On the limiting distribution of service times with rewards

摘要

Let X_0,X_1, X_2,··· be a stationary Markov chain with transition from j to k;. If X_0 ≡0 and P{J_n = k,X_n≤ x|(J_0, X_0),···, (J_(n-1),X_(n-1))} = Q_(J_(n-1,k))(x) (a.s) then {(J_n,X_n);n = 0,1,2,···} is called a (J,X)-process. So, if the previous state is situated, the process has relation only to k and x. Now, on the tandem type queue of single 3-stage service, there are an infinite waiting line in front of the first stage service station, and that, next customer occupies the service station as soon as the first stage service station becomes empty. A queue always exists in front of the first stage, but no queue are allowed to be in front of the other stages.