A key operational problem for those charged with the security of vulnerable facilities (such as airports or art galleries) is the scheduling and deployment of patrols. Motivated by the problem of optimizing randomized, and thus unpredictable, patrols, we present a class of patrolling games. The facility to be patrolled can be thought of as a network or graph Q of interconnected nodes (e.g. rooms, terminals) and the Attacker can choose to attack any node of Q within a given time T: He requires m consecutive periods there, uninterrupted by the Patroller, to commit his nefarious act (and win). The Patroller can follow any path on the graph. Thus the patrolling game is a win-lose game, where the Value is the probability that the Patroller successfully intercepts an attack, given best play on both sides. We determine analytically either the Value of the game, or bounds on the Value, for various classes of graphs, and discuss possible extensions and generalizations.

Abstract:We address the problem of dispatching a vehicle with different product classes. There is a common dispatch cost, but holding costs that vary by product class. The problem exhibits multidimensional state, outcome and action spaces, and as a result is computationally intractable using either discrete dynamic programming methods, or even as a deterministic integer program. We prove a key structural property for the decision function, and exploit this property in the development of continuous value function approximations that form the basis of an approximate dispatch rule. Comparisons on single product-class problems, where optimal solutions are available, demonstrate solutions that are within a few percent of optimal. The algorithm is then applied to a problem with 100 product classes, and comparisons against a carefully tuned myopic heuristic demonstrate significant improvements.

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Patrolling games were recently introduced by Alpern, Morton and Papadaki to model the problem of protecting the nodes of a network from an attack. Time is discrete and in each time unit the Patroller can stay at the same node or move to an adjacent node. The Attacker chooses when to attack and which node to attack, and needs m consecutive time units to carry it out. The Attacker wins if the Patroller does not visit the chosen node while it is being attacked; otherwise the Patroller wins. This paper studies the patrolling game where the network is a line graph of n nodes, which models the problem of guarding a channel or protecting a border from infiltration. We solve the patrolling game for any values of m and n, providing an optimal Patroller strategy, an optimal Attacker strategy and the value of the game (optimal probability that the attack is intercepted).

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This document is the author's final accepted version of the journal article. There may be differences between this version and the published version. You are advised to consult the publisher's version if you wish to cite from it. Abstract-In this paper store-carry and forward (SCF) decision policies for relaying within the cell are developed. The key motivation of SCF relaying stems from the fact that energy consumption levels can be dramatically reduced by capitalizing on the inherent mobility of nodes and the elasticity of Internet applications. More specifically, we show how the actual mobility of relay nodes can be incorporated as an additional resource in the system to achieve savings in the required communication energy levels. To this end, we provide a mathematical programming formulation on the aforementioned problem and find optimal routing and scheduling policies to achieve maximum energy savings. By investigating structural properties of the proposed mathematical program we show that optimal solutions can be computed efficiently in time. The trade-offs between energy and delay in the system are meticulously studied and Pareto efficient curves are derived. Numerical investigations show that the achievable energy gains by judiciously storing and carrying information from mobile relays can grow well above 70% for the macrocell scenario when compared to a baseline multihop wireless relaying scheme that uses shortest path routes to the base station.