next up previous contents
Next: The Poisson process Up: Markov Processes Previous: The Intensity matrix

Birth-Death processes

A useful class of Markov processes when analyzing queueing systems are birth-death processes. The only possible state transitions in this kind of processes are from i to i-1 or from i to i+1. The transition intensity from state i to i+1 is designated $\lambda_i \geq 0$ for $i \geq 0$ and the transition intensity from state i to state i-1 is designated $\mu_i \geq 0$ for $i \geq 1$.

Figure 4: Model graph for a Birth-death process
\epsfig{file=Pictures/,height=10cm,width=3cm, angle=-90} \end{figure}

The state space of the birth-death process is $\{0,1,2,3,....\}$. The intensity matrix will be of tri-diagonal type since there are only two ways of leaving a state. Hence, we have the intensity matrix

\begin{displaymath}\mathbf{Q}= \left( \begin{array}{cccccc}
-\lambda_0 & \lambd...
... \\
\vdots & \vdots & \ & \ & \ & \ddots
\end{array} \right)

As mentioned earlier, certain types of queuing systems are suitably modeled by birth-death processes. The numbers $\{ \lambda_i \}$ and $\{\mu_i \}$ are interpreted as the arrival rate of the queue and service rate of the server, respectively.

Anders Andersson