Generating uniform random numbers in Lua

Standard C random numbers generator used in Lua isn't guananteed to be good for simulation. The words "Markov chain" suggest that you may need a better one. Here's a generator widely used for Monte-Carlo calculations:

local A1, A2 = 727595, 798405  -- 5^17=D20*A1+A2
local D20, D40 = 1048576, 1099511627776  -- 2^20, 2^40
local X1, X2 = 0, 1
function rand()
    local U = X2*A2
    local V = (X1*A2 + X2*A1) % D20
    V = (V*D20 + U) % D40
    X1 = math.floor(V/D20)
    X2 = V - X1*D20
    return V/D40
end

It generates a number between 0 and 1, so r = math.floor(rand()*10) + 1 would go into your example. (That's multiplicative random number generator with period 2^38, multiplier 5^17 and modulo 2^40, original Pascal code by http://osmf.sscc.ru/~smp/)

math.randomseed(os.clock()*100000000000)
for i=1,3 do
    math.random(10000, 65000)
end

Always results in new random numbers. Changing the seed value will ensure randomness. Don't follow os.time() because it is the epoch time and changes after one second but os.clock() won't have the same value at any close instance.

You need to run math.randomseed() once before using math.random(), like this:

math.randomseed(os.time())

From your comment that you saw the first number is still the same. This is caused by the implementation of random generator in some platforms.

The solution is to pop some random numbers before using them for real:

math.randomseed(os.time())
math.random(); math.random(); math.random()

Note that the standard C library random() is usually not so uniformly random, a better solution is to use a better random generator if your platform provides one.

Reference: Lua Math Library