Vessels Arrival Process and its Application to the SHIP/M/ $$\infty$$ ∞ 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], $$t>0$$ 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 $$X_i$$ X i , $$i\in \mathbb {Z}$$ i ∈ Z , which give the actual interarrival time between the $$(i-1)$$ ( 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 $$X_i$$ 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.