Time-changing and truncating K-capacity queues from one K to another

For 0<K'<K≤∞, we obtain a K'-capacity queue from a K-capacity queue through a random time change and a truncation, provided arrivals are Poisson or service is exponential. In the case of an M/G/1/K queue, the time change erases service intervals that begin with more than K' customers in the systems. This constructions yields a straightforward sample path proof of Keilson's result on the proportionality of the ergodic queue length probabilities in M/G/1/K queues. The same approach proves a strengthened result for "detailed" state probabilities. It also reproduces a proportionality result for a vacation model, due to Keilson and Servi. A "dual" construction yields a different kind of proportionality for the G/M/1/K queue.