=Paper=
{{Paper
|id=Vol-2853/paper52
|storemode=property
|title=Models of Distributed Systems Testing Based on Energy Consumption in Behavior
|pdfUrl=https://ceur-ws.org/Vol-2853/paper52.pdf
|volume=Vol-2853
|authors=Oleksandr Martynyuk,Lyudmila Sugak,Hanna Stepova,Victor Antoniuk,Igor Mileiko
|dblpUrl=https://dblp.org/rec/conf/intelitsis/MartynyukSSAM21
}}
==Models of Distributed Systems Testing Based on Energy Consumption in Behavior==
https://ceur-ws.org/Vol-2853/paper52.pdf
Models of Distributed Systems Testing Based on Energy
Consumption in Behavior
Oleksandr Martynyuk a, Lyudmila Sugak b, Hanna Stepova c, Victor Antoniuk d, Igor Mileiko e
a
Odessa National Polytechnic University, Ave. Shevchenko 1, 65044 Odessa, Ukraine
Abstract
Deviations from the standard power consumption often indicate malfunctions of technical
systems. Continuous and periodic, instant and cumulative special analysis of energy
consumption in the behavior of distributed information systems (DIS), compared with the
features of the procedures they execute, can be an essential part of the means of online and
offline testing in operability of DIS. This paper presents a model of behavioral testing and
recognition of DIS, including a formal analysis of energy consumption. The model is based
on behavioral experiments of an automata class and Petri nets, representing a reference DIS
with added characteristics of energy consumption. The model has the features of behavioral
individual and group energy characteristics, cross-sections-states, counters, spatial and
temporal structures of energy consumption. The individuality of the characteristics
presupposes the possibility of identifying the states of the reference model by its energy
consumption, section-states are the subsets of its attainable states/positions identified in the
reference model with characteristic (identifying) energy costs, counters are moving chips,
that register energy consumption, as they move, and finally, energy consumption structures
supposed and achievable topological structures of the movement of counters in the graph of
the behavioral model.The model is designed to determine the energy conditions, used in the
construction of methods of online and offline behavioral testing of the DIS performance,
extended by the analysis of energy consumption.
Keywords 1
Energy consumption, behavioral testing, energy characteristics and identifiers, energy
consumption patterns, energy consumption check
1. Introduction
Energy consumption analysis plays a significant role in monitoring the performance of discrete
systems (DS) [1-3]. Deviations upward or downward from the respectively upper or lower boundary
statistical energy consumption [4-6] may indicate errors, failures, in the operation of the hardware and
software of the DS [7-9], incorrect actions, including unauthorized access and viral attacks [10, 11]. It
is known, that the main power consumption in the DS equipment occurs during state switching [12,
13]. Accordingly, the minimum, average and maximum reference values of energy consumption for
various elementary indivisible commands and micro-operations, executed by them can be determined
for hardware elements and nodes of the DS [14-16]. These values can be used, as the basis for
defining energy state diagrams and their cross-sections (vectors) for firmware of DS equipment, and
on their basis - more general state diagrams and their sections for programs and scenarios of DS
operation. The behavioral nature of microprograms, programs and scenarios of the DS makes it
possible to form quite complex spatial-temporal structures of energy state diagrams, their sections and
IntelITSIS’2021: 2nd International Workshop on Intelligent Information Technologies and Systems of Information Security, March 24–26,
2021, Khmelnytskyi, Ukraine
EMAIL: anmartynyuk@ukr.net (O. Martynyuk); lpsugak@ukr.net (L. Sugak); hanna.suhak@gmail.com (H. Stepova); antoniuk@opu.ua (V.
Antoniuk); mig3@ukr.net (I. Mileiko)
ORCID: 0000-0003-2366-1920 (O. Martynyuk); 0000-0001-5624-7633 (L. Sugak); 0000-0002-7223-8822 (H. Stepova); 0000-0003-0427-
9005 (V. Antoniuk); 0000-0002-8664-7037 (I. Mileiko)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
traces, correlated with the corresponding behavior structures of the microprograms, programs and
scenarios themselves and their derivatives. Accordingly, behavioral analysis, offline and online
testing acquires additional verification capabilities, based on the analysis of events and actions of the
DS together with the analysis of their energy consumption. In the case of an implicit assignment of
the class of properties to be checked, as deviations from the reference values, such behavioral
analysis, offline and online testing can perform research and recognition of such reference values,
both of the events and actions themselves, and their energy consumption.
2. Purpose and objectives of the article research
The purpose of this work is to increase the completeness of the behavioral offline and online
testing of DIS through additional monitoring and analysis of energy consumption in the process of
their main and test functioning. To achieve the goal, the following tasks have been solved: firstly,
studies of known approaches to the analysis of energy consumption for structural and functional
models of DS; second, the development of a decomposition model for analyzing energy costs, based
on an extended Petri net, which has the features of the spatial and temporal structures of localized and
moving energy variables and chips, which determines the conditions for the use of energy costs in
behavioral analysis, check and recognition of DIS; third, the construction of the main steps of
procedures for network and behavioral energy analysis, check and recognition of DIS; fourth, testing
these basic steps for the DIS components.
3. Related works
As you know, the main for the high power consumption of existing equipment are dynamic modes
of operation associated with current transient switching processes between steady states, accumulation
and consumption of electrical energy [5, 6]. A significant part of this energy is released in the form of
heat contact (on various radiators), air-convection and in the form of infrared radiation, and its value
is directly proportional to the average values of the electric currents. That is, the power consumption
of the DS equipment can be analytically estimated using special formal switching models [7-9],
explicitly determined using differential and integral current measurements, integral measurements of
temperature and infrared radiation [5, 6].
Asynchronous, synchronous, clocked single-cycle and multi-cycle devices, both without memory
and with memory, allow you to define variables, their sets, commands, micro-operations, instructions,
operations and functions based on them, for which power consumption is essential, as noted above,
transient processes of all switching, associated with both input variables, synchronizations,
parameters, data/operands, and internal/output variables, data, results. Depending on the
manufacturing technologies for the hardware of the DS, for example, FPGA [17-19], their modes of
operation, for example, critical [20-22], approximation of calculations [23-25], circuit solutions, bit
width, coding, values of commands and operands, it is possible to determine these switches, their
approximate minimum, average maximum numbers with corresponding values of power consumption
[5, 6].
Moreover, for single-cycle devices or devices with low-cycle commands, micro-operations,
instructions, operations and functions, the number of switching transients associated with
internal/output variables, data, results are directly proportional to the number of switching transients,
associated with input variables, synchronizations, parameters, data/operands [7-9].
This circumstance makes it possible in this case to simplify the analysis of energy consumption,
reducing it to an analysis of the numbers of all input switching.
For multi-cycle devices, devices with multi-cycle instructions, micro-operations, instructions,
operations and functions, programs and scenarios for the operation of the DS hardware, considered
indivisibly, the transient switching processes are determined to a greater extent by the sequential logic
of their execution, than by the indicated input switches [17, 21].
When analyzing the energy consumption of multi-cycle devices, a time decomposition behavioral
approach is advisable, based on: firstly, dividing/splitting multi-cycle instructions, micro-operations,
instructions, operations and functions, programs and scripts into simpler low-cycle micro-operations
instructions, instructions, operations and functions; secondly, their temporal sequential behavioral
representation in the form of graph structures, in particular, basic ones – chains, trees, hammocks,
feedback loops, with standard procedures for determining their energy consumption, reduced to the
total (reduced) minimum, average maximum reference values of input switching for one- or few-cycle
instructions, micro-operations, instructions, operations and functions, that make up these structures.
The main features of DIS, in addition to being distributed, include autonomy, asynchrony, greater
separability, the predominance of interactions based on packet switching and messaging, as indivisible
transactions [26, 27], the presence of memory, time delays, lower reliability of functioning [28, 29], a
wide range of power consumption [30]. In the analysis of DIS energy consumption, in addition to these
features, it is essential to take into account the sharply increased program-behavioral, intellectual
implementation of many functions [31-33], which determines the appropriateness of the combined
asynchronous-event, formal-abstract [34], intellectual [35-37], decomposition structural-spatial [38, 39]
and sequential-temporal [40, 41] approaches based on the network and functional nature of DIS
behavior.
Petri nets [42-44], extended with the properties of energy consumption [45], can be quite adequate
formal models in monitoring the performance of DIS, in particular, with the analysis of their energy
consumption, taking into account the noted features.
4. Development of a decomposition model for analyzing energy costs
The functioning of DIS components is accompanied by a certain power consumption, which
depends, as noted above, on the technologies used for their implementation and operating modes. At
the same time, the main emphasis in the analysis during the operation of the DIS can be made on the
characteristics of energy consumption: firstly, instantaneous local energy structures (sections) for the
components of the spatial architecture of DIS and various compositions of these components,
essential for general or special analysis of the system's performance; secondly, dynamic energy
structures (traces) for DIS components, related by direct behavioral interactions in various DIS work
procedures; thirdly, the characteristic dynamic energy structures (traces) of the second type, which
make it possible to identify the reference behavior of DIS components by their energy consumption.
4.1 Local energy properties and instant energy consumption state
diagrams
Petri nets make it possible to efficiently represent asynchronous parallel processes of DIS
functioning, including obtaining an analysis of additional temporal, probabilistic and energy
characteristics, based on extensions [45].
In particular, the extended Petri net, which introduces the characteristics of energy consumption
for events in positions, actions in transitions and transfers in tokens, has the form [45]:
S(f)E=(P, T, X, Y, Ep, Et, F, S, M0), (1)
here P, T are the sets of positions, transitions, respectively, X, Y are the input and output alphabets,
respectively, for events and actions, Ep, Et are the sets of energy consumption of positions and
transitions, F are the incidence relation of positions and transitions, S are the correspondences of
events, actions, energy consumption to positions and transitions, M0 - initial marking.
In this model, the energy consumption for the minimum, average and maximum reference values
of the reduced numbers of one- and multi-circle switching of the events, that have occurred in the
positions, and the actions, performed in the transitions, is localized in the components of the DIS and
can be represented in the form of instantaneous energy structures and their compositions based on the
spatial architecture DIS.
So, on the basis of the entire Petri net S(f)E or some Petri subnet for some of its special
components, as their sub-models, diagrams of attainable instantaneous (at a certain moment of time t)
energy states or markings (sections), marking the above events can be formed positions and transition
actions, type:
Spt ={(pi1,epi1),(pi2,epi2),…, (pik’,epik’),…,(pik,epik)} ≅ {(xi1,epi1),(xi2,epi2),…,(xik’,epik’),…,(xik,epik)}
Stt={(tj1,etj1),(tj2,etj2),…,(tjk’,etjk’),…,(tjk,etjk)} ≅ {(yj1,etj1),(yj2,etj2),…,(yjk’,etjk’),…,(yjk,etjk)} (2)
For a markup diagram and some pair (pik’,epik’), where k’∈{1,2,…, k}, {i1,i2,..,ik}⊆ {1,2,…,nP} and
nP=|P|, the position pik’ is assumed to be ready, that is, the accomplished event xik’ of this position with
the energy consumption epik’, hence (pik’,epik’) ≅ (xik’,epik’). In the case for some pair (tjk’,etjk’), where
k’∈{1,2,…, k}, {j1,j2,..,jk}⊆{1,2,…,nT} and nT=|T|, similarly, the transition tjk’ is assumed in the
completed state, that is, the perfect action yjk’ of this transition with energy costs etjk’, that is, (tjk’,etjk’) ≅
(yjk’,etjk’).
The specified characteristics and relations of the adjacency of energy states in these diagrams
represent the values of the energy consumption of DIS, as well as their dependences and
transformations, selected for analysis in the Petri net model S(f)E.
For DIS components, the minimum, average and maximum values of the reduced numbers of one-
and multiple-cycle switches associated with input variables, synchronizations, parameters,
data/operands and with internal/output variables, data, results for position events and energy state
transition actions, characterize them inherent instantaneous local energy consumption.
Moreover, the energy consumption of transformations between adjacent energy states of the
diagrams is not taken into account in these values of energy consumption. Energy state diagrams
allow one-time, repeatable (periodically) or event-driven execution of an instant local selective
analysis of energy consumption in the functioning of DIS in arbitrary, in particular, some critical
states (sections) of the modeling Petri net S(f)E or its subnetwork without connection with the rest of
the behavior.
For an analysis, that continues in time during unfolding for some points in time (t1, t2, …, tm), the
values of energy consumption for the corresponding selected energy states, it is also possible to form
diagrams of their sequences, having the form:
Sp=(Sp(t1)={(pi1(t1),epi1(t1)),(pi2(t1),epi2(t1)),…, (pik’(t1),epik’(t1)),…,(pik(t1),epik(t1))},
Sp(t2)={(pi1(t2),epi1(t2)),(pi2(t2),epi2(t2)),…, (pik’(t2),epik’(t2)),…,(pik(t2),epik(t2))},
…,
Sp(tm)={(pi1(tm),epi1(tm)),(pi2(tm),epi2(tm)),…, (pik’(tm),epik’(tm)),…,(pik(tm),epik(tm))})
St=(St(t1)={(tj1(t1),etj1(t1)),(tj2(t1),etj2(t1)),…,(tjk’(t1),etjk’(t1)),…,(tjk(t1),etjk(t1))},
St(t2)={(tj1(t2),etj1(t2)),(tj2(t2),etj2(t2)),…,(tjk’(t2),etjk’(t2)),…,(tjk(t2),etjk(t2))},
…,
St(tm)={(tj1(tm),etj1(tm)),(tj2(tm),etj2(tm)),…,(tjk’(tm),etjk’(tm)),…,(tjk(tm),etjk(tm))}) (3)
Diagrams of energy states of modeling Petri nets S(f)E and sequence diagrams based on them can
be used to partially check the operability and correct functioning of DIS components during selective
monitoring and analysis of their energy consumption.
4.2 Moving energy properties and counters, diagrams of energy
consumption behavior
Petri net extensions make it possible to determine dynamic behavioral energy structures (traces) of
energy states (sections), formed by energy-loaded counter counters when modeling the Petri net S(f)E
and associated with the direct functional interaction of DIS components in various DIS operation
procedures.
Model (1), in addition to monitoring and analyzing the values of energy consumption in energy
states, during the formation of events in positions and performing actions in transitions for a static
relation of incidence F and correspondence S, allows monitoring and analysis of dynamic marking
and accumulation of the reduced values of energy consumption in chips-counters, moved when
modeling the Petri net S(f)E, including comparing them with the minimum, average, maximum
reference values.
On the basis of tokens, a set of three-level hierarchies of possible dynamic energy inputs is formed
from the set mPr={mpr1, mpr2,… mprnp} for the objects of the Petri net behavior S(f) - from the
component Petri nets (hierarchy roots) through graph-topological energy structures (hierarchy nodes -
chains, trees - branching and convergence, hammocks, cycles) to positions and transitions of the Petri
net S(f)E (leaves of hierarchies - nodes of the Petri net) of the form [45]:
mpri={(rooti,{(nodei1,{leafi11,leafi12,…,leafi1L1,}),(nodei2,{leafi21,leafi22,…,leafi2L1,}),…,
(nodeiN1,{leafiN11,leafiN12,…, leafi2N1L1,})})}.
Leafi={leafi11,leafi12,…,leafi1L1,}∪{leafi21,leafi22,…,leafi2L1,}∪…∪{leafiN11,leafiN12,…, leafi2N1L1,},
Nodei={nodei1}∪{nodei2}∪…∪{nodeiN1}.
leafij(p)=M(p)=pr2(S(p,x’,ep’))=ep,
leafij(t)=pr2(S(t,y’,et’))=et, (4)
here i∈{1,2,…,np}, for: a) the position p∈P, the current leaf token leafij(t)∈Leafi; b) transition t∈T of
the Petri net S(f)E; c) x’, y’ and ep’, et’ - respectively the events of the position, the transition action,
the previous value of the energy consumption in the leafij(p) tokens that came to the position, p,
leafij(t) the transition t; d) ep, et - values of energy consumption of position p, transition t.
For the initial token leafij(p)0∈Leafi0 of position p∈P in the initial state of the Petri net S(f)E,
leafij(p)0=M0(p). For the node token nodeij∈Node of the graph-topological structure, the root token
rooti of the hierarchy, and the entire Petri net S(f)E in its initial or current state, there is a possible
accumulation of values immediately lower in the hierarchy of tokens [45]:
nodeij=leafij1+leafij2+…+leafijLj,
rooti=nodei1+nodei2+…+nodeiNi
PNEnergy=root1+root2+…+rootR (5)
With an analysis continuing in time during unfolding for some points in time (t1, t2, …, tm), the
values of energy consumption for the corresponding selected graph-topological energy structures, as
well as for energy state diagrams, it is possible to form sequence diagrams. that have the form:
nodeij(t1)=leafij1(t1)+leafij2(t1)+…+leafijLj(t1),
rooti(t1)=nodei1(t1)+nodei2(t1)+…+nodeiNi(t1),
PNEnergy(t1)=root1(t1)+root2(t1)+…+rootR(t1),
nodeij(t2)=leafij1(t2)+leafij2(t2)+…+leafijLj(t2),
rooti(t2)=nodei1(t2)+nodei2(t2)+…+nodeiNi(t2),
PNEnergy(t2)=root1(t2)+root2(t2)+…+rootR(t2),
…,
nodeij(tm)=leafij1(tm)+leafij2(tm)+…+leafijLj(tm),
rooti(tm)=nodei1(tm)+nodei2(tm)+…+nodeiNi(tm),
PNEnergy(tm)=root1(tm)+root2(tm)+…+rootR(tm). (6)
The energy-loaded model defines the conditions for monitoring, analyzing and checking the
correctness of energy consumption. The possibility of energy consumption in the Petri net S(f)E is
determined by the possibility of executing graph-topological structures, as the possibility of executing
all the events necessary for them in positions and actions in transitions. This possibility of execution
is determined on the basis of reaching their three basic states of positions and transitions - waiting,
readiness, execution - in accordance with the mandate (wand) functions of the tokens, moved during
the operation of the Petri net S(f)E. In the Petri net S(f)E, some graph-topological energy structures can
reach the state of partial execution or not execution at all due to the peculiarities of the current
functioning or the incorrectness of the Petri net S(f)E itself. In this case, for the executable elements
Nodei’, Leafi’, mpri’, the inclusion ∅ ⊆ Nodei’⊆ Nodei, ∅ ⊆ Leafi’⊆ Leafi, ∅ ⊆ mpri’⊆ mpri operates.
Then it is true:
nodeij’=leafij1’+leafij2’+…+leafijLj’≤ nodeij,
rooti’=nodei1’+nodei2’+…+nodeiNi’≤ rooti,
PNEnergy’=root1’+root2’+…+rootR’≤ PNEnergy, (7)
Thus, the possible minimum, average, maximum reference values of energy consumption of
graph-topological energy structures in the achieved behavior of the Petri net S(f)E are not exceeded.
Three-level hierarchies of possible dynamic energy consumption of modeling Petri nets S(f)E and
sequence diagrams, based on them, can be used to partially check the operability of DIS components
during selective monitoring and analysis of their energy consumption.
4.3 Spatial and temporal energy structures and identifiers
Behavioral dynamic energy characteristics and structures are part of the behavioral fragments of
the functioning of the Petri net S(f)E, special behavioral characteristic types of which, for example,
presented in the input-output alphabets of events and actions of the Petri nets S(f), can be used to
check and recognize the performance and the correct functioning of the DIS. Therefore, as part of
such characteristic types of behavior - identifiers, check and recognition primitives and fragments of
modeling Petri nets S(f)E [45], complementing them, reference dynamic energy characteristics and
structures can be used to increase the completeness of such behavioral check and recognition.
Obviously, even with correct input-output characteristic fragments of the functioning of the DIS,
the appearance of deviations in the values of the dynamic energy characteristics and structures from
the reference ones, for example, below the minimum and above the maximum values, may indicate
the beginning of the hardware degradation of DIS, leading to disruption of its functioning and
performance.
Moreover, outside the direct correspondence to the input-output characteristic types of behavior of
the modeling Petri nets S(f), behavioral dynamic energy characteristics and structures considered
independently, in a number of cases, can form energy characteristic projections of behavior - energy
identifiers, check and recognition primitives and fragments (special energy graph-topological
structures), represented in the energy alphabets Ep, Et, Em of energy consumption values for events in
positions, actions in transitions, counters in tokens, respectively [45].
The energy alphabets of the Petri net S(f)E can be expanded by combining multiple energy inputs
of events, actions, counters, as a result, extended energy alphabets Ep '= N×Ep, Et' = N×Et and Em '=
N ×Em. The expansion of energy-alphabets allows us to determine the sets of words starting and
ending for combinations of energy consumption of events and actions, let e be the zero step, then:
WE'=WPT'∪ WPP'∪ WTP'∪ WTT', WPT'=(Ep'×Em'×Et')*∪{e}, WPP'=((Ep'×Em'×Et')*∪{e})×Ep',
WTP'=Et'×((Ep'×Em'×Et')*∪{e}), WTT'=Et'×((Ep'×Em'×Et')*∪{e})×Ep'. (8)
Let WE’={w1E’, w2E’,..., wkwE’} be the set of sequential multiple-linear words not marked by
positions and transitions of energy behavior, which is reflected in the energy projection of the
incidence relation F of the Petri net S(f)E. For some wj’∈W’ in the input-output alphabet X'=N×X×Ep
и Y'=N×Y×Et [45] there exist p1,p2∈P and t1,t2∈T, such that either F(p1,wj’)=p2, or F(p1,wj’)=t2, or
F(t1,wj’)=p2, or F(t1,wj’)=t2, and pr3(wj’)=wjE’, that is, the energy word wjE’ is the component wise
third projection of the complete input-output word wj’ of the Petri net S(f)E.
Let PrE={pr1uE, pr2uE,…, prku’E}={PrEp∪PrEt} be the set of verified energy properties of the form of
third projections from, respectively, quadruples and triples of common verifiable properties Pr,
obtained in [45] for the relation F and the quotient, included in F, matching S, that is:
PrE⊆ pr3(F:(P×(X×Ep)→Τ)∪(T×(Y×Et)→P))∪pr3(S:((P→X×Ep)∪(T→Y×Et)))=Ep∪Et. (9)
The first, particular approach to the definition and construction of energy-identifiers is based on
the narrowing of the general identifiers of positions/transitions of the Petri net S(f)E.
Let the set of candidates for energy identifiers be defined as a mapping ϖ based on projections of
common identifiers of the form tijkpp→=(pjtikp,∪jtikip=1kpwjtikipp→’), tijkp→p=(∪jtikip=1kpwjtikip→p’,pjtikp),
tijktt→=(tjtikt,∪jtikit=1ktwjtikitt→’), tijkt→t=(∪jtikit=1ktwjtikit→t’,tjtikt) ∈ Ti, obtained earlier in [45] as two
positions/transitions and their characteristic behavior, that is, the mapping ϖ has view:
eijkpp→=ϖ(tijkpp→)=(pr1(tijkpp→), pr3(pr2(tijkpp→))=(pjtikp, pr3(∪jtikip=1kpwjtikipp→’)=(pjtikp, ∪jtikip=1kpwjtikipp→E’),
eijkp→p=ϖ(tijkp→p)=(pr3(pr1(tijkp→p),pr2(tijkp→p))=(pr3(∪jtikip=1kpwjtikip→p’),pjtikp,)=(∪jtikip=1kpwjtikip→pE’),pjtikp),
eijktt→=ϖ(tijktt→)=(pr1(tijktt→), pr3(pr2(tijktt→))=(tjtikt, pr3(∪jtikit=1ktwjtikitt→’)=(tjtikt, ∪jtikit=1ktwjtikitt→E’),
eijkt→t=ϖ(tijkt→t)=(pr3(pr1(tijkt→t), pr2(tijkt→t))=(pr3(∪jtikit=1ktwjtikit→t’),tjtikt,)=(∪jtikit=1ktwjtikit→t E’),tjtikt,), (10)
The set of energy identifiers Ei={ei1ti, ei2ti,…, eikti}⊆ Eijkpp→∪ Eijkp→p ∪ Eijktt→∪ Eijkt→t assumes for
each eikti 'the property of injectivity, that is, ∀eikti’∈ Ei (|ϖ -1(eikti’)|=1).
Each energy-identifier remains uniquely incident to corresponding position pjtikp and transition tjtikt.
In the second approach, energy-identifiers are determined out of relation to common identifiers on
the basis of their definition of the energy characteristic neighborhood of some position pjtikp or
transition tjtikt. In this case, despite a different synthesis procedure, for example, based on the Rabin-
Scott energy automata, they have the form:
eijkpp→=(pjtikp, ∪jtikip=1kpwjtikipp→E’),
eijkp→p=(∪jtikip=1kpwjtikip→pE’),pjtikp),
eijktt→=(tjtikt, ∪jtikit=1ktwjtikitt→E’),
eijkt→t=(∪jtikit=1ktwjtikit→t E’),tjtikt,), (11)
It should be noted that, in contrast to the functional, input-output representation of DIS behavior
based on automata models, for example, Petri nets S(f), its energy representation in most cases is not
provided by an independent property similar to automata minimality. Consequently, the uniqueness of
only energetic behavioral characteristic neighborhoods of positions and transitions is more difficult to
achieve.
Hence, it is possible to conclude, that monitoring and analysis of dynamic energy characteristics
and structures in the functioning of the DIS, even outside their direct correspondence to the input-
output characteristic types of behavior of the modeling Petri nets S(f), can be used for energy-
consuming check and recognition of the functioning and operability of the DIS. However, it is more
expedient, as noted above, to construct and use energy-identifiers, check and recognition primitives
based on them in monitoring and recognizing the operability and correctness of the functioning of the
DIS in connection with common, input-output identifiers, check and recognizing primitives [37, 41,
45], as their addition, essential for the efficiency of the analysis.
5. Basic steps of procedures for energy analyze, check and recognition
Monitoring and analysis of energy consumption, check and recognition of the operability and
correctness of the functioning of DIS, extended by the analysis of energy consumption, as noted, can
be performed one-time by the operator, periodically, for example, during routine maintenance, event-
driven, when energy costs arise, that require an immediate response.
In the course of such analysis and check, it is possible to independently or comprehensively use
diagrams of local energy states (sections) (Figure 1) and dynamic behavioral energy graph-topological
structures, as well as sequence diagrams based on both of them, as part of more complex check and
recognition of the performance and correct functioning of the modeling Petri net S(f)E.
In any case, for monitoring, analysis, check and recognition of DIS, including energy, the main
steps of the preprocessor and initialization stages are possible, providing for the initial S(f) and energy
S(f)E modeling Petri nets:
1 General structural and functional analysis with the selection at the first step of the basic static
graph-topological structures as part of the base/antibase, nodes, chains, trees, hammocks,
feedback loops.
2 Determination of the local reduced minimum, average, maximum values of potential energy
consumption for all static graph-topological structures for S(f) based on the features of design
and technological solutions for the components of the simulated DIS with obtaining S(f)E.
3 Determination of the necessary diagrams of local energy states and potential dynamic
behavioral energy graph-topological structures, based on the features of DIS functioning.
4 Integration of the reduced minimum, average, maximum values of potential energy
consumption for diagrams of local energy states and potential behavioral energy graph-
topological structures.
5 Definition and integration of the tested functional properties and energy characteristics, as its
special behavioral energy graph-topological structures.
6 Construction and integration of behavioral input-output and energy identifiers, check and
recognition primitives.
7 General initialization of the placement of chips, initial values of information-parametric
structures, initial values of energy consumption of diagrams of local energy states and
potential behavioral energy graph-topological structures, selection of scenarios for monitoring,
analysis, check and recognition of DIS and their initialization.
The stages of monitoring, analysis, recognition, testing of DIS, including complex input-output
and energy, include the possible basic steps (Figure 2) for the energy modeling Petri net S(f)E:
1 Selection and registration of real input-output and energy behavior in the alphabets of S(f)E,
performed in a passive mode for monitoring, analysis, recognition, operational check of the
functioning of real DIS.
2 Selection and modeling of the reference input-output and energy behavior, based on the
reference S(f)E, performed for monitoring, recognition analysis, operational check for DIS
functioning.
3 Comparative analysis of the selected real and reference input-output and energy behavior in
the alphabets S(f)E, performed for monitoring, analysis, recognition, operational check of the
functioning of a real DIS.
4 Complex recognition of input-output and energy identifiers of reference positions/transitions
S(f)E, congruence of identically marked reference positions/transitions, determination of the
registered behavior, based on the congruence of neighborhoods of behavior of the congruences
positions/transitions.
5 Analysis of the completeness of check and recognition of input-output and energy behavior for
the functioning of DIS, as a cover of the check primitives of check and recognition of S(f)E.
6 Recognition and generation of confirmed connecting paths into S(f)E to uncovered check and
recognition primitives, respectively, in the input-output, energy behavior of DIS.
7 Generation of tests for input-output and energy behavior S(f)E, performed in the active mode
according to special procedures, based on deterministic, pseudo-random, evolutionary
optimization search and coverage for testing of the functioning of DIS.
Initial, intermediate and final execution conditions, precedence and compatibility relations,
internal and external parallelism, distribution and separability, exception and error handling for the
presented models and steps are determined, when constructing specific procedures and methods for
monitoring, analysis, recognition, online and offline testing, data structures (Figure 3) depending on
the features of the functioning of the selected class of real DIS, which is the subject of a subsequent
study.
6. Estimates and experiments for the basic steps of the procedures
Representations of Petri nets S(f) by list structures for |P|=np, |T|=nt, |M|=nm, n=np+nt+2nm (here
2nm are two fields with a label index of an energy-loaded type and their number), |Ev|=ne, |X|=nx,
|Ac|=na, |Y|=ny, where X⊆Ev and Y⊆Ac for the transition requires no more than nt memory cells
containing no more than 2np+1t+1a+2Addr (2np+4 conditional fields), for a position - no more than np
conditional memory cells containing no more than 2nt+1p+1e+2Addr (2nt+4 conditional fields). Here
ip is the field with the position index, it is the field with the transition index, 2im are the fields with the
energy load and the number of label instances, ie is the field with the event index, ia is the field with
the action index, 2iAddr are the fields with the address of the next and previous cells list. The upper
bound for the number of conditional fields is defined as:
cS(f)=nt(2np+1t+1a+2m+2Addr)+np(2nt+1p+1e+2m+2Addr)=
=4npnt+(2m+2Addr)(nt+np)+1tnt+1ant+1pnp+1enp ≅ 4npnt+6(nt+np) . (12)
The formulas for the limiting case for the number of conditional cells and the maximum length
when searching in the graph for attainable states of the Petri net S(f) have the form:
cAS(f)= dAS(f)= 2*6ntnp(nt)+2*6npnt(np)=12npnt(np+nt),
cAS(f)multi= dAS(f)multi= 6nt(np2nt+1-1)+6np(nt2np+1-1)=3nt(np2nt+2-2)+3np(nt2np+2-2). (13)
The representation of the Rabin-Scott automaton RSS(f) differs in the order and multiplicity of
positions, transitions, labels, events and actionsб associated with them. In the general case, the upper
bound on the number of fields in conventional cells is not more than:
CRSS(f)p-n=min((2np+1)*min((2np-1), nt),(2np+1)*(2nt-1))=(2np+1)*min(min((2np-1), nt),(2nt-1)). (14)
The total estimate of the fields does not exceed:
CRSS(f)pet-n= (2np+1)*min(min((2np-1), nt),(2nt-1))+Σi=0np(3np*nti). (15)
The estimates represent the upper limits of the applicability of the abstract model, and their
reduction is expedient due to the decomposition of the Petri net model S(f).
Algorithmic, informational and software implementation of basic steps of check and recognition of
correctness of DIS functioning, taking into account energy consumption, was carried out for three-
level DIS as a part (server-administrative), zonal (regional, broker) and point (local, controller,
sensor-actuator) levels.
For levels it is assumed ascending collection, operative processing and generalization of
information, descending management.
Figure 1: Time temperature monitoring of DIS
Figure 2: Basic algorithm of energy check
Figure 3: Normalized model of Data Base for energy check
Approbation of models and basic steps of their analysis and check of correctness of functioning,
taking into account energy consumption for real components of DIS of a class of system "smart
house", showed possibility and expediency of their application. Real estimates of the complexity of
the check analysis, due to functional features and partial certainty of behavior, were lower on 80-87%
than formal upper estimates and amounted to 102-105 conditional vertices with 10-2-10 seconds of
timer, which allows you to perform real-time testing of components of DIS.
7. Conclusion
In the present work, we propose the development of a model of Petri nets in the behavioral check
of DIS, supplemented by the verification of energy consumption. The energetic characteristics added to
positions, transitions and tokens allowed for a comprehensive analysis of the energy-loaded behavior of the
model at three levels of the reference model. The first level is represented by states / positions with
characteristic energy costs. The second level is represented by possible behavioral topological structures with
characteristic power consumption. The third level is topological structures achievable when moving chips with
characteristic, including identifying, power consumption. Check model makes it possible to determine and
save the energy consumption of DIS components, based on a formal analysis of the behavior of model
elements - vertices, graph-topological elements and subnets of the Petri net. This in turn allows you to
determine the steps of the basic behavioral check procedures for DIS components, extended by
checking their energy consumption.
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