Correlated interarrival time Poisson process (CIPP) has been proposed in Proceedings of the Fifth Biennial Conference on Signal Processing and Communications (SPCOM'99), for modeling both the composite arrival process of packets in broadband networks and the individual video source modeling. The CIPP - a generalization of the Poisson process - is a stationary counting process and is parameterized by correlation parameter `ρ', the degree of correlation in adjacent interarrivals and `λ', the intensity of the process. In this paper, we develop the theory for CIPP/M/1 queue and undertake the performance modeling of statistical multiplexer with VBR video traffic in broadband networks using the CIPP/M/1 queue. We first derive the expressions for stationary distributions for queue length and waiting time in a CIPP/M/1 queue. Then, we derive the queuing performance measures of interest. For reasons of feasibility of theoretical performance modeling and realistic compulsions, we propose a deterministic smoothing with random (geometrically distributed) packet sizes. We simulate a queue with (thus smoothed) VBR video trace data as input to compare with the theoretical performance measures derived above. Experimental results show that the CIPP/M/1 queue models well the statistical multiplexer performance with the real-world MPEG-1 VBR video traffic input.