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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.

user68468
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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.

Did
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  • Thank you! I still have some trouble with the notion of having a renewal process with unequal inter-renewal intervals. Do we consider composite intervals ($D^0_k,D^1_k$), which are i.i.d. and hence categorize these as renewal? – user68468 Mar 25 '13 at 21:14
  • Watch out: the set of renewal times is $(S_n)n$ and the time intervals separating them are the intervals $(S_n,S{n+1})$. Their lengthes are distributed alternatively as any $D^0_n$ and as any $D^1_n$. – Did Mar 25 '13 at 23:12
  • Yeah, but my confusion lies in the fact that the $(S_n,S_{n+1})$ intervals are not i.i.d.; instead, they are exponential alternatively with $\lambda_1$ and $\lambda_2$, whereas the definition of a renewal process says that the inter-renewal intervals $(S_n,S_{n+1})$ must be i.i.d. – user68468 Mar 25 '13 at 23:20
  • Watch out again: the i.i.d. things in the basic version are the $S_{n+1}-S_n$, not what you wrote. // OK, so this is an extension of the basic (homogeneous) concept. Great! For alternating renewal processes, see there, starting at page 21. – Did Mar 26 '13 at 06:26