One supplier and multiple consumer model

Question

Consider a kind of game in which there is one card supplier and $n$ player $P=\{p_1,p_2,...p_n\}$ which consumes the cards. suppose the rules of the game are as follows:

1. Card supplier supplies cards at rate $\lambda$, let $X(t)$ be the number of cards at time $t$
2. Each player picks up a card at random times and save all of his cards in variable $K_i$
3. If player $i$ tries to pick up a card at time $t$ and $X(t)=0$ then he must restart from scratch resulting in $K_i=0$ and $P_i=P_i-S$, where $S$ is a constant penalty value.
4. If player $i$ can reach $K_i=A$ then he wins and would be removed from $P$, Where $A$ is a threshold for winning the game. Moreover, $P_i=P_i+R$, where $R$ is a constant profit value.

I want to calculate the Expected value $E[P_i]$. How can I model the problem with the Markov chain? is it posible? or is there any other suggestion that can facilitate my way?