IJPAM: Volume 86, No. 2 (2013)

TIMED PETRI NETS MODELING AND
LYAPUNOV/MAX-PLUS ALGEBRA STABILITY
ANALYSIS FOR A TYPE OF QUEUING SYSTEMS

Zvi Retchkiman Konigsberg
Instituto Politécnico Nacional, CIC
Mineria 17-2, Col. Escandon, Mexico D.F 11800, MEXICO


Abstract. A queuing system, is a dynamical system whose state evolves in time by the occurrence of events at possibly irregular time intervals. Place-transitions Petri nets (commonly called Petri nets) are a graphical and mathematical modeling tool applicable to queuing systems in order to represent its states evolution. Timed Petri nets are an extension of Petri nets, where now the timing at which the state changes is taken into consideration. One of the most important performance issues to be considered in a queuing system is its stability. Lyapunov stability theory provides the required tools needed to aboard the stability problem for queuing systems modeled with timed Petri nets whose mathematical model is given in terms of difference equations. By proving practical stability one is allowed to preassigned the bound on the queuing 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 queuing 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. In this paper, the modeling and stability problem for a type of queueing systems is addressed. Two classes of queues are considered. The first one (one and two servers) working under normal operation, while the second one (one server) having failures in its normal operation due to server breakdown and consumer interruption during service.

Received: January 15, 2013

AMS Subject Classification: 08A99, 93D35, 93D99, 39A11, 08C99, 16Y60, 65F15, 05C50, 15A29, 15A33

Key Words and Phrases: queuing systems, Lyapunov methods, max-plus algebra, timed Petri nets

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DOI: 10.12732/ijpam.v86i2.6 How to cite this paper?
Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 86
Issue: 2