Mysql: Possible to increase the priority of your query on a database?
As has been pointed out, if you are using MyISAM then SELECT HIGH_PRIORITY might give you some advantage... but if you need performance under concurrency, you shouldn't be using MyISAM since the only level of locking available is table-level locking, and table locking occurs implicitly with a significant proportion of queries (hence the rationale for the old high/low priority stuff).

Or you could obtain an explicit lock on the tables involved while you run your queries, but that's probably not going to be a solution.

The question raised by your suggestion that you are "competing with other users" is whether you know for a fact that this is the major cause of your performance issues or, alternatively, whether you are dealing with a query that isn't properly optimized.

For all practical purposes, MySQL does not have any mechanisms for prioritization like this.  Every query is executed with comparable priority, subject to the limits of disk I/O, CPU clock speed and number of cores, available memory (and how it's allocated)... which means if you're not getting a level of performance consistent with the quality of your code, your server needs more of one or more kinds of resources, or you need to scale out with a read replica, or migrate some of the databases to a different machine... or, of course, if the machine isn't dedicated exclusively to running MySQL, then that's potentially problematic as well. 

See also: Is there a way to achieve Workload Management in MySQL?