The mixture tempered space fractional Poisson process (MTSFPP) is a subordinated variant of the Poisson process studied by Gupta et al. [18]. In this paper, we first obtain some key distributional properties of the MTSFPP and derive the asymptotic behavior of its fractional order moments. We introduce a bivariate mixture tempered space fractional Poisson process (BMTSFPP) by time changing the bivariate Poisson process with an independent mixture tempered stable subordinator. We provide explicit expression to the probability law of the BMTSFPP and derive its governing differential equation. The Lévy measure for the BMTSFPP is also formulated. A bivariate shock model based on the BMTSFPP is investigated to predict the failure time of a system subject to two types of random shocks. The system is supposed to fail when the sum of the two types of shocks attains a certain random integer-valued threshold. The analysis focuses on determining the hazard rates, failure density, reliability function, and the probability that the system fails due to a specific type of shock. Some special cases and their ageing notions in the context of new better than used (NBU) and new worse than used (NWU) characterizations are also discussed.
