Solving a constrained convex optimization problem

Question

I have a function with the following form:

P0 = 1 / Sum[(lambda^i)/((mu^i)*i!), {i, 0, B1}] 
P = N[P0*lambda^B1/(mu^B1*B1!)] 

Where mu and lambda are fixed parameters and B1 is unknown.

The code refers to a Markov chain process that I am calculating.

And I need to find the minimum integer value of B1 that corresponds to the constraint P(B1) <= 0.05

Since I am new to the Mathematica, I find it a bit hard to use a correct function. Could you please point me to a correct way of solving such a problem with Mathematica?

Thank you.

UPD

@ulrich-neumann provided a better way to define the function:

F[mu_, lambda_] := (E^(-(lambda/mu)) lambda^B1 mu^-B1 Gamma[1 + B1])/(B1! Gamma[1 + B1, lambda/mu])

And @akku14 has built on top of the previous commentaries to define B1's boundaries:

NMinimize[{(F[1/3, 8] - .05)^2, F[1/3, 8] <= 0.05, B1 > 1, Element[B1, Integers]}, {B1, 5, 50}] to get {0.0000976008, {B1 -> 30}}

Answer 1

First define your function depending on mu,lambda

F[mu_, lambda_] := (E^(-(lambda/mu)) lambda^B1 mu^-B1 Gamma[1 + B1])/(B1! Gamma[1 + B1, lambda/mu])    

Now it's possible to minimize for given parameters (examplary mu=1/3,lambda=8)

NMinimize[{B1, F[ 1/3, 8 ] <= 0.05, B1 > 0,Element[B1, Integers]}, B1 ]
(*{3., {B1 -> 3}}*)

This result seems to be wrong (Thanks @cvgmt`s comment)! Without the integer constraint NMinimize evaluates

NMinimize[{B1, F[ 1/3, 8 ] <= 0.05, B1 > 0 }, B1 ]
(* B1->29.172 *)

The correct solution follows to B1=30 !