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Layered queueing networks describe the simultaneous-resource behaviour of servers that request lower-layer services and wait for them to complete. Layered software systems often follow this model, with messages to request service and receive the results. Their performance has been computed successfully using mean-value queueing approximations. Such systems also have multiservers (which model multi-threaded software processes), multiple classes of service, and what we call deferred operations or "second phases", which are executed after sending the reply message to the requester. Three established MVA approximations for multiclass multiservers are extended to include deferred service, and evaluated within the layered queueing context. Errors ranged from 1% up to about 15%. These servers were then used to model the network file system, as implemented on Linux, to show that the method scales up and gives good accuracy on typical systems, with computation times of a few seconds to a few minutes. This is hundreds of times faster than simulation.