1.1 Model of Resource Sharing with function

Figure: Resource Sharing Model- Mutex1 -

Each process is modelled by a two state automaton ${\cal A}^{(i)}$, the two states beingsleeping and using. We shall let $s{\cal A}^{(i)}$ denote the current state of automaton ${\cal A}^{(i)}$. Also, we introduce the function

$$f = \delta \left ( \sum_{i=1}^{N} \delta ( s{\cal A}^{(i)} = using )  < P \right ) ,$$

where $\delta (b)$ is an integer function that has the value 1 if the boolean $b$ is true, and the value 0 otherwise. Thus the function $f$ has the value 1 when access is permitted to the resource and has the value 0 otherwise. Figure 1 provides a graphical illustration of this model.

The local transition matrix for automaton ${\cal A}^{(i)}$ is :

$${Q}_l^{(i)} = \left ( \begin{array}{cc} - \lambda^{(i)} f & \lambda^{(i)} f \mu^{(i)} & -\mu^{(i)} \end{array}\right ) ,$$

and the overall descriptor for the model, which does not have any synchronizing events, is

$$D =  {\bigoplus_{i=1}^N} _g  Q_l^{(i)} =  \\ \sum_{i=1}... ...mes_g Q_l^{(i)} \otimes_g I_2 \otimes_g \cdots \otimes_g I_2 .$$

The SAN product state space for this model is of size $2^{N}$. Notice that when $P = 1$, the reachable state space is of size $N+1$, which is considerably smaller than the product state space, while when $P = N$ the reachable state space is the entire product state space. Other values of $P$ give rise to intermediate cases.



The textual .san  files describing this model are:

a. N=20, P=1.    b. N=20, P=5.    c. N=20, P=10.   d. N=20, P=15.   f. N=20, P=19.

Where : 
- N  is  the  nomber  of  processes.
- P is  the  number  of  resource.