Exact analysis of a split--merge queue with latent Erlang-factor dependent subtask times

This paper studies a two-server split–merge queue with positively dependent subtask service times modeled through a latent-factor bivariate Erlang construction. An exact characterization of the split–merge completion time is obtained, including explicit formulas for its first two moments and the resulting mean waiting time. Under fixed marginal service-time distributions, independence is shown to stochastically increase the completion time and hence overestimate mean waiting time. Numerical illustrations show that this benchmark gap can be substantial.