Aryanpur DictionartConventionally, such a complex system would be modelled at each abstraction level in isolation, with just sufficient interaction between the levels to ensure a d ic ien t ly accurate mapping of the appropriate characteristics. One of the more interesting obmations that can be made from the study of failures of complex systems is that a very frequent source of s i g " 'systems' problems is at the interface between abstraction levels. What this means is that at precisely where an overall system model d s to be accurate to ~ssess potential problem areas, in the conventional approach it is one of the areas not at d well modelled. As a broad goal, multifaceted modebg aims to overcome this limitation.
The primary aim of this paper is to explore one of these possibilities in the context of a relatively simple FMS example. It should be noted that although the CXBmpk taken is not a complex one, this is due to a high importame being given to providw a tractable exposition of the approach. A secondary aim of this paper is to expand the alpbraic representation of decision-free Pari nets (Timed Event Graphs or TEG's) to " p a s s i d o h o n input from dynarmcal difkcntial equation based models represmting the process In this way, a composite transfer funaion can be derived for the complete system composed of multiple abstraction levels
ALGEBRAIC REPRESENTATION OF TIMED EVENT GRAPHS
The algebraic representation for TEGs is deftned in Cohen etal [ 19891 which builds on the introduction of the model in Cohen etal [1985] It is interesting to note that the basis for the representation was developed by Cuninghame-Green [I9791 who cited the earliest work by Shimbel [1954] and his own early work in 1%0-62. A few other publications at about the same time are also cited dealing with methods of analysis for early communications networks
The model for Fig. 1 is derived using these methods and is modelled on a example taken from Cohen etal [1989] to illustrate the algebraic representation :
[Fig I AnexampleTEGmadel I
where ---> initial token marking I ---> holding lmc in time MILT
Y l
As can be seen from the figure, all places have a single input and output arc - this precludes non-detenninistic decisions and the dual problem of non-deterministic arrival of tokens at a place. Importantly, the Torks' and 'joins' of concurrent processing can be modelled All transitions 'fire' immediately when enabled by sufficient input place tokens, and only places may have hdding time ddays. If transitions with delays are required then these may be equivalently modelled with additional places with delays. Tokens in the initial mark@ are assumed to be available immediately for firing transitions, whereas tokens in places with a holding time must wait for the duration of that holding time before enabling a transition to fire
TO I l l y describe the operation of the model it is sufficient to record the sequence of transition firinss as events and the umes of firings as dntes. This leads n a t u d y to a 2-dimemIonal time-event space on which to map model behaviour - operation can be mewed either as the specification of transition firing times in an event-domnin or transition finng events in a timedomain These representations are duals of one another and can be used independently. or as Cohen etal [1989] argue, two representations can be used simultanmuslv to betta effect.
The representation uses a 'linear' algebra where the operators differ from the usual '+' and 'x' Cohen etal [1989] define this as a form of dioid algebra; in the event-domain the required dioid is 2;, = (Z w {-a) w (+3c), max, +). For notational convenience, the foUowing operators are defined: 0 = max, QD = +, E = -x. e = 0. and it can also be noted that again for convenience, the '8' operator is not shown unless needed to remove any ambiguity.
Any transition x,, x,(n, f) can be thought of as an infomation-state v d b k since it takes on the values of a &-valued infomation- state which can be defined as set of pairs (n, th. A single infonmatiowt.te value corresponds to a a "piece of information" in the terms of Cohen etal[ 19891 which can be interpreted as:
(n, r), 3 at transition I,, the nth event occurred at the earliest at time t .
It is worthwhile illustrating graphically just what this definition for information-state implies; consider the following example.
nts
Fig. 2. Information-State Spacc
Suppose this graph represents the information-state of several inputs to transition I, - the transition effectively processes the union of all the information-states coming from the input places. Graphically, it can be seen that an information-state at (n, t ) dominates (or replaces) all information-states that lie in the south-east cone having (n, 1) as the vertex.
As indicated above, a general information-state can be composed of a number of points in the 2-D time-event plane (see Fig. 2), i.e. for transition xi the current information-state is the set of points given by ((U,. t;).I I E I c N}. To represent this "collection of points of information'' Cohen etal [I9891 introduce the following chnncte&tic function.