  To effectively support the execution of quantum network applications for
multiple sets of user-controlled quantum nodes, a quantum network must
efficiently allocate shared resources. We study traffic models for a type of
quantum network hub called an Entanglement Generation Switch (EGS), a device
that allocates resources to enable entanglement generation between nodes in
response to user-generated demand. We propose an on-demand resource allocation
algorithm, where a demand is either blocked if no resources are available or
else results in immediate resource allocation. We model the EGS as an Erlang
loss system, with demands corresponding to sessions whose arrival is modelled
as a Poisson process. To reflect the operation of a practical quantum switch,
our model captures scenarios where a resource is allocated for batches of
entanglement generation attempts, possibly interleaved with calibration periods
for the quantum network nodes. Calibration periods are necessary to correct
against drifts or jumps in the physical parameters of a quantum node that occur
on a timescale that is long compared to the duration of an attempt. We then
derive a formula for the demand blocking probability under three different
traffic scenarios using analytical methods from applied probability and
queueing theory. We prove an insensitivity theorem which guarantees that the
probability a demand is blocked only depends upon the mean duration of each
entanglement generation attempt and calibration period, and is not sensitive to
the underlying distributions of attempt and calibration period duration. We
provide numerical results to support our analysis. Our work is the first
analysis of traffic characteristics at an EGS system and provides a valuable
analytic tool for devising performance driven resource allocation algorithms.
