StationaryDistribution[ContinuousMarkovProcess[1,{{-a,a},{b,-b}}]]

Mean[%]//Simplify

StationaryDistribution[DiscreteMarkovProcess[1, {{a,1-a}, {1-b,b}}]]

Mean[%]//Simplify

DiscretePlot[PDF[StationaryDistribution[With[{λ=4,μ=60/12,m=10^2},ContinuousMarkovProcess[1,SparseArray[Join[Table[{i,i+1}->λ,{i,m}],Table[{i+1,i}->μ,{i,m}],Table[{i,i}->-λ-μ,{i,2,m}],{{1,1}->-λ,{m+1,m+1}->-μ}]]]]],k],{k,1,20},ExtentSize->0.5,PlotRange->All]