communication channel in real-time. Therefore, the relevant parameters of the safety layer were identified that can be adopted during run-time. Additionally, procedures to estimate the remaining non-changeable parameters were suggested. Finally, an optimization problem were provided and solved.

This approach can serve as an observer procedure of a communication channel in order to adopt the safety parameters and therefore to maintain the required SIL online. The actual algorithm is going to be implemented in a future work. The useful way could be as follows: the in each communication cycle the observer procedure determines the optimal values vopt, nopt, ropt and dopt depending on the current (run-time) values φ and ε. Afterwards, the suitable CRC generator polynomial has to be selected based on the value dopt and ropt. The following communication sequence occurs with selected CRC and with the determined length of messages and the number of messages per second. If the optimization problem goes unresolved within the stated parameter limits, than the deviation from the required safety level shall be signalized to the technical process and, when appropriate, the system has to be transferred to a safe state.

By the mentioned observer algorithm the maximum possible degree of safety with the minimum possible resource consumption can be provided.

In the future, the following steps are planned:
  • proof of the validity of the mentioned model,

  • design and implementation of applications for solving the optimization problem and the observer algorithm,

  • evaluation of the observer function,

  • assessment of the achieved benefits.


  1. 1.

    E/E/PE stands for “electrical and/or electronic and/or programmable electronic” (IEC 61508 2000)


Compliance with ethical standards

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Computer Architecture and System ProgrammingUniversity of KasselKasselGermany

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