Catalytic branching random walk and queueing systems with random number of independent servers

A continuous time branching random walk on the lattice Z in which particles may produce children only at the origin is considered. Assuming that the underlying random walk is symmetric and the offspring reproduction law is critical, we find the asymptotic behavior of the survival probability of the process at time t as t → ∞ and the probability that the number of particles at the origin at time t is positive. We also prove a Yaglom type conditional limit theorem for the total number of particles existing at time t. A relation between the model considered and a queueing system with a random number of independently operating servers is discussed.