## Queueing and Occupancy: The Linear Case

Imagine a parking lot, consisting of a long, linear strip of slots.
Cars enter at one end and leave by the other. Let’s also stipulate
that each arriving car takes *the first available slot* that it
encounters, that is, it will park in the first empty slot that is
nearest to the parking lot entrance. Cars arrive randomly, with a
given, average interarrival time $\tau_A$. Furthermore, cars occupy
each slot for a random amount of time, but all with a common average
dwell time $\tau_D$.

If we number the slots, starting at the entrance, we may now ask the question: what is the probability that the slot with index $n$ is occupied?