Create a Rank column/field based on a filter.

Hi rajatagarwal93 ,

Can you try this measure: 

The data is summarized to find the minimum date for each participant id within a Visit ID

Then the min date is used to order by in the RANK DAX function to calculate the ranking. 

RankDoctor =
VAR __Summarized =
    SUMMARIZE (
        FILTER ( ALL ( RankTable ), RankTable[Role] = "Doctor" ),
        RankTable[Visit ID],
        RankTable[Participant ID],
        "MinDate", MIN ( RankTable[Join time] )
    )
VAR __Rank =
    RANK (
        DENSE,
        __Summarized,
        ORDERBY ( [MinDate], ASC ),
        DEFAULT,
        PARTITIONBY ( RankTable[Visit ID] )
    )
RETURN
    IF ( SELECTEDVALUE ( RankTable[Role] ) = "Doctor", __Rank, 0 )

Rank column is from your question. I had it to refer if my measure was correct. 
RankDoctor is the measure from above formula (the last column) and it matches with the output you had given.