A method for efficiently representing the state equation in a class of max-plus linear systems is proposed. We introduce a construct referred to as 'cell' in which the list of possible longest paths is stored. By imposing interval constraints on the system parameters, we can reduce the complexity of the state equation. The proposed method would be useful in scheduling applications for systems with adjustable system parameters.

In this paper, we derive state equations for linearized discrete-time models of forth-order charge-pump phase-locked loops. We solve the differential equations of the loop filter by using the initial conditions and the boundary conditions in a period. The solved equations are linearized and rearranged as discrete-time state equations for checking stability conditions. Some behavioral simulations are performed to verify the proposed method. By examining the stability of loops with different conditions, we also propose an expression between the lower bound of the reference frequency, the open loop unit gain bandwidth, and the phase margin.

For fixed initial and destination states (i.e., markings), M0 and Md, there exist generally infinite firing count vectors in a Petri net. In this letter, it is shown that all fundamental particular solutions as well as all minimal T-invariants w.r.t. firing count vectors are needed to express an arbitrary firing count vector for the fixed M0 and Md. An algorithm for finding a special firing count vector which is expressed by using the only one specified fundamental particular solution is also given.

A formal necessary and sufficient condition on the general Petri net reachability problem is presented by eliminating all spurious solutions among known nonnegative integer solutions of state equation and unifying all the causes of those spurious solutions into a maximal-strongly-connected and siphon-and-trap subnet Nw. This result is based on the decomposition of a given net (N, Mo) with Md and the concepts of "no immature siphon at the reduced initial marking Mwo" and "no immature trap at the reduced end marking Mwd" on Nw which are both extended from "no token-free siphon at the initial marking Mo" and "no token-free trap at the end marking Md" on N, respectively, which have been both effectively, explicitly or implicitly, used in the well-known fundamental and simple subclasses.

In a cellular system for mobile communications, every service area is divided into a number of cells for utilizing the frequency spectrum efficiently. Service areas for such systems are two dimensional, however, the analysis of the characteristics of the communication traffic for the areas are quite complicated, since the motion of the vehicles in the area can not be predicted precisely. For making the analysis easily, the areas are assumed to be band-shaped like a highway. Furthermore, in the analysis, the traffic offered to a cell is assumed to be stationary. In actual systems, the density of vehicles and the offered communication traffic is not stationary, so that many differences exist between the analysis and the actual systems. This paper presents an analysis method using state equations. The equations represent the transient characteristics of mobile communication traffic when a band-shaped service area is assumed. The transition is made by accidents or congestion, and causes the rapid offered traffic change in a communication system. In the method, numerical analysis is made under the consideration of "handoff" operation. The operation consists of surrendering the channel used in the previous cell and reassigning a new channel when the vehicle crosses the cell boundary. The analytical results are compared with the simulations, and the two results show good agreement. The method presented in this paper can be used for designing the switching system when the offered traffic changes rapidly due to accidents or congestion.