Modeling and Stability Analysis of a Communication Network System

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 sys-tem 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.


Introduction
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 [1] and the references quoted therein.One of the most important performance issues to be considered in a communication network system is its stability.Lyapunov stability theory provides the required tools needed to aboard the stability problem for communication network systems modeled with timed Petri nets whose mathematical model is given in terms of difference equation.By proving practical stability one is allowed to preassigned the bound on the communication network systems dynamics performance.Moreover, employing Lyapunov methods, a sufficient condition for the stabilization problem is also obtained.It is shown that it is possible to restrict the communication network systems 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.The paper is organized as follows.In Section 2, Lyapunov theory for discrete event systems modeled with Petri nets is given.Section 3 presents max-plus algebra.In Section 4, the solution to the stability problem for discrete event systems modeled with timed Petri nets is considered.Finally, in Section 5 the modeling and stability analysis for communication network systems is addressed.Some conclusion remarks are also provided.
× → and define the variation of v relative to (1) by Then, the following result concerns the practical stability of (1).
and ( , ) denote a firing vector where if j t T ∈ is fired then, its corresponding firing vector is [0,..., 0,1, 0,..., 0] T k u = with the one in the th j position in the vector and zeros everywhere else.The nonlinear difference matrix equation describing the dynamical behavior represented by a PN is:  : and consider the matrix difference equation which describes the dynamical behavior of the discrete event system modeled by a PN, see (7).Proposition 9. Let PN be a Petri net.PN is uniform practical stable if there exists a Φ strictly positive m vector such that 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 Stabilization
Definition 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 11) Remark 14.By fixing a particular u, which satisfies (11), the state space is restricted to those markings that are finite.

Basic Definitions
then it is called idempotent.Theorem 17.The max-plus algebra max max ( , , , , ) e ℜ = ⊕ ⊗   has the algebraic structure of a commutative and idempotent semiring.

Let max
n n ×  be the set of n n × matrices with coefficients in max  with the following operations: The sum of matrices max , ⊕ is defined by: ( ⊗ is defined by: the matrix which has its diagonal elements equal to e and all the other elements equal to  .Then, the following result can be stated. Theorem 18.The 5-tuple max max ( , , , , ) , for k m < and m j j = .The path p consists of the nodes In the case when i j = the path is said to be a circuit.A circuit is said to be elementary if nodes k i and l i are different for k l ≠ .A circuit consisting of one arc is called a self-loop.Let us denote by ( , ; ) P i j m the set of all paths from node i to node j of length 1 m ≥ and for any arc ( , ) ( ) i j A ∈  let its weight be given by ij a then the weight of a path ( , ; ) p P i j m ∈ denoted by | | w p is defined to be the sum of the weights of all the arcs that belong to the path.The average weight of a path p is given by (( , ), (( , ) (( , ), ( , ), ( , ), ( , )) as: gives the maximal weight of any path from j to i.If in addition one wants to add the possibility of staying at a node then one must include matrix E in the definition of matrix A + giving rise to its Kleene star representation defined by: be such that any circuit in ( ) A  has average circuit weight less than or equal to  .Then it holds that: Definition 26.Let ( , ) G =   be a graph and , i j ∈  , node j is reachable from node i, denoted as i j  , if there exists a path from i to j.A graph G is said to be strongly connected if i j j i is called irreducible if its communication graph is strongly connected, when this is not the case matrix A is called reducible.
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 [4].

Spectral Theory and Linear Equations
Definition 28.Let is a finite set, the maximum is attained (which is always the case when matrix A is irreducible).In case ( ) is said to be critical if its average weight is maximal.The critical graph of A, denoted by , is the graph consisting of those nodes and arcs that belong to critical circuits in ( ) is irreducible, then there exists one and only one finite eigenvalue (with possible several eigenvectors).This eigenvalue is equal to the maximal average weight of circuits in ( ) G A G a are negative then, the solution is unique.

Max-Plus Recurrence Equations for Timed Event Petri Nets
Definition 32.Let  , with M equal to the maximum number of tokens with respect to all places.Let ( ) i x k denote the kth time that transition i t fires, then the vector

The Solution to the Stability Problem for Discrete Event Dynamical Systems Modeled with Petri
Definition 34.A TPN is said to be stable if all the transitions fire with the same proportion i.e., if there exists q ∈  such that ( ) , 1,..., lim 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 0 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.

Modeling and Stability Analysis of a Communication Network System
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 Figure 1.
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 Ca and Cd respectively (with Ca Cd > ).The incidence matrix that represents the PN model is 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0 1 1 Therefore since there does not exists a Φ strictly positive m vector such that 0 AΦ ≤ the sufficient condition for stability is not satisfied, (moreover, the PN ( TPN ) is unbounded since by the repeated firing of q, the marking in P grows indefinitely).However, by taking [ , , / 2, / 2, / 2]; 0 u k k k k k k = > (but unknown) we get that 0.
T A u ≤ Therefore, the PN is stabilizable which implies that the TPN is stable.Now, let us proceed to determine the exact value of k.From the TPN model we obtain that: which implies * 0 0 0 0 , 0 0 0 0 0 0 .This means that in order for the TPN to be stable and work properly the speed at which the receivers concentrate has to be equal to Ca (the firing speed of transition q) which is attained by taking k Ca = .

Conclusion
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.
with all its elements equal to  and denote by max

1 |
| / | | w p p .Given two paths, as for example, that contains at least one finite element such that: A v v µ ⊗ = ⊗ then, µ is called an eigenvalue and v an eigenvector.Let ( )A  denote the set of all elementary circuits in ( ) average circuit weight.Notice that since ( ) A  . If the communication graph ( ) G A has maximal average circuit weight less than or equal to e, then * Moreover, if the circuit weights in ( ) Mth order recurrence equation.Theorem 33.The Mth order recurrence equation, given by equation state of the system, satisfies the Mth order recurrence equation: hypothesis of theorem (33) are satisfied, and setting ˆ( ) ( ( ), ( 1),..., (

∈
 arbitrary.A an irreducible matrix and λ ∈  its eigenvalue then,
associated to i p P ∈ , called holding time, represents the time that a token must spend in this place until its outputs enabled transitions ,1 ,2 , ,..., It is assumed that at each time k there exists at least one transition to fire.If a transition is enabled then, it can fire.If an enabled transition