Samuel S. Chiu

Optimal Server Location on a Network Operating as an M/G/1 Queue

This paper extends Hakimis one-median problem by embedding it in a general queueing context. Demands for service arise solely on the nodes of a network G and occur in time as a Poisson process. A single mobile server resides at a facility located on G. The server, when available, is dispatched immediately to any demand that occurs. When a demand finds the server busy with a previous demand, it is either rejected Model 1 or entered into a queue that is depleted in a first-come, first-served manner Model 2. Service time for each demand comprises travel time to the scene, on-scene time, travel time back to the facility and possibly additional off-scene time. One desires to locate the facility on G so as to minimize average cost of response, which is either a weighted sum of mean travel time and cost of rejection Model 1, or the sum of mean queueing delay and mean travel time. For Model 1, one finds that the optimal location reduces to Hakimis familiar nodal result. For Model 2, nonlinearities in the objective function can yield an optimal solution that is either at a node or on a link. Properties of the objective function for Model 2 are utilized to develop efficient finite-step procedures for finding the optimal location. Certain interesting properties of the optimal location as a function of demand rate are also developed.

A Method to Calculate Steady-State Distributions of Large Markov Chains by Aggregating States

This paper develops an efficient iterative algorithm to calculate the steady-state distribution of nearly all irreducible discrete-time Markov chains. Computational experiences suggest that, for large Markovian systems more than 130 states, the proposed algorithm can be ten times faster than standard Gaussian elimination in finding solutions to an accuracy of 0.1%. The proposed algorithm is developed in three stages. First, we develop a very efficient algorithm for determining steady-state distributions of a restricted class of Markovian systems. A second result establishes a relationship between a general irreducible Markovian system and a system in the restricted class of Markovian systems. Finally, we combine the two results to produce an efficient, iterative algorithm to solve Markov systems. The paper concludes with a discussion of the observed performance of the algorithm.

LOCATION OF COMPETING FACILITIES IN A USER-OPTIMIZING ENVIRONMENT WITH MARKET EXTERNALITIES

In many location models, it is assumed that customers select the facility they will patronize based on the distance to that facility, or perhaps on a function of distances between the customer and the different available facilities. We consider a location problem in which customers select a facility based not only on travel time or distance to the facility, but also on negative externalities associated with the market share of the facility. A customer from any particular location frequents the facility that minimizes his travel time to that facility plus an externality cost that depends on the aggregated actions of all customers in the system. We consider the case of two competing facilities, each of which wishes to locate to maximize its market share. We specialize our analysis to the case of a tree network. We characterize the optimal facility locations, and develop an O ( n 2 ) algorithm for finding them.

A center location problem with congestion

We introduce a group of facility location problems whose objective involves both congestion and covering effects. For the Stochastic Queue Center problem, a single facility is to be located on a network to minimize expected response time (travel time plus expected queue delay) to the furthest demand point. We demonstrate certain convexity properties of the objective function on a general network, and show how the optimal location can be found using a finite-step algorithm. On a tree network, we characterize the optimal location trajectory as a function of the customer call rate. We compare this problem to the median, center, and Stochastic Queue Median problems. We then consider several different extensions which incorporate probabilistic travel times and/or distribution of demands.

A Unified Family of Single-Server Queueing Location Models

In this paper, we introduce a general class of single-server network location models which includes the median, center, stochastic queue median, stochastic queue center, Lp norm, and other location problems in a single parametric framework. The model incorporates both queueing effects and a cost function similar to a disutility function, which is nonlinear in response time. We develop a number of properties of this class of models, and present a method for evaluating a range of different objective functions for any given problem. The robustness of the median and center objectives as solution concepts is highlighted.

Parametric Facility Location on a Tree Network with an L p -Norm Cost Function

This paper examines the optimal location of a single facility on a tree network with the objective being to minimize the sum of weighted distances from each node measured by an L p -norm-based cost function. Our goal is to trace the trajectory of the optimal location when the L p -norm parameter p varies from one to infinity. Convexity of the objective function, when p is fixed, allows us to perform the parametric analysis. We characterize possible trajectory paths, and show that some surprising trajectory results can occur.

Locating the two-median of a tree network with continuous link demands

Typical formulations of thep-median problem on a network assume discrete nodal demands. However, for many problems, demands are better represented by continuous functions along the links, in addition to nodal demands. For such problems, optimal server locations need not occur at nodes, so that algorithms of the kind developed for the discrete demand case can not be used. In this paper we show how the 2-median of a tree network with continuous link demands can be found using an algorithm based on sequential location and allocation. We show that the algorithm will converge to a local minimum and then present a procedure for finding the global minimum solution.

Technical Note-A Dominance Theorem for the Stochastic Queue Median Problem

In this note we prove a dominance property of the set of Hakimi medians for the Stochastic Queue Median SQM problem on a network. The SQM minimizes the average response time of a mobile server to a random request for service which occurs in accordance with independent Poisson processes from nodes of a network. We constructively find two strictly positive ranges of the Poisson arrival rate over which a subset of the Hakimi medians will dominate all other locations on the network. These ranges are at the extremes of the arrival rates in which the SQM problem is feasible. Specifically, these intervals are server utilization ranges covering zero and one, respectively.

Trajectory Analysis of the Stochastic Queue Median in a Plane with Rectilinear Distances

In this paper we analyze the trajectory of stochastic queue median (SQM) location problem in a planar region with a rectilinear travel metric. The location objective is to minimize expected response time to customers (that is, travel time plus queue delay). We introduce a methodology for parametric analysis of planar location problems which is potentially applicable to other location problems as well. Using the methodology, we demonstrate strong parallels between our planar SQM problem and the same problem on a tree network. We show how the optimal SQM location must occur in a certain region of the plane. Given a mild regularity condition, we develop trajectory results for the optimal location as a function of the customer call rate, and we derive a simple necessary and sufficient ratio condition which characterizes points on the optimal trajectory, and present an algorithm for finding that trajectory. We also analyze the problem in the degenerate case when the regularity condition is violated. Finally, we extend our results to the planar stochastic expected queue median problem, which incorporates stochastic travel times.

Establishing Continuity of Certain Optimal Parametric Facility Location Trajectories

Parametric elements arise naturally in many facility location problems. In this note we establish that if the objective function for a single-server parametric location problem is strictly convex in a certain sense, and if the objective function possesses certain reasonable continuity properties, then the parametric trajectory of the optimal location is continuous.