Passenger Inter-arrival times A number (n) Cruise ship passengers disembark at a port of call. n people will arrive on 5400 second interval prior to the ship's departure. then the average inter-arrival time is 5400/n = 54 seconds.  Assume that all passengers are anti-social singles and that they do not return in groups.  At most 1 passenger is arriving per second.
Questions: Why would the inter-arrival times of passengers be modeled with a Poisson distribution?  What are the limitations, issues or assumptions of such a model?
 A: We begin with a discussion of connections between Poisson and exponential distributions.
Suppose that $N_t$ random events occur in the time interval $(0,t],$ where $t>0,$ and that $N_t \sim \mathsf{Pois}(\lambda t).$ The probability of seeing none of these Poisson events in $(0,t]$ is $P(N_t = 0) = e^{-\lambda t}.$
Then let $X$ be the waiting time, starting at $t = 0,$ until we see the first event.
$$P(X > t) = 1 - F_X(t) = P(N_t = 0) = e^{-\lambda t},$$
for $t > 0.$ The density function $f_X$ of the CDF $F_X$ is found by
differentiation, so that, for $t > 0,$
$$f_X(t) = F_X^\prime(t) = \lambda e^{-\lambda t}.$$
Thus $X$ has an exponential distribution with rate $\lambda.$ That is,
$X \sim \mathsf{Exp}(\text{rate} = \lambda).$ One can show, using
integration by parts, that $E(X) = SD(X) = 1/\lambda.$
One could run this argument backwards to show that exponential times
between passenger arrivals imply a Poisson distribution for the number
of passengers arriving in a given time interval. (See @whuber's link, just now posted, for additional detail.)
The exponential distribution is commonly used to model waiting 
times---perhaps sometimes even when its use does not really match reality.
Temptations to use the exponential model are (a) its mathematical
simplicity and (b) its no-memory property:
$$P(X > s+t|X > s) = \frac{P(X > s+t, X > s)}{P(X > s)} =
\frac{P(X > s+t)}{P(X > s)}\\ 
= \frac{\exp[-\lambda(s+t)]}{\exp[-\lambda s]} = e^{-\lambda t}
= P(X > t).$$
Thus, in order to use an exponential model, we need not take into
account how long a passenger has already been waiting for 'arrival' in order to
find that passenger's additional waiting time to arrival. Also, as you say,
there is a Poisson assumption that customers arrive independently of ona
another.
In any one application of Poisson-exponential modeling, it is unlikely
that all of the independence and no-memory assumptions are exactly true.
However, the exponential distribution has been widely and often successfully
used to model the behavior of real-life waiting time and queueing ('waiting line') situations.
