Pranešimas XVII mokslinės kompiuterininkų konferencijos sekcijoje „K8. Statistiniai metodai, optimizavimas ir prognozavimas“ „Kompiuterininkų dienos – 2015“, Panevėžyje, KTU PTVF 2013-09-19

1. Equivalent states search
algorithm for models with
continuous time
Dalius Makackas and Regina Miseviciene
Kaunas University of Technology, Faculty of Informatics,
Kaunas, Lithuania

2. Introduction
• A real-time system’s accuracy depends not only on the logical result of
computations, but also on the time at which the results are produced
• The most commonly used are following formal notations: Time Petri Nets,
Discrete Event System Specification (DEVS) , Timed Automata, Piece-linear
Aggregate (PLA), Finite State Machine and others.

3. Problem
• Verifying correctness of real time systems, reachable states analysis method is
amply developed and used. However, this method, described in scientific
literature, cannot avoid the endless increase of reachable state space.
• This presentation presents an equivalent nodes search algorithm enabling to
reduce the number of nodes in the reachable states graph.

4. A goal of presentation
• A goal of this presentation is to present an algorithm, which transforms
the graph of the infinite reachable states into the graph with the finite node
numbers.
• The algorithm is designed for real-time systems specified by Piece-linear
aggregate method.

5. This presentation is organized following
• Formal definition of Piece-linear aggregate;
• Transformation algorithm of the reachable states graph for the continuous
time models
• Illustrative example

6. Piece-Linear Aggregate Method (1)
A system specified by the Piece-linear aggregate method is understood as a set of
interacting piece-linear aggregates. Each aggregate is defined by a set of states
...},{ 21 zzZ  , a set of input signals ...},{ 21 xxX  , a set of output signals
...},{ 21 yyY  , a set of internal E  and external E events, a set of transition
ZZEH : and output YZEG : operators.

7. Piece-Linear Aggregate Method (2)
The aggregate method generates time-point sequences ...},{ 10 ttT  and state
)...}(),({ 10 tztz transitions in these time points. The state ))(),(()( tztvtz v consists of
two components: discrete )(tv and continuous )(tzv . Each element )(twi of a
continuous component )...)(),(()( 21 twtwtzv  indicates a time when an event ie
occurs. The event changes j elements of discrete and continuous component of state
according to the law: ))(,()( tzthth v
j
v
j  , ))(,()( tzthth w
j
w
j  .

8. Piece-Linear Aggregate Method (3)
In the aggregate model it is also defined the concept of the operation. This function
takes the following values:
   
t.timeatpasiveisit
t;timeatendedit
t;timeatactiveisit
teOtOO ee







,1
,0
,1
,

9. Behavior definition
     ,,,,,, 222111000 IesIesIes (8)
Other notation is used:
       
     33221100
3210
IeIeIeIe
ssss , (9)

10. Equal behaviors
Two behaviors     ,,,,, 1
2
1
1
1
1
1
1
1
0
1
0
1
01 sIesIes and     ,,,,, 2
2
2
1
2
1
2
1
2
0
2
0
2
02 sIesIes
are called equal when 21
jj ss  ( 1
je matches 2
je ), and their occurrence
intervals are equal 21
jj II  for all j .

11. Definitions
Definition 3. The behavior  is called periodical when its traces are periodical.
The system detailed functioning can be shown by the graph when all the behaviors in
the set  are periodical or finite.
Definition 4. Two states is and js , are called equivalent in the behavior
             ,,,,,,,,,,,,,,, 21112111111000  jjjjjjjiiiiiii sIesIessIesIesIesIes ,
when    2121 :, TTITIT ji   .

12. The search algorithm of the equivalent states
The system behavior  is chosen
For the state is from the behavior  the
next equivalent state js , with the same
discrete component is determined.
Checking whether the same active
operations and corresponding continuous
components values coincide.

13. An Example
1
Mass service system
2
Queue
Two-channel mass service system
The system specification consists of the components:
 a set of inputs X and a set of outputs Y ;
 a set of events EEE  , where E ;  321 ,, eeeE  , 1e - a new message
arrived, 2e - a first channel service is completed, 3e - a second channel service is
finished;
 controlling sequences  2101 ,, e ,  2102 ,, e ,  2103 ,, e ;
 a discrete component     tnt  , where   tn is a number of messages in a queue.
 a continuous component         tewtewtewtz ,,,,, 321 ;
a parameter s - is a maximum length of the queue.

14. Reachable state tree fragment

15. Equivalent states
  2111 ;,,6,4;0:3 Rtt  , where   64 01021 tttR
 43 121  ttt
  6133 ;,,6,4;0:9 Rtt  , where
    4364 12101061 ttttttR   64 131 ttt
 43 343  ttt .

16. Transformed graph
5
21l
22l 23l
51l
8
4
41l
7
2
3
31l
6
1
11l
61l
62l
10

17. Conclusions
• Verifying correctness of real time systems, a reachable states graph analysis method
is widely developed. Because of the infinite number of the nodes in the reachable
states graph is impossible to analyze the graph.
• Presented algorithm enables to minimize the nodes number of the reachable state
graph using the equivalent relationships. Using the relations, the reachable states
graph can be transformed into the graph with the finite nodes number.
• It is especially important for continuous time models when transforming the graph
into the smaller one and solving the same problems as in the original graph.