  The BB84 QKD protocol is based on the idea that the sender and the receiver
can reconcile a certain fraction of the teleported qubits to detect
eavesdropping or noise and decode the rest to use as a private key. Under the
present hardware infrastructure, decoherence of quantum states poses a
significant challenge to performing perfect or efficient teleportation, meaning
that a teleportation-based protocol must be run multiple times to observe
success. Thus, performance analyses of such protocols usually consider the
completion time, i.e., the time until success, rather than the duration of a
single attempt. Moreover, due to decoherence, the success of an attempt is in
general dependent on the duration of individual phases of that attempt, as
quantum states must wait in memory while the success or failure of a generation
phase is communicated to the relevant parties. In this work, we do a
performance analysis of the completion time of the BB84 protocol in a setting
where the sender and the receiver are connected via a single quantum repeater
and the only quantum channel between them does not see any adversarial attack.
Assuming certain distributional forms for the generation and communication
phases of teleportation, we provide a method to compute the MGF of the
completion time and subsequently derive an estimate of the CDF and a bound on
the tail probability. This result helps us gauge the (tail) behaviour of the
completion time in terms of the parameters characterising the elementary phases
of teleportation, without having to run the protocol multiple times. We also
provide an efficient simulation scheme to generate the completion time, which
relies on expressing the completion time in terms of aggregated teleportation
times. We numerically compare our approach with a full-scale simulation and
observe good agreement between them.
