A Paradox of the Average Waiting Time for the Case of a Single Bottleneck on the Commuters’ Route

Average waiting time is considered as one of the basic performance indicators for a bottleneck zone on a route for commuter traffic. It turns out that the average waiting time in a queue remains paradoxically unchanged regardless of how fast the queue dissolves for a single bottleneck problem. In this study, the paradox is verified theoretically for the deterministic case with constant arrival and departure rates. Consistent results with the deterministic case have also been obtained by simulation runs for which vehicle interarrival time is a random variable. Results are tabulated for interarrival times which have uniform, triangular, normal, and exponential distributions along with a statistical verification of the average waiting time paradox.

An Uncertain Single-Server Queueing Model

In the queueing theory, the interarrival times between customers and the service times for customers are usually regarded as random variables. This paper considers human uncertainty in a queueing system, and proposes an uncertain queueing model in which the interarrival times and the service times are regarded as uncertain variables. The busyness index is derived analytically which indicates the service efficiency of a queueing system. Besides, the uncertainty distribution of the busy period is obtained.

Scheduling and simulation of maritime traffic in congested waterways: an application to the Strait of Istanbul

The aim of this study is to develop a simulation model which is capable of mimicking actual vessel arrival patterns and vessel entrance decisions (which are made based on expert opinions generally) on congested, narrow waterways. The model is tested on the transit traffic pattern in the Strait of Istanbul. Based on a heuristic scheduling algorithm, this model decides entrance times and vessel types on the strait. The model, with different policies for day and night traffic, is run for a period of seven years with 20 replications for each year. The performance measures of the model are: average interarrival times, number of vessels passed and entrance times for each successive vessel pair in both traffic directions. The model results are congruent with the actual results of performance measures. Therefore, it may be deduced that the proposed algorithm can be a guide for operators regarding scheduling decisions in congested, narrow waterways.

Measuring Discrepancies Between Poisson and Exponential Hawkes Processes

Poisson processes are widely used to model the occurrence of similar and independent events. However they turn out to be an inadequate tool to describe a sequence of (possibly differently) interacting events. Many phenomena can be modelled instead by Hawkes processes. In this paper we aim at quantifying how much a Hawkes process departs from a Poisson one with respect to different aspects, namely, the behaviour of the stochastic intensity at jump times, the cumulative intensity and the interarrival times distribution. We show how the behaviour of Hawkes processes with respect to these three aspects may be very irregular. Therefore, we believe that developing a single measure describing them is not efficient, and that, instead, the departure from a Poisson process with respect to any different aspect should be separately quantified, by means of as many different measures. Key to defining these measures will be the stochastic intensity and the integrated intensity of a Hawkes process, whose properties are therefore analysed before introducing the measures. Such quantities can be also used to detect mistakes in parameters estimation.

Ruin Probabilities of Continuous-Time Risk Model with Dependent Claim Sizes and Interarrival Times

In this paper we investigate an insurance continuous-time risk model when the claim sizes and inter-arrival times are m-dependent random variables. We provide an upper exponential bound for the ruin probability.