3. Dining Philosophers

The example is a simplification of the classical dining philosophers problem, that also represents a particular case of the resource sharing problem. Let us assume for this example that $N$ philosophers (clients) are sitting around a table and $N$ forks are disposed between every pair of philosophers. The basic routine is that the philosophers think for a given time $t$ before picking up the right fork then the left fork, can then eat for a time $s^{3}$. They then put down the right fork then the left fork and return to thinking. As it is not ergodic we introduce a small change considering one left-handed philosopher (the others are right-handed philosophers) which pick up and put down the left fork before the right one. It suppresses the absorbing state where all the philosophers have taken posession of the right hand fork.

The proposed SAN model for right-handed and left-handed philosophers is shown in Figure(?). States (0, 1, ......., 5) of automata $A^{(n)}$describe the state of the philosopher. For example we take right-handed philosopher:

1. (0) : the philosopher is thinking.
2. (1) : the philosopher has finished thinking and wishes to pick up the right fork.
3. (2) : the philosopher has the right fork and wishes to pick up the left fork.
4. (3) : the philosopher has the both forks and is eating.
5. (4) : the philosopher has finished eating and will put down the right fork.
6. (5) : the philosopher has replaced the right fork and he will put down the left fork.

We note $x^{(ni)}$ the curent state of automata $A^{(n)}$ where $i=(0, 1,........., 5)$ is its state. The functions ($f_{n}$ and $g_{n}$) represent the mechanism of access restriction to the forks (right-left).As running example we choose the first philosopher is Left-handed and the others are Right-handed philosophers.

Figure: Philosophers modele of :
 (a)Right-handed philosopher                           (b)Left-handed philosopher

The textual .san  files describing this model are:

a. N=3.    b. N=4.    c. N=6    d. N=8.

Where : 
- N  is  the  nomber  of  philosophers