Alternating distributions of inter-arrival times in a stochastic process I have some difficulty in deciding if this is a form of {alternating} renewal process or not. The description of the problem is as follows. 
-> There are 2 sources, which emit 0 and 1 respectively, with rates $\lambda$1 and $\lambda$2. (The 2 processes are Poisson.)
-> These two sources are combined to form a new process X(t).
-> The counting process N(t) counts the number of times the values on X(t) 'flips', up to time t. [A flip is defined as the event that the present value of X(t)=1, given X(t-1)=0, or X(t)=0, given X(t-1)=1.)]
Is N(t) a renewal process? I understand that the distribution of the next inter-arrival time given that {X(t-1)=0,X(t)=1} is different from the distribution when {X(t-1)=1, X(t)=0}. But I'm unable to understand whether this is, in fact, a form of a renewal process, possibly, an alternating renewal process. 
Thank you.
 A: Name the two regimes as 0 and 1 and adapt the notations accordingly. The, due to the lack of memory of both exponential distributions, this is indeed an alternating renewal process. If the process counts a 0 at time $t$, nothing happens until time $t+s$, where $s$ is exponentially distributed with parameter $\lambda_1$, and at time $t+s$ the process counts a 1. Likewise, if the process counts a 1 at time $t$, nothing happens until time $t+s$, where $s$ is exponentially distributed with parameter $\lambda_0$, and at time $t+s$ the process counts a 0. 
Equivalently, considering for definiteness that at time $t=0$ one starts to wait for a 0, consider $(S_n)_{n\geqslant0}$ defined by $S_0=0$, $S_{2n}=\sum\limits_{k=1}^nD^0_k+D^1_k$ and $S_{2n+1}=S_{2n}+D^0_{n+1}$, where, for $i=0$ and $i=1$, $(D^i_n)_{n\geqslant1}$ is i.i.d. and exponentially distributed with parameter $\lambda_i$, and $(D^0_n)_{n\geqslant1}$ and $(D^1_n)_{n\geqslant1}$ are independent. (For example, $S_5=D^0_1+D^1_1+D^0_2+D^1_2+D^0_3$.) Then, for every $n\geqslant0$, $[N(t)=n]=[S_n\leqslant t\lt S_{n+1}]$, as usual.
For alternating renewal processes, see there, starting at page 21.
