I am struggling with the following problem and am hoping someone can give me some insights.
Consider a regenerative process like the busy period ($BP$) in an M/G/1 queue. As an example, it is well known that the busy period of an M/G/1 queue is an alternating renewal process, which is also a known to be regenerative process.
From my understanding of regenerative processes, at the beginning of a busy period, if say, the random variable $X_1$ represents the number of customers served during that busy period, and $X_2$ represents the number of customers served during a subsequent busy period, then $X_1$ and $X_2$ are independent.
It is also my understanding that functions of random variables of regenerative processes are independent of another between cycles. For instance, if $Z_1$ represents the maximum customers served during $BP_1$, and $Z_n$ is the number of customers served during $BP_n$, then $Z_1$ and $Z_n$ are statistically independent. Furthermore, it is my understanding (taking this one step further) the random variable during a busy period $R_1$ = $Z_{1_{\max}}$- $Z_{1_{min}}$ and for some busy period $n$ $R_n$ = $Z_{n_{\max}}$- $Z_{n_{min}}$ that $R_1$ and $R_2$ are statistically independent as well.
I have simulated the above scenarios extensively and have found that what is claimed above is confirmed (according to my simulations).
My only issue concerns itself with the Min ($Z_{min}$) process during a busy period where $Z_{min}$ is the number of customers served. The problem is that during a busy period the minimum customers served is always 1. So for all busy periods, $Z_{min}$ will always be 1. Thus, the min process is correlated with subsequent busy which is at odds with the concept that functions of regenerative processes are statistically independent.
So the main question is:
1) Is the min process a regenerative process in this case? If so, how can it be correlated?