Are you trying to set up an alert system? For a simple "heartbeat", do a HTTP request to the Tomcat port.
For more elaborate monitoring, you can set up JMX and/or SNMP to view JVM stats. We run Nagios with the SNMP plugin (bridges to JMX) to check Tomcat memory usage and request thread pool size every 10-15 minutes.
http://tomcat.apache.org/tomcat-6.0-doc/monitoring.html
Update (2012):
We have upgraded our systems to use "monit" to check the tomcat process. I really like it. With very little configuration it automatically verifies the service is running, and automatically restarts if it is not. (sending an email alert). It can integrate with the /etc/init.d scripts or check by process name.