Continuous time

Consider a model of resource sharing - Mutex1 [Fer92].
In this model, $N$ distinguishable processes share a resource. Each process alternates between sleeping state and using state. The number of processes concurrently use is bounded by $P$, where $1\leq P \leq N$.
We shall let $\lambda_i$ be the rate at which proces $i$ awakes from the sleeping state wishing to access the resource, and $\mu_i$, the rate at which this same process releases the resource after possession of this one.

Figure 5.1: Resource Sharing Model - Mutex1

Take for example, a Mutex1 model, with $N=3$ processes, $P=1$ resource, and rates values :

• $\lambda=6$.
• $\mu=9$.

We obtain as generator matrix :

$$\mathbf{Q}= \left(\begin{tabular}{cccc} -18 & 6 & 6 & 6 \\ 4 & -4 & 0 & 0 \\ 4 & 0 & -4 & 0 \\ 4 & 0 & 0 & -4 \\ \end{tabular} \right)$$

Note that we are :

• $\Lambda_{max}=18$.
• $\textrm{dimension}=4$.
• $\textrm{nz\_number}=10$.
• $\pi=(\frac{2}{11},\frac{3}{11},\frac{3}{11},\frac{3}{11})$.

Consider this'n3p1.marca' importation file :

Then, uniformization step, gives the equivalent discrete^5.1 time Markov chain :

And it follows the Aliasing data file :

Note that it is ESSENTIAL for simulation, to provide a 'simu' file, where Aliasing data are stored by column !!!

You're now ready for Perfect Simulation.