Vessels arrival process and its application to the SHIP/M/infinity queue

In modeling of port dynamics it seems reasonable to assume that the ships arrive on a somewhat scheduled basis and that there is a constant lay period during which, in a uniform way, each vessel can arrive at the port. In the present paper, we study the counting process $N(t)$ which represents the number of scheduled vessels arriving during the time interval $(0,t]$, $t>0$. Specifically, we provide the explicit expressions of the probability generating function, the probability distribution and the expected value of $N(t)$. In some cases of interest, we also obtain the probability law of the stationary counting process representing the number of arrivals in a time interval of length $t$ when the initial time is an arbitrarily chosen instant. This leads to various results concerning the autocorrelations of the random variables $X_i$, $i\in {\mathbb Z}$, which give the actual interarrival time between the $(i-1)$-th and the $i$-th vessel arrival. Finally, we provide an application to a stochastic model for the queueing behavior at the port, given by a queueing system characterized by stationary interarrival times $X_i$, exponential service times and an infinite number of servers. In this case, some results on the average number of customers and on the probability of an empty queue are disclosed.