For large queueing network analysis the general computational approach is to utilize decomposition to facilitate computational tractability. To accomplish this individual analysis the input and output streams must be characterized. This usually is done via two-parameter characterizations: the process mean and a variance measure (most commonly the squared coefficient of variation SCV). In most approaches independent and identically distributed (i.i.d.) approximations are used. For multiple input streams and/or multiple (identical) servers, the assumptions of i.i.d. times between arrivals and, similarly, i.i.d. times between departures are particularly theoretically and computationally inaccurate. In this paper we develop a generator for the background multidimensional continuous time Markov chain associated with the inter-departure times for the associated multi-stream and multi-server Markovian queues (where inter-arrival times and service times are Coxian). This generator allows for the computation of the moments of the departure process and the lag-k correlations between successive k-separated departures.