ABSTRACTIn many ‘smart city’ applications, congestion arises in part due to the nature of signals received by individuals from a central authority. In the model of , each agent uses one out of multiple resources at each time instant. The per-use cost of a resource depends on the number of concurrent users. A central authority has up-to-date knowledge of the congestion across all resources and uses randomisation to provide a scalar or an interval for each resource at each time. In this paper, the interval to broadcast per resource is obtained by taking the minima and maxima of costs observed within a time window of length r, rather than by randomisation. We show that the resulting distribution of agents across resources also converges in distribution, under plausible assumptions about the evolution of the population over time.
On the interface of transportation and control theory, there has been a recent interest in load balancing @cite_0 @cite_37 . These schemes, however, rely on simple randomisation, without modelling heterogenous agent behaviour and actions and without allowing for the same information to be provided to all agents. On the interface of transportation and behavioral science, Ben- @cite_6 have studied the effects of information on drivers. In a number of subsequent papers @cite_3 @cite_55 and the dissertation of Bottom @cite_34 , fixed points have been used to study deterministic scalar signaling with deterministic response of the population. See @cite_7 for an extensive survey. In contrast, our analysis can be seen as a study of a probabilistic counter-part of fixed points, which allows for the uncertainty in the response of the population.
@cite_37 @cite_7 @cite_55 @cite_3 @cite_6 @cite_0 @cite_34
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We study the question of whether a large population of agents in a traffic network is able to converge to an equilibrium quickly. To that end, we consider a round-based variant of the Wardrop model. Every agent is allowed to reroute its traffic once in a while with the aim of finding a path with minimal latency. As a first result we find that using a replication policy which allows agents to imitate others gives rise to a bicriterial approximate equilibrium very quickly. In particular, the time bound depends logarithmically on the ratio between minimum and maximum latency but is otherwise independent of the network size. In the single-commodity case, this bicriteria approximate equilibrium has an intuitive interpretation as a state in which almost all agents are almost happy. This kind of approximate equilibrium, however, is transient. In order to reach a global approximation, we need to add an exploration component which enables the agents to explore the strategy space independently of the other agents. Although it can be shown that, when used exclusively, exploration policies imply an exponential lower bound, applying exploration carefully allows the population to approximate the global Wardrop equilibrium in polynomial time. Since the distributed and concurrent fashion of our policies bears the risk of oscillating behavior, we must take into account the steepness of the latency functions. We show that the relevant parameter is elasticity, a parameter closely related to the polynomial degree. This improves significantly over earlier results which depend on the absolute slope and therefore have a pseudopolynomial flavor. This paper re-visits Shayevitz & Feder's recent ‘Posterior Matching Scheme’, an explicit, dynamical system encoder for communication with feedback that treats the message as a point on the [0,1] line and achieves capacity on memo-ryless channels. It has two key properties that ensure that it maximizes mutual information at each step: (a) the encoder sequentially hands the decoder what is missing; and (b) the next input has the desired statistics. Motivated by brain-machine interface applications and multi-antenna communications, we consider developing dynamical system feedback encoders for scenarios when the message point lies in higher dimensions. We develop a necessary and sufficient condition — the Jacobian equation — for any dynamical system encoder that maximizes mutual information. In general, there are many solutions to this equation. We connect this to the Monge-Kantorovich Optimal Transportation Problem, which provides a framework to identify a unique solution suiting a specific purpose. We provide two examplar capacity-achieving solutions — for different purposes — for the multi-antenna Gaussian channel with feedback. This insight further elucidates an interesting relationship between interactive decision theory problems and the theory of optimal transportation. Traditionally, traffic assignment models, both for within-day static and dynamic demand, have been formulated following an equilibrium approach in which a state ensuring internal consistency between demand (flows) and costs is sought. However, equilibrium analysis is significant under some assumptions on its “representativeness” (coincidence or closeness with the actual attractor of the system) and analytical properties, such as existence, uniqueness, and stability. Moreover, transients due to modifications of demand and or supply cannot be simulated through equilibrium models, nor can a statistical description of the state of the system, i.e. means, modes, moments and, more generally, frequency distributions of flows over time be obtained. In this paper, interperiodic (day-to-day) dynamic modeling of transportation networks is addressed following two different approaches, namely deterministic and stochastic processes. In both cases several theoretical results are shown by making use of a formal framework covering most models discussed in the literature as well as some possible extensions. Most of the results reported can be extended to cover within-day dynamic models but these models are not explicitly dealt with. Within the framework of deterministic processes the relevance of day-to-day dynamic models for demand supply interaction in comparison with the traditional user equilibrium approach is discussed, and conditions for coincidence of fixed-point attractors and equilibrium states are stated. Conditions for existence and uniqueness of fixed-point attractors are proposed, generalizing and extending those presented in the literature for user equilibrium. Conditions for stability of both fixed-points and equilibrium states were formulated by making use of results from non-linear dynamic system theory. Moreover, it is possible to devise a new family of “dynamic” algorithms which simulate the system convergence to a fixed-point in order to obtain an equivalent equilibrium state, as opposed to conventional “optimisation” algorithms. In this case the fixed-point stability analysis can be viewed as a convergence analysis for the algorithms specified this way. Conditions for stochastic process regularity are proposed ensuring, among other things, existence and uniqueness of a stationary probability distribution of system states. These conditions generalize and extend results presented in the literature to a wider class of possible dynamic models. Relationships between a deterministic process, together with corresponding fixed-points or equilibrium states, and stochastic probability distribution are also briefly addressed. Finally, some numerical examples confirming theoretical results are reported for a small test network. The ubiquitous deployment of mobile and sensor technologies has led to both the capacity to observe human behavior in physical (offline) settings as well as to record it. This provides researchers with a new lens to study and better understand the individual decision processes that were previously unobserved. In this paper, we study decision making behavior of 11,196 taxi drivers in a large Asian city using a rich data set consisting of 10.6 million fine-grained GPS trip records. These records include detailed taxi GPS trajectories, taxi occupancy data (i.e., whether a taxi was occupied with a passenger or was vacant) and taxi drivers’ daily incomes. This capacity to use data where occupancy of the taxi is known is a distinctive feature of our data set and sets this work apart from prior work which has attempted to study driver behavior. The specific decision we focus on pertains to actions drivers take to find new passengers after they have dropped off their current passengers. In particular, we study the role of information derivable from the GPS trace data (e.g., where passengers are dropped off, where passengers are picked up, longitudinal taxicab travel history with fine-grained time stamps) observable by or made available to drivers in enabling them to learn the distribution of demand for their services over space and time. We conduct our study using a heterogeneous Bayesian learning model. We find strong heterogeneity in individual learning behavior and driving decisions, which is significantly associated with individual economic outcomes. Drivers with higher incomes benefit significantly from their ability to learn from not only demand information directly observable in the local market, but also aggregate information on demand flows across markets. Interestingly, our policy simulations indicate information that is noisy at the individual level becomes valuable after being aggregated across various spatial and temporal dimensions. Moreover, the value of information does not increase monotonically with the scale and frequency of information sharing. Finally, our study has important welfare implications in that efficient information sharing leads to an income increase among all drivers, instead of a redistribution of income between different types of drivers. Our work allows us not only to explain driver decision making behavior using these detailed behavioral traces, but also to prescribe information sharing strategy for the firm in order to improve the overall market efficiency. The problem of transporting patients or elderly people has been widely studied in literature and is usually modeled as a dial-a-ride problem (DARP). In this paper we analyze the corresponding problem arising in the daily operation of the Austrian Red Cross. This nongovernmental organization is the largest organization performing patient transportation in Austria. The aim is to design vehicle routes to serve partially dynamic transportation requests using a fixed vehicle fleet. Each request requires transportation from a patient's home location to a hospital (outbound request) or back home from the hospital (inbound request). Some of these requests are known in advance. Some requests are dynamic in the sense that they appear during the day without any prior information. Finally, some inbound requests are stochastic. More precisely, with a certain probability each outbound request causes a corresponding inbound request on the same day. Some stochastic information about these return transports is available from historical data. The purpose of this study is to investigate, whether using this information in designing the routes has a significant positive effect on the solution quality. The problem is modeled as a dynamic stochastic dial-a-ride problem with expected return transports. We propose four different modifications of metaheuristic solution approaches for this problem. In detail, we test dynamic versions of variable neighborhood search (VNS) and stochastic VNS (S-VNS) as well as modified versions of the multiple plan approach (MPA) and the multiple scenario approach (MSA). Tests are performed using 12 sets of test instances based on a real road network. Various demand scenarios are generated based on the available real data. Results show that using the stochastic information on return transports leads to average improvements of around 15 . Moreover, improvements of up to 41 can be achieved for some test instances. In the traffic assignment problem, first proposed by Wardrop in 1952, commuters select the shortest available path to travel from their origins to their destinations. We study a generalization of this problem in which competitors, who may control a nonnegligible fraction of the total flow, ship goods across a network. This type of games, usually referred to as atomic games, readily applies to situations in which the competing freight companies have market power. Other applications include intelligent transportation systems, competition among telecommunication network service providers, and scheduling with flexible machines. Our goal is to determine to what extent these systems can benefit from some form of coordination or regulation. We measure the quality of the outcome of the game without centralized control by computing the worst-case inefficiency of Nash equilibria. The main conclusion is that although self-interested competitors will not achieve a fully efficient solution from the system's point of view, the loss is not too severe. We show how to compute several bounds for the worst-case inefficiency that depend on the characteristics of cost functions and on the market structure in the game. In addition, building upon the work of Catoni and Pallotino, we show examples in which market aggregation (or collusion) adversely impacts the aggregated competitors, even though their market power increases. For example, Nash equilibria of atomic network games may be less efficient than the corresponding Wardrop equilibria. When competitors are completely symmetric, we provide a characterization of the Nash equilibrium using a potential function, and prove that this counterintuitive phenomenon does not arise. Finally, we study a pricing mechanism that elicits more coordination from the players by reducing the worst-case inefficiency of Nash equilibria. Vehicular communications are becoming an emerging technology for safety control, traffic control, urban monitoring, pollution control, and many other road safety and traffic efficiency applications. All these applications generate a lot of data which should be distributed among communication parties such as vehicles and users in an efficient manner. On the other hand, the generated data cause a significant load on a network infrastructure, which aims at providing uninterrupted services to the communication parties in an urban scenario. To make a balance of load on the network for such situations in the urban scenario, frequently accessed contents should be cached at specified locations either in the vehicles or at some other sites on the infrastructure providing connectivity to the vehicles. However, due to the high mobility and sparse distribution of the vehicles on the road, sometimes, it is not feasible to place the contents on the existing infrastructure, and useful information generated from the vehicles may not be sent to its final destination. To address this issue, in this paper, we propose a new peer-to-peer (P2P) cooperative caching scheme. To minimize the load on the infrastructure, traffic information among vehicles is shared in a P2P manner using a Markov chain model with three states. The replacement of existing data to accommodate newly arrived data is achieved in a probabilistic manner. The probability is calculated using the time to stay in a waiting state and the frequency of access of a particular data item in a given time interval. The performance of the proposed scheme is evaluated in comparison to those of existing schemes with respect to the metrics such as network congestion, query delay, and hit ratio. Analysis results show that the proposed scheme has reduced the congestion and query delay by 30 with an increase in the hit ratio by 20 . In this paper, we consider a class of dynamic vehicle routing problems, in which a number of mobile agents in the plane must visit target points generated over time by a stochastic process. It is desired to design motion coordination strategies in order to minimize the expected time between the appearance of a target point and the time it is visited by one of the agents. We propose control strategies that, while making minimal or no assumptions on communications between agents, provide the same level of steady-state performance achieved by the best-known decentralized strategies. In other words, we demonstrate that inter-agent communication does not improve the efficiency of such systems, but merely affects the rate of convergence to the steady state. Furthermore, the proposed strategies do not rely on the knowledge of the details of the underlying stochastic process. Copyright © 2007 John Wiley & Sons, Ltd. Wireless vehicle-to-vehicle (V2V) and vehicle-toinfrastructure (V2I) communication holds great promise for significantly reducing the human and financial costs of vehicle collisions. A common characteristic of this communication is the broadcast of a device's core state information at regular intervals (e.g., vehicle speed and location or traffic signal state and timing). Unless controlled, the aggregate of these broadcasts will congest the channel under dense traffic scenarios, reducing the effectiveness of collision avoidance applications that use transmitted information. Active congestion control using distributed techniques is a topic of great interest for establishing the scalability of this technology. This paper defines a new adaptive congestion control algorithm that can be applied to the message rate of devices in this vehicular environment. While other published approaches rely on binary control, the LInear MEssage Rate Integrated Control (LIMERIC) algorithm takes advantage of full-precision control inputs that are available on the wireless channel. The result is provable convergence to fair and efficient channel utilization in the deterministic environment, under simple criteria for setting adaptive parameters. This “perfect” convergence avoids the limit cycle behavior that is inherent to binary control. We also discuss several practical aspects associated with implementing LIMERIC, including guidelines for the choice of system parameters to obtain desired utilization outcomes, a gain saturation technique that maintains robust convergence under all conditions, convergence with asynchronous updates, and using channel load to determine the aggregate message rate that is observable at a receiver. This paper also extends the convergence analysis for two important cases, i.e., measurement noise in the input signal and delay in the update process. This paper illustrates key analytical results using MATLAB numerical results and employs standard NS-2 simulations to demonstrate the performance of LIMERIC in several high-density scenarios. In this paper we investigate the potential benefits of coordinated congestion control for multipath data transfers, and contrast with uncoordinated control. For static random path selections, we show the worst-case throughput performance of uncoordinated control behaves as if each user had but a single path (scaling like log(log(N)) log(N) where N is the system size, measured in number of resources). Whereas coordinated control gives a throughput allocation bounded away from zero, improving on both uncoordinated control and on the greedy-least loaded path selection of e.g. Mitzenmacher. We then allow users to change their set of routes and introduce the notion of a Nash equilibrium. We show that with RTT bias (as in TCP Reno), uncoordinated control can lead to inefficient equilibria. With no RTT bias, both uncoordinated or coordinated Nash equilibria correspond to desirable welfare maximising states. Moreover, simple path reselection polices that shift to paths with higher net benefit can find these states. It is well known that in a network with arbitrary (convex) latency functions that are a function of edge traffic, the worst-case ratio, over all inputs, of the system delay caused due to selfish behavior versus the system delay of the optimal centralized solution may be unbounded even if the system consists of only two parallel links. This ratio is called the price of anarchy (PoA). In this paper, we investigate ways by which one can reduce the performance degradation due to selfish behavior. We investigate two primary methods (a) Stackelberg routing strategies, where a central authority, e.g., network manager, controls a fixed fraction of the flow, and can route this flow in any desired way so as to influence the flow of selfish users; and (b) network tolls, where tolls are imposed on the edges to modify the latencies of the edges, and thereby influence the induced Nash equilibrium. We obtain results demonstrating the effectiveness of both Stackelberg strategies and tolls in controlling the price of anarchy. For Stackelberg strategies, we obtain the first results for nonatomic routing in graphs more general than parallel-link graphs, and strengthen existing results for parallel-link graphs, (i) In series-parallel graphs, we show that Stackelberg routing reduces the PoA to a constant (depending on the fraction of flow controlled). (ii) For general graphs, we obtain latency-class specific bounds on the PoA with Stackelberg routing, which give a continuous trade-off between the fraction of flow controlled and the price of anarchy, (iii) In parallel-link graphs, we show that for any given class L of latency functions, Stackelberg routing reduces the PoA to at most α + (1 - α) · ρ(L), where α is the fraction of flow controlled and ρ(L) is the PoA of class L (when α = 0). For network tolls, motivated by the known strong results for nonatomic games, we consider the more general setting of atomic splittable routing games. We show that tolls inducing an optimal flow always exist, even for general asymmetric games with heterogeneous users, and can be computed efficiently by solving a convex program. Furthermore, we give a complete characterization of flows that can be induced via tolls. These are the first results on the effectiveness of tolls for atomic splittable games. In this paper, we study the performance characteristics of simple load sharing algorithms for heterogeneous distributed systems. We assume that nonnegligible delays are encountered in transferring jobs from one node to another. We analyze the effects of these delays on the performance of two threshold-based algorithms called Forward and Reverse. We formulate queuing theoretic models for each of the algorithms operating in heterogeneous systems under the assumption that the job arrival process at each node in Poisson and the service times and job transfer times are exponentially distributed. The models are solved using the Matrix-Geometric solution technique. These models are used to study the effects of different parameters and algorithm variations on the mean job response time: e.g., the effects of varying the thresholds, the impact of changing the probe limit, the impact of biasing the probing, and the optimal response times over a large range of loads and delays. Wherever relevant, the results of the models are compared with the M M 1 model, representing no load balancing (hereafter referred to as NLB), and the M M K model, which is an achievable lower bound (hereafter referred to as LB). In many urban areas where traffic congestion does not have the peak pattern, conventional traffic signal timing methods does not result in an efficient control. One alternative is to let traffic signal controllers learn how to adjust the lights based on the traffic situation. However this creates a classical non-stationary environment since each controller is adapting to the changes caused by other controllers. In multi-agent learning this is likely to be inefficient and computationally challenging, i.e., the efficiency decreases with the increase in the number of agents (controllers). In this paper, we model a relatively large traffic network as a multi-agent system and use techniques from multi-agent reinforcement learning. In particular, Q-learning is employed, where the average queue length in approaching links is used to estimate states. A parametric representation of the action space has made the method extendable to different types of intersection. The simulation results demonstrate that the proposed Q-learning outperformed the fixed time method under different traffic demands. This paper contains a quantitative evaluation of probabilistic traffic assignment models and proposes an alternate formulation. First, the concept of stochastic-user-equilibration (S-U-E) is formalized as an extension of Wardrop's user-equilibration criterion. Then, the stochastic-network-loading (S-N-L) problem (a special case of S-U-E for networks with constant link costs) is analyzed in detail and an expression for the probability of route choice which is based on two general postulates of user behavior is derived. The paper also discusses the weaknesses of existing S-N-L techniques with special attention paid to Dial's multipath method and compares them to the suggested approach. The proposed model seems reasonable and does not exhibit the inherent weaknesses of the logit model when applied to sets of routes which overlap heavily. The discussion is supported by several numerical examples on small contrived networks. The paper concludes with the discussion of two techniques that can be used to approximate the link flows resulting from the proposed model in large networks. We consider the problem of resource allocation in downlink OFDMA systems for multi service and unknown environment. Due to users' mobility and intercell interference, the base station cannot predict neither the Signal to Noise Ratio (SNR) of each user in future time slots nor their probability distribution functions. In addition, the traffic is bursty in general with unknown arrival. The probability distribution functions of the SNR, channel state and traffic arrival density are then unknown. Achieving a multi service Quality of Service (QoS) while optimizing the performance of the system (e.g. total throughput) is a hard and interesting task since it depends on the unknown future traffic and SNR values. In this paper we solve this problem by modeling the multiuser queuing system as a discrete time linear dynamic system. We develop a robust H∞ controller to regulate the queues of different users. The queues and Packet Drop Rates (PDR) are controlled by proposing a minimum data rate according to the demanded service type of each user. The data rate vector proposed by the controller is then fed as a constraint to an instantaneous resource allocation framework. This instantaneous problem is formulated as a convex optimization problem for instantaneous subcarrier and power allocation decisions. Simulation results show small delays and better fairness among users. Deterministic timed systems can be modeled as fixed point problems (B. Roscoe and G. Reed, 1988), (R. K. Yates, 1993), (E. A. Lee, 1999). In particular, any connected network of timed systems can be modeled as a single system with feedback, and the system behavior is the fixed point of the corresponding system equation, when it exists. For delta-causal systems, we can use the Cantor metric to measure the distance between signals and the Banach fixed-point theorem to prove the existence and uniqueness of a system behavior. Moreover, the Banach fixed-point theorem is constructive: it provides a method to construct the unique fixed point through iteration. In this paper, we extend this result to systems modeled with the superdense model of time (O. , 1992), (Z. Manna and A. Pnueli, 1993) used in hybrid systems. We call the systems we consider eventually delta-causal, a strict generalization of delta-causal in which multiple events may be generated on a signal in zero time. With this model of time, we can use a generalized ultrametric (Generalized ultrametric spaces, I, 1996) instead of a metric to model the distance between signals. The existence and uniqueness of behaviors for such systems comes from the fixed-point theorem of (S. Priess-Crampe and P. Ribenboim, 1993), but this theorem gives no constructive method to compute the fixed point This leads us to define petrics, a generalization of metrics, which we use to generalize the Banach fixed-point theorem to provide a constructive fixed-point theorem. This new fixed-point theorem allows us to construct the unique behavior of eventually delta-causal systems. This paper builds upon the scalable admission control schemes for CDMA networks developed in F. (2003, December 2004). These schemes are based on an exact representation of the geometry of both the downlink and the uplink channels and ensure that the associated power allocation problems have solutions under constraints on the maximal power of each station user. These schemes are decentralized in that they can be implemented in such a way that each base station only has to consider the load brought by its own users to decide on admission. By load we mean here some function of the configuration of the users and of their bit rates that is described in the paper. When implemented in each base station, such schemes ensure the global feasibility of the power allocation even in a very large (infinite number of cells) network. The estimation of the capacity of large CDMA networks controlled by such schemes was made in these references. In certain cases, for example for a Poisson pattern of mobiles in an hexagonal network of base stations, this approach gives explicit formulas for the infeasibility probability, defined as the fraction of cells where the population of users cannot be entirely admitted by the base station. In the present paper we show that the notion of infeasibility probability is closely related to the notion of blocking probability, defined as the fraction of users that are rejected by the admission control policy in the long run, a notion of central practical importance within this setting. The relation between these two notions is not bound to our particular admission control schemes, but is of more general nature, and in a simplified scenario it can be identified with the well-known Erlang loss formula. We prove this relation using a general spatial birth-and-death process, where customer locations are represented by a spatial point process that evolves over time as users arrive or depart. This allows our model to include the exact representation of the geometry of inter-cell and intra-cell interferences, which play an essential role in the load indicators used in these cellular network admission control schemes. The authors show how to design truthful (dominant strategy) mechanisms for several combinatorial problems where each agent's secret data is naturally expressed by a single positive real number. The goal of the mechanisms we consider is to allocate loads placed on the agents, and an agent's secret data is the cost she incurs per unit load. We give an exact characterization for the algorithms that can be used to design truthful mechanisms for such load balancing problems using appropriate side payments. We use our characterization to design polynomial time truthful mechanisms for several problems in combinatorial optimization to which the celebrated VCG mechanism does not apply. For scheduling related parallel machines (Q spl par C sub max ), we give a 3-approximation mechanism based on randomized rounding of the optimal fractional solution. This problem is NP-complete, and the standard approximation algorithms (greedy load-balancing or the PTAS) cannot be used in truthful mechanisms. We show our mechanism to be frugal, in that the total payment needed is only a logarithmic factor more than the actual costs incurred by the machines, unless one machine dominates the total processing power. We also give truthful mechanisms for maximum flow, Q spl par spl Sigma C sub j (scheduling related machines to minimize the sum of completion times), optimizing an affine function over a fixed set, and special cases of uncapacitated facility location. In addition, for Q spl par spl Sigma w sub j C sub j (minimizing the weighted sum of completion times), we prove a lower bound of 2 spl radic 3 for the best approximation ratio achievable by truthful mechanism. We examine how the selfish behavior of heterogeneous users in a network can be regulated through economic disincentives, i.e., through the introduction of appropriate taxation. One wants to impose taxes on the edges so that any traffic equilibrium reached by the selfish users who are conscious of both the travel latencies and the taxes will minimize the social cost, i.e., will minimize the total latency. We generalize previous results of Cole, Dodis and Roughgarden that held for a single origin-destination pair to the multicommodity setting. Our approach, which could be of independent interest, is based on the formulation of traffic equilibria as a nonlinear complementarity problem by Aashtiani and Magnanti (1981), We extend this formulation so that each of its solutions will give us a set of taxes that forces the network users to conform, at equilibrium, to a certain prescribed routing. We use the special nature of the prescribed minimum-latency flow in order to reduce the difficult nonlinear complementarity formulation to a pair of primal-dual linear programs. LP duality is then enough to derive our results. In 1991, D. J. Bertsimas and G. van Ryzin introduced and analyzed a model for stochastic and dynamic vehicle routing in which a single, uncapacitated vehicle traveling at a constant velocity in a Euclidean region must service demands whose time of arrival, location and on-site service are stochastic. The objective is to find a policy to service demands over an infinite horizon that minimizes the expected system time (wait plus service) of the demands. This paper extends our analysis in several directions. First, we analyze the problem of m identical vehicles with unlimited capacity and show that in heavy traffic the system time is reduced by a factor of 1 m2 over the single-server case. One of these policies improves by a factor of two on the best known system time for the single-server case. We then consider the case in which each vehicle can serve at most q customers before returning to a depot. We show that the stability condition in this case depends strongly on the geometry of the region. Several pol...