# Search Results

###### Mixed Vehicle Flow At Signalized Intersection: Markov Chain Analysis

## Abstract

We assume that a Poisson flow of vehicles arrives at isolated signalized intersection, and each vehicle, independently of others, represents a random number *X* of passenger car units (PCU’s). We analyze numerically the stationary distribution of the queue process {*Zn*}, where *Zn* is the number of PCU’s in a queue at the beginning of the *n*-th red phase, *n* → *∞*. We approximate the number *Y _{n}* of PCU’s arriving during one red-green cycle by a two-parameter Negative Binomial Distribution (NBD). The well-known fact is that {

*Zn*} follow an infinite-state Markov chain. We approximate its stationary distribution using a finite-state Markov chain. We show numerically that there is a strong dependence of the mean queue length

*E*[

*Zn*] in equilibrium on the input distribution of

*Y*and, in particular, on the ”over dispersion” parameter

_{n}*γ*=

*Var*[

*Y*]

_{n}*/E*[

*Y*]. For Poisson input,

_{n}*γ*= 1.

*γ >*1 indicates presence of heavy-tailed input. In reality it means that a relatively large ”portion” of PCU’s, considerably exceeding the average, may arrive with high probability during one red-green cycle. Empirical formulas are presented for an accurate estimation of mean queue length as a function of load and

*g*of the input flow. Using the Markov chain technique, we analyze the mean ”virtual” delay time for a car which always arrives at the beginning of the red phase.

###### Using Deficit Functions for Crew Planning in Aviation

## Abstract

We use deficit functions (DFs) to decompose an aviation schedule of aircraft flights into a minimal number of periodic and balanced chains (flight sequences). Each chain visits periodically a set S of airports and is served by several cockpit crews circulating along the airports of this set. We introduce the notion of ”chunks” which are a sequence of flights serviced by a crew in one day according to contract regulations. These chunks are then used to provide crew schedules and rosters. The method provides a simplicity for the construction of aircraft schedules and crew pairings which is absent in other approaches to the problem.

###### Failure Development in a System of Two Connected Networks

We consider a pair of networks *A *and *B *which are subject to failures of their components. In *A, *edges are subject to failure, and *A *fails when it disintegrates into several isolated clusters each containing a single terminal. Edges of *A *fail in random order and their failure moments follow Poisson process. After *A *has failed, terminal α of *A *causes a failure (”attacks”) on *R*
_{α} randomly chosen non terminal nodes of network *B*. All edges incident to an attacked node are erased. The ”attacks” take negligible time. Network *B *failure takes place if it loses its terminal connectivity. We study the probability that *B *will be in failure state at moment *t *as a function of *t *and *R * = ∑ R_{α} The main formal tools which we use are the D-spectra (signatures) of networks *A *and *B *and de Moivre’s combinatorial formula.

###### On The Problem of Constructing Routes, Part I: Preface

## Abstract

This is a preface of the translation of the 1967 paper by Linis and Maksim, “On the problem of constructing routes” (in Russian) (in the Proceedings of the Institute of Civil Aviation Engineering, Issue 102, pp. 36-45). It marks 50-year to the deficit function (DF) model initially developed in this 1967 work; the DF model then paved the way to further research of vehicle-fleet management in terms of optimal routing and scheduling. The merit of this translation is to describe the roots of the DF modelling to enable further studies to emerge with more contributions.