Hi Martin,
Sheila is right! Note that there is no such a state as (S,0), because as soon as there are 0 customers in the shop, then Greta goes immediately to the restaurant. If we are in state (R,0) and we do an exponential jump with rate \gamma, Greta goes to the shop, sees 0 customers and immediately returns to the restaurant in state (R,0), so the Markov chain does not actually leave that state.
The remaining balance equations are derived in the usual way: you look at each state and write
"rate of jumping out" x "probability of being in that state" = sum of ("rate of jumping in" x "probability of being in the state from where I jump in").
The jumping rates are described in the lines above the balance equations.
Let me know if it's not clear :)
Matteo
Sheila is right! Note that there is no such a state as (S,0), because as soon as there are 0 customers in the shop, then Greta goes immediately to the restaurant. If we are in state (R,0) and we do an exponential jump with rate \gamma, Greta goes to the shop, sees 0 customers and immediately returns to the restaurant in state (R,0), so the Markov chain does not actually leave that state.
The remaining balance equations are derived in the usual way: you look at each state and write
"rate of jumping out" x "probability of being in that state" = sum of ("rate of jumping in" x "probability of being in the state from where I jump in").
The jumping rates are described in the lines above the balance equations.
Let me know if it's not clear :)
Matteo