Modelling and Simulation of Adaptive Multi-Agent Systems with
Stochastic Nets-within-Nets
Michael K
¨
ohler-Bußmeier
1 a
and Lorenzo Capra
2 b
1
University of Applied Sciences Hamburg, Berliner Tor 7, D-20099 Hamburg, Germany
2
Dipartimento di Informatica, Universit
`
a degli Studi di Milano, Via Celoria 18, Milan, Italy
Keywords:
Adaptive Systems, Multi-Agent Systems, Stochastic Petri Nets, Nets-within-Nets, Simulation.
Abstract:
This study centers on self-adapting multi-agent systems modeled utilizing the SONAR framework. Our key
focus is on forecasting the costs and benefits of adaptation during execution within the MAPE loop (monitor,
analyse, plan, and execute). Analyzing these adaptation processes is intricate due to SONAR enabling second-
order activities, such as structural adaptation involving agent interaction protocols or the organizational net-
work itself. We forecast these dynamic processes using a stochastic run-time model (e.g., the environment has
a stochastic representation). Since SONAR is conceptualized with HORNETS (a nets-within-nets formalism),
we necessitate “probabilistic” HORNETS. To illustrate our approach’s effectiveness, we showcase a small case
study of a self-modifying MAS organization and provide an analysis of adaptation dynamics.
1 INTRODUCTION
This research focuses on self-adaptive systems, par-
ticularly multi-agent systems (MAS) (Weiß, 1999)
and their analysis (cf. (Capra and K
¨
ohler-Bußmeier,
2024)). Our application domain, that is, MAS
for cyberphysical systems (Leitao and Karnouskos,
2015), allows structural modifications at run-time.
Therefore, we use a specification framework, called
SONAR (K
¨
ohler-Bußmeier et al., 2009), which en-
ables the modification, addition, and deletion of
components (e.g., agent interaction protocols) us-
ing a MAPE-Loop (monitor, analyse, plan, and ex-
ecute) (Weyns, 2020). On the one hand, self-
modification is powerful. However, it makes anal-
ysis, such as predicting adaption costs and benefits,
challenging. It becomes even more demanding when
the MAPE loop (Donckt et al., 2018).
The SONAR MAPE-Loop is specified as a Petri
net, which has organizations and interaction proto-
cols as tokens. Since protocols are Petri nets, we also
use Nets-within-nets (Valk, 2003), which allow Petri
nets to be tokens of other nets. The MAS organiza-
tion is specified using HORNETS (K
¨
ohler-Bußmeier,
2009), which are a specialization of algebraic Petri
a
https://orcid.org/0000-0002-3074-4145
b
https://orcid.org/0000-0002-1029-1169
nets (Reisig, 1991) with net operators and nesting.
This research seeks to facilitate the examination
of self-adaptive systems using a digital twin of the
entire system. To achieve a functional model suit-
able for real-time analysis, the twin model inte-
grates stochastic (probabilistic) components to ab-
stract, for instance, the agents’ decision-making pro-
cesses. Consequently, our twin model adopts HOR-
NETS enhanced with probabilistic parameters.
Using runtime models for the analysis and plan-
ning of self-adaptation is an increasingly popular re-
search domain; cf. (Simeoni et al., 2004; Bencomo
et al., 2014; Calinescu et al., 2012; Bencomo et al.,
2019). In this context, we illustrate our method for
run-time analysis through a basic example where the
MAS is able to make structural changes.
This article has the following structure: In Sec-
tion 2, we present the formalism SONAR and its se-
mantics, the SONAR-MAPE loop. We will explain
how structural modifications of the MAS are carried
out at run-time through second-order teamwork. In
Section 3 we will present the digital twin model that
is used during the analysis and planning phase of the
MAPE-Loop to predict the benefit of the adaptions.
We will shortly explain the underlying Petri net for-
malism named HORNETS. Section 4 presents the case
study – implemented using RENEW (Kummer et al.,
2004). The results of the analysis are given in Sec-
tion 5. The work ends with a conclusion.
Köhler-Bußmeier, M. and Capra, L.
Modelling and Simulation of Adaptive Multi-Agent Systems with Stochastic Nets-within-Nets.
DOI: 10.5220/0013019700003837
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 16th International Joint Conference on Computational Intelligence (IJCCI 2024), pages 313-320
ISBN: 978-989-758-721-4; ISSN: 2184-3236
Proceedings Copyright © 2024 by SCITEPRESS – Science and Technology Publications, Lda.
313

2 MAS-ORGANIZATIONS AND
THE SONAR-MAPE-LOOP
Multi-agent systems (MAS) constitute a network of
interacting agents. Unlike general distributed sys-
tems, MAS assumes that agents are autonomous enti-
ties, each pursuing its own goals. Traditional concepts
in multi-agent systems, such as reasoning, coordina-
tion, negotiation, etc., are considered bottom-up ap-
proaches (Weiß, 1999). To structure MAS as a whole,
these concepts are complemented by top-down con-
cepts such as roles, hierarchies, positions, etc. All
these concepts are encompassed by that of an organi-
zation (Org-MAS) (Dignum and Padget, 2013).
More than a decade ago, the SONAR framework
(K
¨
ohler-Bußmeier et al., 2009) was introduced, a for-
malism based on Petri nets to specify and analyze
Organizational Multi-Agent Systems (Org-MAS). In
addition, a SONAR model is employed to create an
execution engine. The SONAR engine establishes a
general execution loop called the SONAR MAPE loop
(K
¨
ohler-Bußmeier and Sudeikat, 2024), where a task
initiates the formation of a team. This team then for-
mulates a team plan (through negotiation), which is
carried out in a distributed fashion by the team agents:
organization → Team → Plan
As a special feature, the execution of the team plan
may involve transformation statements that modify
the SONAR-model at run-time. This enables coopera-
tive, self-organized adaption of the organization:
organization
→ Team
→ Plan
⇝ Transformation at run-time
→ new organization
→ ...
The formalism of SONAR (K
¨
ohler-Bußmeier
et al., 2009) defines a notion of well-formedness
(which we will not introduce here) to guarantee that
each task may be handled by at least one team, etc.
Furthermore, transformations in well-formed organi-
zations preserve well-formedness.
3 THE DIGITAL TWIN MODEL
USING HORNETS
The MAPE-Loop is nested within itself as it includes
a digital twin. During the planning step, a model of
the entire loop is used to predict the benefits of poten-
tial adjustments. We have opted for the Nets-within-
Nets paradigm (Valk, 2003) to represent this recur-
sive structure, employing a Petri net formalism where
the tokens themselves are Petri nets. Nets-within-nets
can be seen as the Petri net perspective on contex-
tual change, in contrast to the process algebra view of
the Ambient Calculus (Cardelli et al., 1999) or the π-
calculus (Milner et al., 1992). Here, we have selected
HORNETS (K
¨
ohler-Bußmeier, 2009) since the orga-
nization model SONAR is based on Petri nets. These
nets become net-tokens in our MAPE-Loop. Espe-
cially, we make use of the ability of HORNETS to per-
form algebraic operations on these net-tokens. Our
Sonar-MAPE-Loop utilizes this functionality to ad-
just the structure of the organization net or the work-
flows (modeled as net-tokens) during runtime.
3.1 Hornets
In the following, we give a rather informal introduc-
tion into HORNETS –for more details cf. (K
¨
ohler-
Bußmeier, 2009; K
¨
ohler-Bußmeier, 2014a).
We will illustrate the main concepts of HORNETS
using the example given in Figure 1. For simplicity,
the set of net types only contains the workflow net:
K = {WFN}. We have one operator ∥ for the par-
allel composition: ∥ ∈ Σ
WFN
2
,WFN
. The operator is
interpreted by I as the usual AND operation between
the workflow nets. We have the universe with three
object nets: U
WFN
= {N
1
, N
2
, N
3
}. All places of the
system-net have the same type. The structure of the
system net
b
N and the object nets is given in the usual
way, as shown in Fig. 1. This EHORNET does not
use communication channels, that is, all events oc-
cur autonomously. In the initial marking, we consider
a EHORNET with two nets N
1
and N
2
as tokens (as
shown on the left): µ
0
=
b
p[N
1
, v]+
b
q[N
2
, s].
To model a run-time adaption, we combine N
1
and
N
2
resulting in the net N
3
= (N
1
∥N
2
). This modifi-
cation is modelled by system net transition
b
t of the
HORNET. The net token generated on
b
r has the struc-
ture of N
3
and its marking is obtained as a transfer
from the token on v in N
1
and the token on s in N
2
into N
3
:
b
p[N
1
, v]+
b
q[N
2
, s]
θ
1
−→
b
r

(N
1
∥N
2
), s + v

This transfer is possible since all the places of N
1
and
N
2
are also places in N
3
and the tokens can be trans-
ferred in an obvious way.
Assume a fixed many-sorted predicate logic Γ =
(Σ, X, E, Ψ) with operators Σ, variables X, equations
E, and predicates Ψ.
Definition 1. An elementary HORNET (EHORNET) is
a tuple EH = (
b
N, U, I , k, Θ, µ
0
) such that:
1.
b
N is an algebraic net, called the system net.
2. (U, I ) is a finite net-theory for the logic Γ.
ECTA 2024 - 16th International Conference on Evolutionary Computation Theory and Applications
314

mt
b
p


*
x
mt
b
q
H
H
Hj
y
b
t
-
(x∥y)
m
b
r
Z
Z
Z}


=
N
1
i
i
1
-
a
- i
u
-
b
- ir
v
-
c
- i
f
1
N
2
i
i
2
-
d
- ir
s
-
e
- i
f
2
N
3
net-token produced on
b
r by
b
t
h
i
3
-


@
@R
@
@R


- h
f
3
h
i
1
-
a
- h
u
-
b
- hr
v
-
c
- h
f
1
h
i
2
-
d
- h
s
r -
e
- h
f
2
Figure 1: Modification of the Net-Token’s Structure.
3. k :
b
P → K is the typing of the system-net places.
4. Θ is the set of nested events.
5. µ
0
∈ M
H
is the initial marking.
We refer to (K
¨
ohler-Bußmeier, 2009; K
¨
ohler-
Bußmeier, 2014a) for the formal definition of the fir-
ing rule.
The reachability graph RG(EH) = (V, E, µ
0
) is de-
fined in the usual way. The nested markings are the
nodes, that is, V = M
H
, and the firing rule µ
θ
−→ µ
′
generates the edges (µ, θ, µ
′
) ∈ E.
For stochastic HORNETS, we equip the model
with a rate function Λ : Θ → R
>0
that assigns firing
rates Λ(θ) to events θ ∈ Θ.
For stochastic Petri nets (SPN) (Marsan, 1990),
we normalize over all transitions enabled in a given
marking to obtain probabilities from these rates. The
same idea is applied to the nested events of EHOR-
NETS in the following. Let En(µ) := {θ ∈ Θ | µ
θ
−→}
be the set of all events enabled in the nested marking
µ. Then, the probability of firing θ ∈ En(µ) is pro-
portional to its rate Λ(θ) ∈ R
>0
. For EHORNETS we
define the firing probability as:
Pr
µ
(θ) :=
Λ(θ)
∑
θ∈En(µ)
Λ(θ)
(1)
For an arc (µ, θ, µ
′
) in the reachability graph
RG(EH) = (V, E, µ
0
) we define its probability as:
Pr((µ, θ, µ
′
)) := Pr
µ
(θ) (2)
These probabilities turn the reachability graph into a
(discrete) Markov chain.
3.2 The Digital Twin of the
SONAR-MAPE-Loop
The Digital Twin of our SONAR-MAPE-Loop is used
during the planning stage of the MAPE-Loop. The
twin model allows us to evaluate different adaptation
options based on their costs and advantages. Since
planning must come before ongoing activities, the
digital twin version is more simplified. The dig-
ital twin of our SONAR-MAPE-Loop is illustrated
in Fig. 2. In contrast to the original loop (K
¨
ohler-
Bußmeier and Sudeikat, 2024), we utilize the follow-
ing abstractions:
• We incorporate environmental assumptions us-
ing a stochastic model, specifically a distribution
P[TASKS = p
0
] to generate initial tasks p
0
.
• We replace the decision logic of organizational
position agents (OPAs), which resolves the con-
flicts between the team formation operators op,
with a distribution P[ACT
p
= op].
• In the workflow nets (WFN) (Aalst, 1997), which
specifies the agent interaction protocols, we simu-
late the decision logic for xor choices, again, with
a stochastic distribution P[XOR
p
= choice
k
].
• We also incorporate the agents’ decision logic and
their learning capabilities (i.e., parts that are ex-
ternal from the organization’s viewpoint) using a
stochastic model.
For more details on the SONAR-MAPE-Loop, the
reader is referred to (K
¨
ohler-Bußmeier and Sudeikat,
2024).
4 THE CASE STUDY:
SELF-MODIFYING
ORGANIZATIONS
In our case study, we consider a scenario involving
co-learning, which means that learning occurs at both
the individual agent level and the organizational level.
For simplicity, we will focus on the latter. The sce-
nario is based on the well-known battle of sexes from
game theory, in which two agents must choose be-
tween two actions, labeled a and b. They receive a
positive reward if they choose the same action, and
Modelling and Simulation of Adaptive Multi-Agent Systems with Stochastic Nets-within-Nets
315

Figure 2: The SONAR-Mape-Loop (adapted from (K
¨
ohler-Bußmeier and Sudeikat, 2024)).
zero otherwise. In this game, the first agent prefers the
action a, while the second prefers b. If we assume that
the reward for the preferred outcome is three times
higher than for the other, then the payoff matrix be-
low specifies the game:
Here, we have two Nash equilibria for pure strate-
gies: (a, a) and (b, b). It can be shown that this game
has an equilibrium unique for mixed strategy, where
both agents choose their preferred action p = 75% of
the time.
a b
a (3, 1) (0, 0)
b (0, 0) (1, 3)
The situation described is known as a coordination
game because agents would benefit from coordinating
their actions in advance. In this scenario, the social
welfare (which is the sum of the individual payoffs) is
3+1 = 1+3 = 4 in both cases. However, in uncoordi-
nated games, a mixed strategy is the best choice, and
agents only coordinate in 0.75 · 0.25 + 0.75 · 0.25 =
0.375 of the cases, leading to an expected payoff of
(0.75 · 0.25 + 0.75 · 0.25) · 4 = 1.5. The ratio
3+1
1.5
=
2.66.., called price-of-anarchy, measures the need for
an external coordination mechanism.
For MAS, this mechanism is termed an organiza-
tion (Dignum and Padget, 2013). The SONAR-Org-
MAS of this scenario is depicted in Fig. 3. The pri-
mary purpose of the SONAR-Org-Model is to assem-
ble a team in response to certain triggers. It deter-
mines the workflow net used for the response and the
agents involved and sets constraints on the agents’ de-
cisions, represented by parameters µ and δ. The Org-
Model has three organizational agents O
0
, O
1
, and
O
2
. Manages two tasks: The first task triggers the
team formation process for the workflow net (shown
in Fig. 4) representing the battle-of-sexes interaction
P. The protocol P defines the interaction of two roles
R
1
and R
2
. The model specifies that R
1
is assigned to
O
1
and R
2
to O
2
.
The choice between options a and b depends on
the probability, prob, which is calculated as c · p
org
+
(1 − c) · p
agent
, where p
agent
represents the agents’
decision logic and p
org
represents the organizational
constraint. For simplicity, set p
agent
and p
org
both to
75%, which represents the probability of choosing op-
tion a based on the Nash equilibrium. Therefore, we
obtain prob = 75%, regardless of the value of the or-
ganizational impact c.
The right side of the organization handles a task
that triggers the adaptation. The second task in Fig. 3
initiates the team formation for a second-order WFN
(cf. Fig. 5), which alters the role fragment so that
the second agent always chooses option a (that is,
we force the second agent to deviate from the opti-
mal mixed strategy). This structural modification is
formalized in the upper block (modify WFN) of the
WFN. Furthermore, this second-order WFN extends
the original organization model to incorporate this
modification, as formalized in the lower block (modify
organization model). It is worth noticing that we need
a protocol here to synchronize the necessary elemen-
tary transformation steps. The shaded area of the or-
ganization net in Fig. 3 shows the nodes that will be
added by the second order protocol. Since each agent
block receives new nodes, the second-order protocol
designates three roles to the agents O
0
, O
1
, and O
2
.
We use the syntax of the Java-based Petri net tool
RENEW (Kummer et al., 2004) to execute the sce-
nario. The sources are available at https://github.com/
ECTA 2024 - 16th International Conference on Evolutionary Computation Theory and Applications
316

Figure 3: The SONAR-organization (The shaded nodes are those elements that are added by the second-order workflow given
in Fig. 5.).
Figure 4: Stochastic WFN: Battle-of-the-Sexes (The shaded transition will be deleted by the second-order workflow given in
Fig. 5.).
koehler-bussmeier/bos. The nets given in that repos-
itory are used to obtain the simulation results. A de-
tailed description of the model is given at the reposi-
tory, too.
5 ANALYSIS OF THE MODEL
To simplify the presentation, we use the price-of-
anarchy to quantify the benefit of an adaptation for
this case study, and the costs are measured by the
number of transitions in the second-order protocol.
In our application area of cyber-physical systems
(CPS), we would evaluate models using key perfor-
mance indicators (KPI) like throughput, occupation,
etc. As mentioned above, the HORNET model is in-
herently stochastic. For example, we have a random-
ized choice between the options in the WFN shown in
Figure 4.
We conducted a stochastic simulation of our
model to predict the effects of adaptation. The model
can be found at github.com/koehler-bussmeier/bos.
We utilized the Java-based Petri net tool RENEW
(Kummer et al., 2004) to carry out the scenario. In
the simulation, we initiated the first-order workflow
(Fig. 4) 100 times to ensure statistical stability. Sub-
sequently, we implemented an organizational adap-
tation protocol (specified by the second-order work-
flow in Fig. 5). This adaptation results in a modified
version of the first-order workflow, where the second
agent persistently chooses option a, and in an adjust-
ment in the organization so that this protocol would
always be adopted from that point on. Finally, we
generate another 100 trigger events. Typically, we
would repeat the simulation multiple times to estab-
lish error ranges, but for this illustrative scenario, we
decided to omit this step.
The simulation results (given in the left columns
of Table 1) are in good alignment with the analyti-
cal results (right columns): In the initial configuration
of this example, we can expect (a, a) in 0.75 · 0.25 =
18, 75% of the cases, with an outcome of (3, 1); for
symmetry reasons, we have the same probability for
Modelling and Simulation of Adaptive Multi-Agent Systems with Stochastic Nets-within-Nets
317

Figure 5: Second-order WFN: Team-based modification of the workflow given in Fig. 4 and the organization given in Fig. 3 .
Table 1: Simulation Results (left) and Expected Values (right).
outcome before and . . . after adaptation before (expected) after (expected)
(a, a) 20% 78% 18, 75% 75%
(a, b) or (b, a) 62% 22% 62.50% 25%
(b, b) 18% 0% 18, 75% 0%
(b, b). In 62.5% of times, the agents will choose op-
posite options.
After the adaptation has taken place, we expect
that in 0.75 · 1 of interactions, the agents will play
(a, a) and in 0.25 · 1 of interactions, the agents will
play (b, a). This leads to the expected social welfare
of 0.75·(3 + 1) +0.25 · (0 +0) = 3. So, the new price
of anarchy is
3+1
3
= 1.33.., which is a substantial im-
provement from the initial value of
3+1
1.5
= 2.66...
Note that restricting the agents’ options is not al-
ways beneficial: Removing option a for the second
agent results in a welfare of 0.25 · (3 + 1) + 0.75 ·
ECTA 2024 - 16th International Conference on Evolutionary Computation Theory and Applications
318

(0 + 0) = 1. This leads to a new price-of-anarchy of
3+1
1
= 4, which is worse. The costs of this transfor-
mation would be the same.
Although anticipated and illustrated by a straight-
forward example, these results demonstrate the abil-
ity to forecast the costs and benefits of structural
changes in complex organizations through the quan-
titative modeling and simulation of distributed agents
using a formal digital-twin approach.
6 CONCLUSION
Our research explores self-modifying systems within
the context of Org-MAS and MAPE-Loop. We use
the formalism HORNETS for the twin model to predict
the cost-benefit ratio of adaptation. Since the model
includes stochastic elements, we require an appropri-
ate analysis tool. In this study, we examine a simple
case study using the battle of sexes game to demon-
strate that the quantitative analysis of adaption using
HORNETS is feasible and beneficial for the designer.
In our current research, we are focusing on
converting elementary, two-levels, HORNETS into
stochastic Symmetric Petri Nets (SSN) (that can be
assessed within the GREATSPN framework (Am-
parore et al., 2016)) by mimicking net-tokens through
an emulator (Camilli and Capra, 2021). Given the
lower level of abstraction in SSN, we also aim to de-
scribe the semantics of HORNETS employing an ad-
vanced algebraic framework, such as Maude (Clavel
et al., 2007). This extends the approach we ap-
plied to Elementary Object Systems (EOS) (Capra
and K
¨
ohler-Bußmeier, 2023b; Capra and K
¨
ohler-
Bußmeier, 2023a).
In general, such a translation is quite complex due
to the nested nature of Nets-within-Nets. (Note, that
in general there is no equivalent unnested nets, since
the reachability problem is undecidable for Nets-
within-Nets (K
¨
ohler-Bußmeier, 2014b), while it is de-
cidable for place transition nets; so we need at least
colored tokens and algebraic inscriptions.) However,
we observed that our SONAR-MAPE-Loop has an in-
teresting structural property, that is, the system-net
does not combine or distribute net-tokens of the same
net type, i.e., we have no forks or joins, which are usu-
ally the source of complexity for HORNETS (K
¨
ohler-
Bußmeier, 2017).
For a similar subclass of EOS, called General-
ized State Machines (GSM), we have already shown
in (K
¨
ohler-Bußmeier, 2014b) that this structural sub-
class can be translated into an equivalent unnested
Petri net. In ongoing work, we extend this idea to ob-
tain an equivalent unnested algebraic Petri net from a
fork/join-free EHORNET. This translation would al-
low symbolic methods to rely on established and effi-
cient analysis methods.
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