Investigate the positive discrete-time MJS with disturbance input, time-varying delay, and uncertain transition probabilities together for the first time, where the transition probability matrix is described by a polytope set.

Provide a necessary and sufficient condition of stochastic stability with regard to the resulting error dynamic system.

Present the exact computation of l_∞-gain index for the resulting stochastically stable error dynamic systems firstly, and derive a necessary and sufficient condition of the l_∞-gain performance correspondingly.

Provide a sufficient condition on the existence of a desired observer to solve the state estimation problem, such that the error dynamic system is stochastically stable with a prescribed l_∞-gain performance.

All the proposed conditions are expressed in linear programming forms, which can be solved effectively by using some existing softwares, and the observer gain matrices can be directly obtained by an effective algorithm.