I have a problem to solve where N users chooses value with uniform distribution between 3 and 7 initially. Then every second, every user decrements its value (like if 7 is chosen 7, then it becomes 6, if 3, then it becomes 2).
I want to calculate expected number of users reaching state when they have the value of 1.
I have solved for getting probability $P_1$ to be in state with value of 1 using Markov chain, but how can I know the number of users every second that reach the value of 1 ? After getting to the value of 1 each user again chooses a number between 3 and 7 randomly and process continues.
It is not at all clear from your following statement:
exactly what it is that you want to calculate. Given the scenario you have described, however, the state of every user can be represented by a $7$-state Markov chain with transition matrix $$ P=\pmatrix{0&0&\frac{1}{5}&\frac{1}{5}&\frac{1}{5}&\frac{1}{5}&\frac{1}{5}\\ 1&0&0&0&0&0&0\\ 0&1&0&0&0&0&0\\ 0&0&1&0&0&0&0\\ 0&0&0&1&0&0&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&1&0}\ . $$ and initial distribution $$ \pi_1=\pmatrix{0&0&\frac{1}{5} &\frac{1}{5} &\frac{1}{5} &\frac{1}{5} &\frac{1}{5}}\ . $$ The distribution $\ \pi_t\ $ of a user's state at time $\ t\ $ is given by $$ \pi_t=\pi_1P^{t-1}\ . $$ If there are $\ N\ $ users at the start, then the expected number $\ e_{tj}\ $ of users in state $\ j\ $ at time $\ t\ $ is given by $$ e_{tj}=N\pi_{tj}\ . $$ If you want an explicit formula for $\ \pi_{t1}\ $, you can get it in terms of the eigenvalues of $\ P\ $, which are the roots of its characteristic equation: $$ x^7-\frac{1}{5}\left(x^4+x^3+x^2+x+1\right)=0\ . $$ The stationary distribution of the chain is $$ \pi_\infty=\pmatrix{\frac{1}{5} &\frac{1}{5} &\frac{1}{5} &\frac{4}{25} &\frac{3}{25} &\frac{2}{25} &\frac{1}{25}}\ , $$ so for sufficiently large $\ t\ $, the expected number of users in sate $\ 1\ $ will be $$ N\pi_{t1}\approx \frac{N}{5}\ . $$