In view of the problem of bandwidth constraints in networked control systems, the stabilization of system with stochastic communication logic is studied. The stochastic communication logic is introduced based on Poisson process with time-dependent intensity, under which the function of the Poisson intensity and the estimation error is obtained. Then the update time of system state having Markov characteristic is presented. Moreover, the networked control system is modeled as a Markovian jump linear system. The sufficient condition of almost surely stability and second moment stability for the system is proposed in term of stochastic stability theory. Finally, a numerical example shows the effectiveness of the proposed design approach．