A duality theorem of Heathcote exhibiting a relationship between first passage times of the queue length in the GI/M/1 queue and the busy period of its dual M/G/1 queue is generalized to the phase type queues GI/PH/1 and PH/G/1. The phase type distributions include a number of well-known models such as generalized Erlang and hyperexponential as special cases and form a versatile class with a number of interesting closure properties.
"A Duality Theorem for Phase Type Queues." Ann. Probab. 8 (5) 974 - 985, October, 1980. https://doi.org/10.1214/aop/1176994625