Ergodic Birth and Death Process is Time Reversible

Theorem

Every ergodic birth and death process is time reversible.

Interpretation

Since a birth and death process can only move between neighboring states ($n$ and $n+1$), any transition from $n$ to $n+1$ must eventually be followed by a transition from $n+1$ back to $n$. In the long run, the rates in both directions must balance.

Proof Sketch

In any length of time $t$, the number of transitions from $i$ to $i+1$ must equal (to within 1) the number from $i+1$ to $i$, since the process can only go from $i$ to $i+1$ and must return through $i+1$. As $t\to \infty$, the rates become equal:

${\lambda }_i{\pi }_i={\mu }_{i+1}{\pi }_{i+1}$

which is exactly the time reversibility condition $P_i{q}_{i,i+1}=P_{i+1}{q}_{i+1,i}$ (equivalently, ${\pi }_i{\lambda }_i={\pi }_{i+1}{\mu }_{i+1}$).

Important Consequence

The output process of an M/M/s queue (with $\lambda <s\mu$) in steady state is a Poisson process with rate $\lambda$. This follows because the M/M/s process is a birth-death process (hence time reversible), and going backward in time, the points where the process decreases by 1 (departures) must form a Poisson process, just as the arrival points do going forward.

Related

• Time Reversible CTMC
• Birth and Death Process
• Truncated Time Reversible CTMC