I simulated a system using discrete-events simulation, more particularly next-event simulation. The system is M/M/1 (infinite FIFO waiting room). Whenever the system is jobless, the server shutdowns. When a job arrives, the server restarts if it was off, but it cannot serve until the restart is finished. Restart time is defined by an exponential distribution.

• $\lambda$ is the arrival rate (expected inter-arrival is $1/\lambda$)
• $\mu$ is the service rate (expected service time is $1/\mu$)
• $\theta$ is the restart rate (expected service time is $1/\theta$)

The simulation stops at time $\tau$. Jobs that entered before that time are guaranteed to be served. $\mu > \lambda$ is guaranteed.

I want to observe :

• $\mathbb{E}[B]$, the expected service time
• $\mathbb{E}[S]$, the expected sojourn time (departure - arrival)
• $P_{SETUP}$, the probability that the server is restarting when a client arrives
• $P_{OFF}$, the probability that the server is off when a client arrives

I know the theoretical values: $$\mathbb{E}[S] = \frac{1 / \mu}{1 - \rho} + \frac{1}{\theta}$$ $$P_{OFF} = (1 - \rho) \frac{1/\lambda}{1/\lambda + 1/\theta}$$ $$P_{SETUP} = (1 - \rho) \frac{1/\theta}{1/\lambda + 1/\theta}$$

I've programmed this system. Now, I'm wondering how many simulations is enough so that I can conclude the measures obtained from my simulations are representative of the real system.

I'm wondering if I could draw $n$ samples, then $n_2$ samples, with $n_2 = 2 * n$ and test the null hypothesis $H_0 : v = v_2, H_1 : v \neq v_2$. With $v$ any of the above measures from the first batch of $n$ simulation and $v_2$ the same measure from the second batch of $n_2$ simulations. But I fear that it will only prove that doubling the number of simulations is not significant.

I've also seen that we can do the test with only one sample if we have the population measure (and I do) but this gets me nonsensical results. By example if I do

scipy.stats.ttest_1samp(np.random.exponential(1, 1000000), 1)

I get a statistic and pvalue very different from one run to the other and very rarely less than 0.05 which I think is my purpose.