A Traffic Model for Networked Devices in the Building ... - CiteSeerX

A Traffic Model for Networked Devices in the Building Automation J¨orn Pl¨onnigs, Mario Neugebauer, Klaus Kabitzsch Institute for Applied Computer Science Dresden University of Technology D-01062 Dresden {jp14, mn7, kk10}@inf.tu-dresden.de

Abstract Traffic models play a decisive part in the performance evaluation and design of communications systems. But the growing diversification of devices in large networks asks for automated generated traffic models. In this paper we present a generic device model which is suited for automated generation in the domain of building automation. The traffic is deduced successive for common devices from measurements and simulations using classification of the devices by their traffic behaviour and automated parameterization of the device models.

1. Introduction and Related Work To design a large network infrastructure it is necessary to have at least an idea of the future utilization of the network. The developer uses his experience, measurements or simple calculations to approximate the utilization. But, with increasing network size reliable decisions become difficult. First, large networks cannot be calculated by hand. Second, they are uneconomic to prototype for measurements. Third, the projection from experience is dangerous as the interaction of the system increases and varies with the network. Last, the bus itself begins to affect the system behaviour. For example closed control cycles are sensitive to jitter of the transmit time, which variance rises with the bus utilization [8, 13]. Such complications let Tanenbaum state the rule, that “Avoiding Congestion is Better than Recovering from it.” [21]. But, how to avoid a situation, if it is difficult to identify. To tell the truth, the methods to estimate the load are well known. It can be calculated using for example simple superposition [15], queuing theory [1] for mean value analysis or Network Calculus [9] for maximum approximation. Beside the analytic methods also simulative approaches can be used [20, 22] to predict the utilization. Nevertheless, independent of the chosen approach the quality of the result corresponds to the quality of the used traffic model. Hence, for a realistic prediction of the load you will need a realistic traffic model. But traffic has the bad habit to behave random, as every road user knows.

This makes realistic traffic models difficult to establish. The traffic in a network is generated by messages which originate from the connected devices. It can be described by the number of messages per time unit, called arrival rate λ, and the bitsizes m of the messages. To characterize the traffic, it can be abstracted by generalizing the individual messages and analyzing distributions or otherwise it can be modeled in detail following each message on its way through the network and identifying the source and target. The theory prefers for analytic simplicity the first case, generalized poisson arrivals [3, 19]. But generalization is inadvisable in networks with a high variance and correlation of the arrival rates of the individual devices. Both results in burst periods with a temporary strong increased arrival rate which can overload the network. Numerous studies have analyzed measurements and confirmed such burst periods and self-similarities for LAN and WAN networks [19, 10]. One reason for this behavior is the dependence on the arbitrarily acting human in this internet dominated network class. The resulting traffic models are even more complex because of multiple routes in switched networks and changing source target relations. OPNET Guru [18] as representative software package in this area uses a wide range of hardware and user models to emulate and finally predict the performance of a future IT-infrastructure. Jasperneite [7] found similar burst periods in an investigated industrial automation process. He identified different kinds of mixed distributions of the arrival rate and explained them with background knowledge of the measured technical process. He skipped a more general classification and stated as conclusion of his network calculus approach in [23] the demand for automated traffic model extraction. Another specific traffic modeling approach uses Tomura et al. [22] in their simulation. They describe the process as state charts and generate the traffic model during their discrete event simulation. Such specific traffic models of the devices may be very accurate for the individual system but are not easy to transfer to other systems because approaches for mapping like classification are missing. This is comprehensible in the

diversity of industrial processes. In building automation the processes are not quite that multifarious as in industry automation. First, strong standardization of devices and basic functions (e.g. [11]) pre-classifies the diversity of devices. Second, the traffic is not affected by internet traffic but mainly determined by known physical processes and therefore comprehensible. Further, steady communication relationships and known functional interactions of the devices make the traffic model applicable for years [2]. Last, the communication model and the used devices can be extracted from available design databases [4]. We apply these advantages in a general traffic model for the domain of building automation. As the network size is increasing we focus on automated model generation from design databases. We decompose the traffic to its sources - the devices. The resulting general device model is particularly suitable for load prediction as it is flexible to resizing. It has the following demands for automatic modelling:

design db (e.g. LNS)

standard device device descriptions profiles

XIF functional links parameter

λ device db

device templates

estimation

Figure 1. Overview of the procedure to estimate λ

3. Basic Device Model Let us state that devices are the only traffic sources in a network. This will neglect other network elements like routers. They are not a source of the traffic in our definition because they bundle the arrival rate of the device messages that pass them. This bottleneck function need to be detailed in the network model. A device is described in the domain of building automation usually as black box with a visible set of input and output variables. Figure 2 illustrates for example the graphical representation of a device in LonWorks [6]. To describe the functional interaction between multiple devices the output variables are connected by bindings to other input variables (see Figure 7). This is done with the help of a design tool which stores the connection information in the network as well as in a separate design database.

D1 The general device model, needs to be adjustable to all common cases. D2 Simple classification schemes are necessary for automatic assignment. D3 The parameterization has to be deduced mainly from the design databases with only little adjustment of experts. The next section introduced the process to generate the traffic model of a network. Afterwards, the device model will be derived regarding the given demands. In Section 4.4 it will be parameterized exemplary for typical processes from measurements and simulations. Finally, the proposed method will be applied to a case study of a single room lighting control in Section 7.

device

2. Overview

i1 i2

The process to estimate the arrival rate λ of a network is pictured in Figure 1. We use an existing design database to extract the device types, their functional interaction and parameterization. A separate device database is consulted for a template describing the arrival rate behavior of each device type used in the network. These device templates are derived from standardized device profiles [11], device descriptions [12] or from the design database [4],[5]. To receive more precise results a device engineer can detail theses templates further. The model will be finally adjusted using the parameterization extracted from the design database. We use the resulting arrival rate model to predict the load of a future network [15]. The necessary network model and message sizes can be gathered from the design database as well [16]. In this paper we will capitalize the basing general arrival rate device model.

i3

inputs

v4,1 * v4,2 * v4,3 *

+

gain

i4

outputs

Figure 2. Device in Lon with shaded internal λ behavior The grey part of Figure 2 already suggests that internal connections between inputs and outputs may exist. Like a router not every device is a causal source of an arrival rate. Many devices are only generating a message at an output after a stimulating message at an input, for example a PIDcontroller which only produces a new plant input after it gets an updated plant output. We call them λ-processors and detail them in Section 5. The actuator which receives the plant output from the PID-Controller adjusts the plant accordingly. Because the incoming message has no further direct influence on the 2

network the device is called λ-sink. The remaining device class is the λ-source, which creates messages not as reaction on incoming telegrams but on changes of its environment or a timer. Sensors are the most important members of this class. These three device types differ in their causal context (Figure 7). To summarize the first classification scheme L- according to demand D2 we identified

SS state signal (e.g. on/off) or a SC continuous signal (e.g. 5.3◦ C). Second, the signal determinability compose the classification EED deterministic (e.g. physical signals) ES stochastic (e.g. human influences).

LS λ-sources

The corresponding tree for the classification of the devices is summarized in Figure 3.

LP λ-processors LI λ-sinks

arrival rate λ

All three types should be combined in one general device model to accomplish demand D1. We introduce the model time continuous for two reasons. First, it is more precise in analysis, as the arrival rate of a dynamical process can change with time. Second, it can map burst periods, for example when a single alarm message causes a chain reactions over multiple λ-processors. However, let us summarize the time continuous device model: The arrival rate λo (t) of a device output o depends as well on an external source λsrco (t) as on the arrival rate λi (t) of a device input i with the gain voi (t) and the processing time τ (s. Figure 2). This equals in matrix representation λ(t) =V(t) λ (t − τ) + λsrc (t)   v1,1 (t) . . . v1,Ni (t)   .. .. .. with V(t) =  . . . . vNo ,1 (t) · · · vNo ,Ni (t)

Causal context

λ-source

λ-processor

λ-sink

time-triggered

time-triggered

event-triggered

event-triggered

Trigger class

Determination

stochastic

deterministic

state signal

state signal

continuous

continuous

Signal type

Figure 3. Classification of the devices by their arrival rate Fortunately most deterministic event-triggered λ-sources are continuous (ED/SC) whereas many stochastic event-triggered λ-sources send state signals (ES/SS). We will detail them in these common combinations as the methods are transferable to the less usual ones as well. Stochastic event-triggered λ-sources are mainly caused by humans who act arbitrarily, as generally known, and are therefore difficulty to determine. In most cases only generalized observations or rough guess from experts (D3) can be used for modeling. Fortunately their impact on the traffic is only small because they use state signals which are only transmitted on change. The state signal g(t) can be for example the occupancy state of a room (n-persons) or the position of a light switch (on/off). It changes if somebody enters or leaves the room or switches the light on or off. In both examples the arrival rate is obvious much smaller than of a time triggered lambda source which sends 10 messages per second (TE = 100ms). In the next sections we will focus deterministic eventtriggered λ-sources. Their determinism is given by known physical processes in the building. To characterize them we analyze different continuous measurements and simulations.

(1)

The source arrival rate λsrco (t) will be examined in the next sections and the gain of the voi (t) of the λ-processors in Section 5.

4. λ-Sources 4.1. Further classification and simple cases Though a λ-source generates messages without stimulus at an input, the messages usually arise not random. The creation can be triggered by a timer or by a change at a sensor of the device. The second level classification scheme T- is then TT time-triggered λ-sources, TE event-triggered λ-sources. Time-triggered λ-sources produce a constant arrival rate. If the timer has an elapse time TE the arrival rate is exactly (2) λsrco (t) = TE−1 = const.

4.2. Continuous deterministic λ-sources Each continuous physical signal is transmitted discrete over a network and need therefore to be sampled. The classical control theory recommends equidistant sampling with a constant sampling period TD . Hence, the resulting

with no variance theoretically. The elapse time TE can be gathered from the device database or design database. Event-triggered λ-sources cannot be handled that easy. Their behavior depends mainly on the signal they observe. First, the signal value can be either a 3

arrival rate is equivalent to a time-triggered λ-sources with TE = TD . But, with equidistant sampling many redundant messages are generated if the sampled signal f (t) is not changing. In building automation systems the send-ondelta concept is used to avoid redundancies and reduce network load [11][17]. Using the send-on-delta concept a message is only created if the sampled signal f (t) changes more than a significant δ since the last transmitted message value f (t − Ti ). The condition to generate a new message after Ti is then δ ≤ | f (t) − f (t − Ti )| .

it is impossible to say which outside temperature we will have exactly one year later, but what we can state is, that it is usually the same as this year because it changes periodic with a known characteristic. To identify this characteristic dynamic we generalize the arrival rate for a universal prediction corresponding to the demand D1 and D3. The expectancy E[λ] is an adequate description. It can be estimated from the expectancy of the absolute rise E[| f 0 |] with  
E[λ] ≈λsrco E f 0    (5) 1 E[| f 0 |] 1 ; max ; ≈ min TL TU δ

(3)

In periods where no changes occur in the process no messages are exchanged. Due to long inter message times the receiver might suspect the device does not work properly. To avoid such a dubious situation, the parameter max-send-time TU ≥ Ti defines a maximum time between two messages. To reduce the load impact of a loose sensor contact a minimum inter message time can be defined with min-send-time TL ≤ Ti . These three parameters send-on-delta δ, min-send-time TL and max-send-time TU are influencing and limiting the inter message time Ti and therefore the arrival rate λsrco (t) = Ti (t)−1 . For a known continuous signal f (t) follows    δ Ti (t) = max TL ; min TU ; 0 | f (t)|    0
| f 0 (t)| 1 1 (4) λsrco f (t) = min ; max ; TL TU δ d f (t) with f 0 (t) = . dt

This approximation results from Equation (4) for a constant | f 0 (t)| = E[| f 0 |] and is valid for rises with a little variance D [| f 0 |] and on the condition of δ/TU  E[| f 0 |]  δ/TL and δ/TU  E[| f 0 |]  δ/TL . Where these limits arise from will be introduced later, please be patient. The expectancy of the absolute rise E[| f 0 |] can be estimated calculating the arithmetic mean of a series of samples { f1 , . . . , fN }   1 N | fn − fn−1 | E f 0 = ∑ N n=2 tn − tn−1

(6)

or using the time continuous rise f (t) for a time period [0, T ]   1 Z T 0 f (t) dt. (7) E f 0 = T 0 But, the expectancy E[λ] disregards bursts of messages which are neither uncommon for industry processes [7] nor in building automation. For instance, imagine what happen to all occupancy sensors in an office building if all employees come to work at the same time. All occupancy states will change at the same time and result in a burst of messages. The rest of the day nothing comparable happens. This example has two aspects of burst. First, each device has a period of many messages in the morning and less for the rest of the day independent of the other. This can be modeled by analyzing the frequency of each period. If it is unimportant when periods with many and less messages occur the frequency of each period results in the probability density functions (pdf) φ (λ) of the arrival rate. The second aspect is the superposition of bursts due to correlated causes like in our example a fixed time to start work. Such strong correlated situations stand out by a temporary raised arrival rate at all devices. We model them by special cases which focus a significant period of time, like the morning, with a multiplied arrival rate. We assume less correlated λ-sources as common case which makes it possible to estimate a generalized pdf φ (λ) of the arrival rate for classes of the same device without loss of generality. But how does the distribution affect the arrival rate and where arise the boundaries in Equation (5) from? In

To fulfill the demand D1 of a simple device model equidistant sampling and time-triggered λ-sources are included as special case in this λ-source model with TL = TU = TE = TA . State signals are mapped with the condition that | f 0 (t)| = |g0 (t)|. The send-on-delta δ needs to be whole-numbered in this case as the the state signal |g0 (t)| is. The parameters TL , TU , δ are obtained from the device database or from the design database if a nonstandard parameter is used. Default values can be taken further from standards [11]. If a parameter is not defined for an output the corresponding part of the equation is omitted with TU = ∞ or TL = 0. This leaves the estimation of the absolute rise | f 0 (t)| for the next section. 4.3. The absolute rise of continuous processes The last unknown is the time continuous process f (t). The bad news is, that it usually stays unknown for a future process as it is affected by too many disturbances to be predicted correctly. The good news is, that we are only interested in the dynamic | f 0 (t)| which has a characteristic behavior. Quasi, 4

0.15

4

λ(ω) φ(ω)

The expectancy of the arrival rate is therefore

Z∞

3.5

E[λ] =

3

−∞ Z∞

ω frequency

2.5 2 1.5

λ(ω) in Msg/s

0.1

= 0

λφ (λ) dλ =

Z∞

λ (ω) φ (ω) dω

−∞

(9)

  1 1 ω min ; max ; φ (ω) dω. TL TU δ 

0.05

The approximation (5) ignores that a part of the distribution it generalizes will be limited by TL−1 and TU−1 . Thats why it becomes incorrect when the expectancy get near them.

1 0.5 0 0

0.5

1

1.5

2 2.5 ω in units/s

3

3.5

0 4

0.15

φ(ω) (exp) φ(λ)

Figure 4. Exponential pdf φ (ω) and Equation (4) with δ = 1, TL = 0.5, TU = 5, µe = 1 frequency

0.1

Figure 4 the pdf φ (| f 0 |) = φ (ω) of an exponentially distributed rise ω = | f 0 | is displayed as a dashed line. The solid line represents the corresponding transfer function of the arrival rate. It can be calculated from Equation (4) with | f 0 (t)| = ω. The graph visualize the behavior of Equation (4). On the one hand it transfers the rise linearly to the arrival rate which is visible in the middle part of the curve. On the other hand it limits the result by the reciprocal min-send-time TL−1 as maximum and the reciprocal max-send-time TU−1 as minimum which applies to the shaded parts on the right and on the left.

0.05

0 0

λ = TU−1 TU−1 < λ < TL−1 . λ = TL−1 otherwise

1

1.5 2 2.5 ω in units/s, λ in msg/s

3

3.5

4

Figure 5. Exponential pdf φ (ω) and the resulting pdf φ (λ) ( δ = 1, TL = 0.5, TU = 5, µe = 1)

Now the pdf of the absolute rise represents the frequency of each absolute rise ω. This frequency will now be transposed to the corresponding arrival rate λ (ω). In the linear part this means only a linear shift by φ (λ) = φ (ω/δ). In the left shaded part, where the rise ω is not big enough and the max-send-time reacts faster, the arrival rate will be limited to TU−1 . This means that the frequencies of all values of ω ≥ δ/TU will superpose at TU−1 . The analog happens to the opposite case on the right when the send-on-delta produces more messages than the min-sendtime allows. Now the frequency of all ω ≤ δ/TL superpose at TL−1 . This results in the pdf φ (λ) of the arrival rate with

 P (ω ≥ δ/TU )     φ (ω = λδ) φ(λ) =  P (ω ≤ δ/TL )    0

0.5

4.4. Common cases In this section we will analyze typical values of the absolute rise | f 0 | to make the data available for the analytical solution. But before it is necessary to define the last classification scheme. The introduction already mentions a correlation of the arrival rate with the bus load in closed-loop controls. This correlation emerges not only at high load. For example, a sensor measures a sinking lighting and notifies the lighting control about it. In response the controller raises the light in the room using a lamp. The new lighting value is again transmitted by the sensor to the controller. So the second telegram send by the sensor can be interpreted as reaction on the first one. Consequently, the arrival rate of the sensor depends partly on itself even if actually the dynamics of the physical value determine it. We examine this special case separately and call it λ-circles. Open loop controls or other switch chains are substituted by λ-sequences in which the arrival rate of a λsource does not depend on the arrival rate of any λ-sink which is following causal. These two cases form the classification scheme C-

(8)

It has a skewed characteristic of the rise between TU−1 < λ < TL−1 and combines the truncated densities in two point distributions at the limits λ = TU−1 and λ = TL−1 (Figure 5).

CS λ-sequence and CC λ-circles. 5

We will first examine the λ-sequences and then detail the λ-circles.

Value 1. Sky covering 2. Wind direction 3. Wind speed 4. Precipitation 5. Air pressure 6. Temperature 7. Rel. Humidity 8. Direct radiation 9. Diffuse radiation 10. Lighting

Unit ◦

m/s mm/h hpa ◦C W/m2 W/m2 lx

max ω 1.39 −04 9.72 −02 2.69 −03 1.92 −03 7.50 −04 1.17 −03 5.83 −05 1.08 +01 2.48 +00 1.12 +03

E[ω] 1.67 −05 1.73 −02 2.90 −04 1.36 −05 9.00 −05 1.85 −04 8.45 −06 9.70 −03 5.38 −03 1.33 +00

D2 [ω] 5.83 −10 6.45 −04 7.77 −08 4.71 −09 6.65 −09 3.04 −08 6.35 −11 2.83 −03 4.35 −04 3.32 +01

analysis of the pdf in Equation (9). It has the solution       1 δ δ δ E[λ] = P Ω≤ +I −I TU TU TL TU   1 δ + P Ω≥ TL TL with ( x 1 − e− µe 0 ≤ x P (Ω ≤ x) = 0 otherwise ( x 1 − (µe + x) e− µe 0 ≤ x I (x) = δ 0 otherwise

(10)

under the assumption of an exponential distributed absolute rise. For λ-sources we assume that the signal is send-ondelta sampled. In this case the arrival rate depends on the dynamic of the close-loop system beneath. The stability of such systems is difficulty to estimate ([8]) as it is altered by the dead time (transport time) impact of the bus. Further, changes of the setpoint and disturbance affect the dynamic. We used a simulation of a single room to investigate the coherences. First results indicate that the distribution of the rise of a stable control loop resemble exponential. This is comprehensible as the dynamic is ruled by the disturbances which are mostly external changes of weather. For example the lighting of a room depends often on the sun outside and if a cloud dims the sun it becomes darker in a room. But the outside lighting is reduced by the number, size and position of windows, the kind of glass used and the attached sunblind that can be open or close. These options render a general statement about the dynamic impossible. But considering D1 we assume that a value in a stable control loop has the same or a lower dynamic as the greatest disturbance which is usually the corresponding external weather value from Table 1. With this worst-case-assumption the resulting arrival rate is at least larger than the realistic one and an underestimation is prevented.

Table 1. Characteristics of weather values (min ω = 0)

Weather values are very representative λ-sequences. On the one hand they are independent of any actor in the building and on the other hand they have a large influence on the internal values, like the temperature or lighting of a room. That’s why we studied the measurement of one year for 10 weather values. Table 1 lists the resulting characteristics of the rise ω. The pdf of the absolute rise of the examined weather values is similar to an exponential distribution (compare Figure 6). This indicates that small rises and idle periods are much more common than large steps. The advantage of the exponential distribution is that it has only one degree of freedom the expectancy µe . It can be set corresponding to the expectancy of the rise µe = E[ω] which is listed in Table 1 or easily approximated by a user according to Demand D3. Now the easiness of the approximation (5) can be combined with the accurate

5. λ-processors A λ-processors generates messages at an output o in reaction on arriving messages at an input i. If an arriving message always results in a new created message, the gain between input i and output o

frequency

1.5 1 0.5

voi = 0 0 1 2 x / xmean

3 4 1.

5.

7.

4.

6.

3.

9. exp.

2.

10.

λo = 1. λi

(11)

This constant gain can be assumed as a default for the most event-triggered close and open loop controller or any data processing unit in the network. But some devices reduce the incoming arrival rate. A single-step-controller for example only activates its actuator signal if their input fi (t) exceeds a threshold So . The

8.

dataset

Figure 6. pdf of weather values 6

actuator signal is deactivated again if the input falls below the threshold. So, the output has two states ’on’ and ’off’ which corresponds to the logical results of the condition fi (t) > So . The input signal is thus transferred in a state signal go (t) = z ( fi (t)) which controls the output. The arrival rate can be computed from this state signal using Equation (4). By the same procedure the arrival rate can be estimated for every transfer function u ( fi ) which applies to an input signal fi . The send-on-delta, the min-send-time or maxsend-time can be parameterized according to the requirements. The following Table 2 lists the gain voi (t) from an input i to an output o of typical λ-processors. The number of state changes of the condition C per time unit is expressed for simplification as H (C).

7. Case study As case study we use a single office room with an lighting control as shown in Figure 7. The lighting of a room results from internal light sources like lamps and the sun as external source.

voi (t) 0 but λsrco (t) =

1 TD

λ-source

1 1 TL ; max



1 TU

; | fo0 (t)| δ−1



λ-chain



d1

i1 i2

louver control l1

i3 i4

d4

i5

louver drive l5

d6

i6

l2

1 λi H ( f i (t) ≥ Si ) ≤ 1 d2

1 λi H (SiL

λ-sink

λ-processor

weather station

min

≥ fi (t) ≥ SiU ) ≤ 1

λ-circle

λ-processor time-triggered (period TD ) event-triggered (forwarding) send-on-delta (output fo = u ( fi )) single-stepcontroller (threshold Si ) multi-step-controller (SiU ≥ SiL )

The loss can be caused by many errors like collision or overflow of queues. Both probabilities can be generalized from simulations [14] or computed with a fix point approach in a later queuing analysis as performed in [1]. Finally, all device models in the network can be computed sequently using the introduced set of classifications and equations. It is best to begin with the λ-sources and then to follow the bindings over the λ-processors until the last λ-source is reached. Causal loops are uncommon so that the equations form a solvable linear system of equations. For simplicity it can be assumed that causal loops do not occur.

Table 2. Gain voi (t) and impact λsrco (t) for typical λ-processors

d5 i7

l3

illumination sensor d3

i9

i11

i10

l6

d7

i12

illumination control

lamp

i8 l4

occupancy sensor

Figure 7. Exemplary lighting control

Attention should be paid to the fact that the gain voi of a multi-step-controller is usually much smaller than the gain of an event-triggered PID-controller because they only send a value update if their input fi passes their threshold Si . Therefore, they can replace PID-controllers to reduce the network load.

In our example the external sunlight is absorbed by a sunblind d6 which is controlled by the louver control d4 . The louver control deactivates the sunblind during frost. It gets the outside lighting and temperature from the weather station d1 . A second lighting sensor d2 is installed in the office. It measures the lighting in the room every 100ms and sends the value to the lighting control d5 . It will correct a reduced lighting using the lamps d7 as long as at least one person occupies the room which is detected by the occupancy sensor d3 . The device models of the λ-sources are (see Equation (1))

6. Consolidation of the model With the gain from Section 5 and the source arrival rates from Section 4 the device model of Equation (1) can be solved completely. Only the arrival rate at the variable inputs needs to be determined. This can be done by using the binding information stored in the design database. A binding represents a connection between an output and input variable. In case of a lossless transmission each message generated at an output will arrive at the input and the arrival rates of the two bound variables are equal λi = λo ,

if o is bound to i.

d1 : λi1 = λsrc; i1 , d2 : λi2 = λsrc; i2 , d3 :

The occupancy sensor is a state signal stochastic eventtriggered λ-source (TE/ES/SS). It sends two states (occupied/free room) to the lighting control. However, the occupancy depends on the arbitrariness of the human and therefore it is modeled stochastically.

(12)

In case of a lossy bindings with poi the probability of loss the arrival rate at the input is λi = (1 − poi ) λo ,

if o is bound to i.

λi7 = λsrc; i7 , λi8 = λsrc; i8 .

(13) 7

Let’s assume the office is occupied between 8.00 and 18.00 o’clock and it changes in average 3.4 times a day. It ¯ of 3.4 Messages per day (Msg/d) results in an average λ following

the assumption of exponential distribution supplies better results especially for the temperature where the assumption is met better. The Equation (9) produces the best results compared to the simulation, not surprisingly as it uses higher moments. The last row in Table 4 further compares the calculation times. The approximations (5) and (10) provide of course the fastest processing. More time-consuming are the extensive analysis from a pdf using Equation (9) and the simulation. The arrival rate of the λ-processors can be calculated using the device models from Equation (1)    vi5 , i3 λi3 d4 : λi5 = + λsrc;i5 ,  vi5 , i4  λi4  vi11 , i9 λi9 d5 : λi11 = + λsrc;i11 . vi11 , i10 λi10

¯ src; i = 3.4Msg/d = 3.94 −5 Msg/s. λ 8 The lighting sensor is a time-triggered λ-source (TT) with λsrc; i7 = 100−1 Msg/ms = 10Msg/s. The fact, that the lighting sensor forms a λ-circle with the lighting control and lamp, has no influence in this example. But, if the lighting sensor in the room would be replaced with a send-on-delta sampled sensor it would have a comparable dynamic like the lighting sensor outside in the weather station with the same parameterization. The remaining weather station d1 is a continuous, deterministic event-triggered λ-source (TE/ED/SC). An examined test device had the standard parameterization listed in Table 3 which has been extracted from the device database. Max-send-time Min-send-time Send-on-delta

Temperature i1

Lighting i2

TU; i1 = 60s TL; i1 = 20s δi1 = 0.01◦ C

TU; i2 = 60s TL; i2 = 20s δi2 = 20Lux

The louver and lighting control work both event-triggered and have a gain according to Table 2 with vi5 , i3 = vi5 , i4 = vi11 , i9 = vi11 , i9 = 1. Further they contain no source so that λsrc;i5 = 0,

In our example we derived the bindings from Figure 7 and define the arrival rate of two bound variables (e.g. i1 → i3 ) as equal and get

Table 3. Parameterization of an examined device

λi3 = λi1 , λi4 = λi2 , λi6 = λi5 , λi9 = λi7 , λi10 = λi8 , λi12 = λi11 . Using the results of the pdf from Equation (9) in Table 4 we receive finaly

To compare the introduced methods the arrival rate will be estimated from the weather values in Table 1 using the approximation (5), the Equation (10) with assumption of exponential distribution and (9) with the pdf φ (ω) of the analyzed weather values. The results are compared to a simulation which implements the send-on-delta-concept. It has been verified against the mentioned devices and can process a dataset of one year in a few seconds. Table 4 contains the results of the calculation and simulation. ¯ src in Msg/s λ Temperature i1 Lighting i2 Calc. time in s

eq. (5) 0.0185 0.0500 0.0

eq. (10) 0.0229 0.0370 0.0

eq. (9) 0.0227 0.0277 0.35

λsrc;i11 = 0.

λi6 = λi5 = λi3 + λi4 = λi1 + λi2 = 0.0554Msg/s λi12 = λi11 = λi9 + λi10 = λi7 + λi8 = 10Msg/s. From the resulting arrival rate we can now approximate for example the load of the channel. Let’s say all Messages have the same size of 10 Bytes with unacknowledged message service ([15]) than we can superpose the rates induced in the channel to the channel load L
L = λi1 + λi2 + λi5 + λi7 + λi8 + λi11 10Byte = 20.11 · 10Byte/s = 1608.87bit/s.

Sim. 0.0226 0.0280 17.9

This equals a utilization of 2% of a common 78kbit/s bus.

8. Conclusion

Table 4. Comparison of the results for different methods

We proposed in this paper a traffic model for networked devices in the building automation. It should be further useable for automated modeling and so three demands have been established in Section 1. The traffic model uses a generic device model which has been introduced in Section 3 and satisfies demand D1. Using the classification scheme L–, T–, E–, S– and C–

The quality of the results show the same trend though the temperature and the lighting have very different dynamics (compare Figure 6). The approximation (5) has in both cases the greatest differences to the simulation which is assumed to be realistic. The Equation (10) with 8

the most suitable method to estimate the arrival rate can be chosen for each device by simple decisions. This fulfils demand D2. The model further offers multiple levels of detail. The given time continuous model enables very precise modeling for example for simulation or model based diagnostic. Otherwise, the approximation (5) offers a rough estimation. We commonly use the pdf approximation (9) to predict the load as it combines fast processing and high detail. Besides, we offer the exponential approximation (10) for easy and accurate manual input as each user can estimate the mean rise. All necessary specific parameters can be extracted from the design database which satisfies the demand D3. As all conditions are complied with, the proposed approach enables automatic modeling and subsequent detailed analysis. The traffic model generator is implemented and running. As the test with simulations succeeds further work will now deal with the validation on existing networks.

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9. Acknowledgement The project the present report is based on was promoted by the Federal Ministry of Education and Research under the registration number 13N8177. The authors bear all the responsibility for contents.

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