We start out with the balance equations - total input to the queue (i.e. write rate, including copies) is A, and remaining traffic at the end of the queue is A-1:

eq1 = A * Exp[-α/A] == A-1

eq2 = Map[(#/(-A))+1&,eq1]

eq3 = Simplify[Map[1/#&,eq2]]

With a change of variables, Mathematica is able to give us a solution:

eq4 = eq3/.{α/A->t,A->α/t}

soln = Solve[eq4,t,InverseFunctions->True][[1]]

And now change back and we’ve solved for A, as seen in the graph below:

(Subscript[A, lru])[α_]=α/t/.soln

Plot[(Subscript[A, lru])[α],{α,1.03,1.5},AxesOrigin->{1,0}, PlotStyle-> Directive[Thick],TicksStyle->Directive[14]]