Distinct count for multiple fact tables in the same cube

I'm fairly new to working with SSAS, but have been working with DW environments for many years.
I have a cube which has 4 fact tables.  The central fact table is Encounter and then I also have Visit, Procedure and Medication.  Visit, Procedure and Medication all join to Encounter on Encounter Key.  The relationship between Encounter
and Procedure and Encounter and Medication are both an optional 1 to 1.  The relationship between Encounter and Visit is an optional 1 to many.
Each of the fact tables join to the Patient dimension on the Patient Key.  The users are looking for a distinct count of patients in all 4 fact tables.  
What is the best way to accomplish this so that my cube does not talk all day to process?  Please let me know if you need any more information about my cube in order to answer this.
Thanks for the help,

Hi Andy,
Each distinct count measure cause an ORDER BY clause in the SELECT sent to the relational data source during processing. In SSAS 2005 or later, it creates a new measure group for each distinct count measure(it's a technique strategy for improving perormance).
Besides, please take a look at the following distinct count optimization techniques:
Create Customized Aggregations
Define a Processing Plan
Create Partitions of Equal Size
Use Partitions Comprised of a Distinct Range of Integers
Distribute the Hash of Your UserIDs
Modulo Function
Hash Function
Choose a Partitioning Strategy
For more detail information, please refer to the article below:
Analysis Services Distinct Count Optimization:
In addition, here is a good article about SSAS Best Practices for your reference:
If you have any feedback on our support, please click
Hope this helps.
Elvis Long
TechNet Community Support

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