In this work, the modeling and stability problem for a communication network system is addressed. The communication network system consists of a transmitter which sends messages to a receiver. The proposed model considers two possibilities. The first one, that messages are successfully received, while in the second one, during the sending process the transmitter breaks down and as a result the message does not reach the receiver. Timed Petrinets is the mathematical and graphical modeling technique utilized. Lyapunov stability theory provides the required tools needed to aboard the stability problem. Employing Lyapunov methods, a sufficient condition for stabilization is obtained. It is shown that it is possible to restrict the communication network system state space in such a way that boundedness is guaranteed. However, this restriction results to be vague. This inconvenience is overcome by considering a specific recurrence equation, in the max-plus algebra, which is assigned to the timed Petri net graphical model.
In this work, the modeling and stability problem for a communication network system is addressed. The communication network system consists of a transmitter which sends messages to a receiver. The proposed model considers two possibilities. The first one, that messages are successfully received, while in the second one, during the sending process the transmitter breaks down and as a result the message does not reach the receiver. A communication network system can be considered as a dynamical system whose state evolves in time by the occurrence of events at possibly irregular time intervals, (therefore belong to the class of dynamical systems known as discrete event systems). Place-transitions Petri nets (commonly called Petri nets) are a graphical and mathematical modeling tool that can be applied to the communication network system in order to represent its states evolution. Petri nets are known to be useful for analyzing the systems properties in addition of being a paradigm for describing and studying information processing systems. Timed Petri nets are an extension of Petri nets, where now the timing at which the state changes are taken into consideration. This is of critical importance since it allows considering useful measures of performance as for example: how long does the communication network system spends at a given state etc. For a detailed discussion of Petri net theory see [
NOTATION:
where
Definition 1. The n vector valued function
Definition 2. The system (1) is said to be practically stable, if given
Definition 3. A continuous function
Consider a vector Lyapunov function
Then, the following result concerns the practical stability of (1).
Theorem 4. Let
for
imply the practical stability properties of system (1).
Corollary 5. In Theorem (4): if
Definition 6. A Petri net is a 5-tuple,
Definition 7. The clock structure associated with a place
The positive number
Definition 8. A timed Petri net is a 6-tuple
Notice that if
Let
Let
where if at step k,
Let
Proposition 9. Let PN be a Petri net. PN is uniform practical stable if there exists a
Moreover, PN is uniform practical asymptotic stable if the following equation holds
Lemma 10. Let suppose that Proposition (9) holds then,
Remark 11. Notice that since the state space of a TPN is contained in the state space of the same now not timed PN, stability of PN implies stability of the TPN.
Lyapunov StabilizationDefinition 12. Let PN be a Petri net. PN is said to be stabilizable if there exists a firing transition sequence with transition count vector u such that system (7) remains bounded.
Proposition 13. Let PN be a Petri net. PN is stabilizable if there exists a firing transition sequence with transition count vector u such that the following equation holds
Remark 14. By fixing a particular u, which satisfies (11), the state space is restricted to those markings that are finite.
NOTATION:
Definition 15. The set
Definition 16. A semiring is a nonempty set R endowed with two operations
Theorem 17. The max-plus algebra
Let
Theorem 18. The 5-tuple
Definition 19. Let
Definition 20. A matrix
Definition 21. Let
Let
Definition 22. A path from node i to node j is a sequence of arcs
Let us denote by
Theorem 23. Let
Definition 24. Let
Lemma 25. Let
Definition 26. Let
Remark 27. In this paper irreducible matrices are just considered. It is possible to treat the reducible case by transforming it into its normal form and computing its generalized eigenmode see [
Definition 28. Let
Let
average circuit weight. Notice that since
Definition 29. A circuit
Theorem 30. If
Theorem 31. Let
Definition 32. Let
Theorem 33. The Mth order recurrence equation, given by equation
With any timed event Petri net, matrices
state of the system, satisfies the Mth order recurrence equation:
Definition 34. A TPN is said to be stable if all the transitions fire with the same proportion i.e., if there exists
This means that in order to obtain a stable TPN all the transitions have to be fired q times. It will be desirable to be more precise and know exactly how many times. The answer to this question is given next.
Lemma 35. Consider the recurrence relation
Proof. Let v be an eigenvector of A such that
Now starting with an unstable TPN, collecting the results given by: proposition (13), what has just been discussed about recurrence equations for TPN at the end of subsection (3.3) and the previous lemma (35) plus theorem (30), the solution to the problem is obtained.
In this section, the modeling and stability analysis for a communication network system is addressed. The communication network system consists of a transmitter which sends messages through a communication channel to a receiver. The proposed model considers two possibilities. The first one, that messages are successfully received, while in the second one, during the sending process the transmitter breaks down and as a result the message does not reach the receiver. Consider a communication network system whose TPN model is depicted in
Where the events (transitions) that drive the system are: q: receivers connect to the communication network, s: messages are sent, b: the transmitter breaks, r: the transmission is restored, d: the message has been successfully received. The places (that represent the states of the queue) are: A: receivers concentrating, P: the receiver is waiting for a message to be sent, B: the message is being received, D: transmitter breaks down, I: the transmitter is idle. The holding times associated to the places A and I are
Therefore since there does not exists a
leading to:
Therefore,
This paper studies the modeling and stability problem for communication network systems using timed Petri nets, Lyapunov methods and max-plus algebra. The results obtained are consistent to what was expected.
Zvi Retchkiman Königsberg, (2015) Modeling and Stability Analysis of a Communication Network System. Journal of Computer and Communications,03,176-183. doi: 10.4236/jcc.2015.311028