Dispatcher Queue : Queue in which the requests for the various work process types are collected. The dispatcher queue contains a queue for dialog requests, a queue for spool requests and a queue for lock requests, depending on which work process types are configured in this instance. The profile parameter rdisp/elem_per_queue specifies how many requests the queue can contain.

How to check the status of current dispatcher queue :

1.go to transaction SM51 and select the server for which you want to see the queue ,now  go to "Server Name" > "Information" > "Queue Information".

Above information you can get it at OS level also by using DPMON utility.