离散时间排队MAP/PH/3

本文研究具有马尔可夫到达过程的离散时间排队MAP/PH/3,系统中有三个服务台,每个服务台对顾客的服务时间均服从位相型分布。运用矩阵几何解的理论,我们给出了系统平稳的充要条件和系统的稳态队长分布。同时我们也给出了到达顾客所见队长分布和平均等待时间。     

On the uniqueness of solutions to the Poisson equations for average cost Markov chains with unbounded cost functions

We consider the Poisson equations for denumerable Markov chains with unbounded cost functions. Solutions to the Poisson equations exist in the Banach space of bounded real-valued functions with respect to a weighted supremum norm such that the Markov chain is geometrically ergodic. Under minor additional assumptions the solution is also unique. We give a novel probabilistic proof of this fact using relations between ergodicity and recurrence. The expressions involved in the Poisson equations have many solutions in general. However, the solution that has a finite norm with respect to the weighted supremum norm is the unique solution to the Poisson equations. We illustrate how to determine this solution by considering three queueing examples: a multi-server queue, two independent single server queues, and a priority queue with dependence between the queues.     

The maximin HAZMAT routing problem

The hazardous material routing problem from an origin to a destination in an urban area is addressed. We maximise the distance between the route and its closest vulnerable centre, weighted by the centre’s population. A vulnerable centre is a school, hospital, senior citizens’ residence or the like, concentrating a high population or one that is particularly vulnerable or difficult to evacuate in a short time. The potential consequences on the most exposed centre are thus minimized. Though previously studied in a continuous space, the problem is formulated here over a transport (road) network. We present an exact model for the problem, in which we manage to significantly reduce the required variables, as well as an optimal polynomial time heuristic. The integer programming formulation and the heuristic are tested in a real-world case study set in the transport network in the city of Santiago, Chile.     

Multi-objective vehicle routing problems

Routing problems, such as the traveling salesman problem and the vehicle routing problem, are widely studied both because of their classic academic appeal and their numerous real-life applications. Similarly, the field of multi-objective optimization is attracting more and more attention, notably because it offers new opportunities for defining problems. This article surveys the existing research related to multi-objective optimization in routing problems. It examines routing problems in terms of their definitions, their objectives, and the multi-objective algorithms proposed for solving them.     

Ordered interval routing schemes

An Interval Routing Scheme (IRS) represents the routing tables in a network in a space-efficient way by labeling each vertex with an unique integer address, and the outgoing edges at each vertex with disjoint subintervals of these addresses. An IRS that has at most k intervals per edge label is called a k-IRS. In this paper, we propose a new type of interval routing scheme, called an Ordered Interval Routing Scheme (OIRS), that uses an ordering of the outgoing edges at each vertex and allows non-disjoint intervals in the labels of those edges. We show for a number of graph classes that using an OIRS instead of an IRS reduces the size of the routing tables in the case of optimal routing, i.e., routing along shortest paths. We show that optimal routing in any k-tree is possible using an OIRS with at most 2^k−1 intervals per edge label, although the best known result for an IRS is 2^k+1 intervals per edge label. Any torus has an optimal 1-OIRS, although it may not have an optimal 1-IRS. We present similar results for the Petersen graph, k-garland graphs and a few other graphs.     

A multi-stop routing problem

The problem of determining the sequence of stops and the amount of load to carry in each segment route, named the Multi-Stop Routing Problem (MSRP) is addressed. A 0/1 mixed integer linear program and formulation refinements which facilitate the solution process are presented. Since the constraint set of the MSRP includes 0/1 mixed rows, valid inequalities for this type of regions are presented. Then these results are applied to the constraint set of the routing problem, presenting additional valid inequalities. In addition, polynomial separation algorithms associated with the valid inequalities are given, computational results are also included.     

Multi-objective evacuation routing in transportation networks

In this paper, the optimal design and analysis of evacuation routes in transportation networks is examined. An methodology for optimal egress route assignment is suggested. An integer programming (IP) formulation for optimal route assignment is presented, which utilizes M/G/c/c state dependent queueing models to cope with congestion and time delays on road links. M/G/c/c simulation software is used to evaluate performance measures of the evacuation plan: clearance time, total travelled distance and blocking probabilities. Extensive experimental results are included.     

Rich vehicle routing problems: From a taxonomy to a definition

Over the last years, several variants of multi-constrained Vehicle Routing Problems (VRPs) have been studied, forming a class of problems known as Rich Vehicle Routing Problems (RVRPs). The purpose of the paper is twofold: (i) to provide a comprehensive and relevant taxonomy for the RVRP literature and (ii) to propose an elaborate definition of RVRPs. To this end, selected papers addressing various cases are classified using the proposed taxonomy. Once the articles have been classified, a cluster analysis based on two discriminating criteria is performed and leads to the definition of RVRPs.     

The pyramidal capacitated vehicle routing problem

This paper introduces the pyramidal capacitated vehicle routing problem (PCVRP) as a restricted version of the capacitated vehicle routing problem (CVRP). In the PCVRP each route is required to be pyramidal in a sense generalized from the pyramidal traveling salesman problem (PTSP). A pyramidal route is defined as a route on which the vehicle first visits customers in increasing order of customer index, and on the remaining part of the route visits customers in decreasing order of customer index.     

The effects of triangle inequality on the vehicle routing problem

Routing problems often utilize experimental networks to represent real world scenarios. However most ignore the inclusion of triangle inequality violations, a phenomenon resulting from delays or rounding errors within a network. This work evaluates the effect of both frequency – the number of violations – and severity – the degree of intensity of a violation – of triangle inequality and evaluates both solution quality and solution time based on Simulated Annealing, Ant Colony Optimization and Savings Algorithm methods. Findings indicate that while both frequency and severity degrade solution quality, increased levels of frequency and severity together result in significant adverse affects to solution quality. Solution time, however, is not impacted by the presence of triangle inequality violations within the network. This information should encourage practitioners to identify delays and maintain the presence of triangle inequality violations in a network to ensure accuracy of solution quality.     

The Busy Period and the Waiting Time Analysis of a MAP/M/c Queue with Finite Retrial Group

Abstract

We concentrate on the analysis of the busy period and the waiting time distribution of a multi-server retrial queue in which primary arrivals occur according to a Markovian arrival process (MAP). Since the study of a model with an infinite retrial group seems intractable, we deal with a system having a finite buffer for the retrial group. The system is analyzed in steady state by deriving expressions for (a) the Laplace–Stieltjes transforms of the busy period and the waiting time; (b) the probabiliy generating functions for the number of customers served during a busy period and the number of retrials made by a customer; and (c) various moments of quantites of interest. Some illustrative numerical examples are discussed.     

Continuity of the M/G/c queue

Consider an M/G/c queue with homogeneous servers and service time distribution F. It is shown that an approximation of the service time distribution F by stochastically smaller distributions, say F _n , leads to an approximation of the stationary distribution π of the original M/G/c queue by the stationary distributions π _n of the M/G/c queues with service time distributions F _n . Here all approximations are in weak convergence. The argument is based on a representation of M/G/c queues in terms of piecewise deterministic Markov processes as well as some coupling methods.      

Periodic solution to the time-inhomogeneous multi-server Poisson queue

We derive the periodic family of asymptotic distributions and the periodic moments for number in the queue for the multi-server queue with Poisson arrivals and exponential service for time-varying periodic arrival and departure rates, and time-varying periodic number of servers. The method is a straight-forward application of generating functions.     

Computing inter-site distances for routing and scheduling problems

Routing and scheduling problems have received considerable attention in the literature in terms of model building and algorithm development. On these fronts, progress has been substantial. However, one often neglected (yet critical) aspect concerning the use of these models and algorithms is their data requirements. In particular, the distance matrix yielding the shortest distance between each pair of sites (nodes) represents a major portion of the data required by all such problems. Yet, such data are seldom available with the degree of accuracy desired and often are not available at all.This paper describes an efficient method for obtaining this distance matrix that is based on the underlying road structure for the geographic region in question. Thus, the distances obtained reflect ‘actual’ distances. Finally, the paper presents some brief computational experience and discusses an implementation concerning the routing of environmental inspectors in the state of Pennsylvania.     

Branch-and-cut algorithms for the split delivery vehicle routing problem

In this paper we present two exact branch-and-cut algorithms for the Split Delivery Vehicle Routing Problem (SDVRP) based on two relaxed formulations that provide lower bounds to the optimum. Procedures to obtain feasible solutions to the SDVRP from a feasible solution to the relaxed formulations are presented. Computational results are presented for 4 classes of benchmark instances. The new approach is able to prove the optimality of 17 new instances. In particular, the branch-and-cut algorithm based on the first relaxed formulation is able to solve most of the instances with up to 50 customers and two instances with 75 and 100 customers.     

City-courier routing and scheduling problems

This research seeks to propose innovative routing and scheduling strategies to help city couriers reduce operating costs and enhance service level. The strategies are realized by constructing a new type of routing and scheduling problem. The problem directly takes into account the inherent physical and operating constraints associated with riding in city distribution networks, which makes the problem involve multiple objectives and visiting specified nodes and arcs. Through network transformations, this study first formulates the city-courier routing and scheduling problem as a multi-objective multiple traveling salesman problem with strict time windows (MOMTSPSTW) that is NP-hard and new to the literature, and then proposes a multi-objective Scatter Search framework that seeks to find the set of Pareto-optimal solutions to the problem. Various new and improved sub-procedures are embedded in the solution framework. This is followed by an empirical study that shows and analyzes the results of applying the proposed method to a real-life city-courier routing and scheduling problem.     

An algorithm for node-capacitated ring routing

A strongly polynomial time algorithm is described to solve the node-capacitated routing problem in an undirected ring network.     

Rich routing problems arising in supply chain management

The purpose of this paper is to provide basic models for highly relevant extensions of the classical vehicle routing problem in the context of supply chain management. The classical vehicle routing problem is extended in various ways. We will especially focus on extensions with respect to lotsizing, scheduling, packing, batching, inventory and intermodality. The proposed models allow for a more efficient use of resources, while explicitly taking into account interdependencies among the subproblems. The contribution of this survey is twofold: (i) it provides an overview of recent and suitable literature for the interested scholar and (ii) it presents six integrative models for the above mentioned extensions.