On Time-Changed Linear Birth–Death–Immigration Process

We introduce and study a time-changed variant of the linear birth–death process under an immigration effect. Here, the time is changed via an inverse stable subordinator. It is shown that the state probabilities of this immigration model are governed by a system of fractional differential equations. We obtain the explicit expressions for its transient probabilities in three different cases of its birth, death and immigration rates. Also, some particular cases of this process are studied in detail.