Vickrey's Model Research Articles

Fog-Integrated Cloud Architecture enabled multi-attribute combinatorial reverse auctioning framework

Fog computing is an emerging service-oriented market in conjunction with Cloud computing to fulfill the resource demand of mobile users as well as IoT users for real-time applications. Auctioning in Fog computing is highly challenging due to mobility, dynamic pricing, real-time demand in comparison to Cloud based auctioning models. Further, due to users’ mobility and limited Fog resources, existing reverse auction techniques developed for Cloud computing model cannot directly be applied for the resource procurement in Fog-Integrated Cloud Architecture (FICA). Therefore, a reverse auction-based model which includes customer, auctioneer, Fog provider, Cloud provider, and Fog & Cloud provider together as auction participants, is proposed in this work. The proposed model, for resource provisioning using a multi-attribute combinatorial reverse auction, is named as Fog-Integrated Cloud Auctioning Model (FICAM). FICAM pricing scheme includes three types of resources depending on their requirement i.e., local Fog, remote Fog, and Cloud. A truthful, robust, and fair algorithm for resource allocation is proposed considering response time, data source mobility requirements, and Fog resource limitations. To encourage providers to bid truthfully, the Vickrey model is extended. FICAM also introduces a new algorithm for resource procurement in which instead of giving all resources of the bundle, only the required resources at a time are given to the customer with the bundle discount. The discount is based on a certain threshold in the ratio of the availed amount of resources to the offered amount of resources. Rigorous experimentation exhibits that the proposed model offers a low resource procurement cost in polynomial time as compared to other state of the art algorithms.

Parking Management Efficiency Analysis through Various Charge Schemes for Day-Long Commuting considering Elastic Travel Demand

In this paper, we investigate the travel pattern of the day-long commuting in a bisection bottleneck network and the efficiency of pricing schemes with elastic travel demand. We extend the Vickrey model to morning and evening commutes and allow commuters to arrive at workplace late and depart from workplace early. The parking searching time is considered in the morning commute. Next, we derive the independent morning and evening commuting travel patterns without road toll and parking fee. Then, we propose three pricing regimes: duration dependent parking fees; optimal time-varying road tolls; optimal time-varying road tolls with duration dependent parking fees. We compare the efficiency of the four schemes with elastic demand. Theoretical analysis and the numerical examples show that optimal time-varying road toll is the most efficient pricing scheme. Charging a duration dependent fee neither improves nor deteriorates the scheme of time-varying road toll, if the toll rates are appropriately set. The regime of duration dependent parking fee only is less efficient than the regime of independent morning and evening commuting travel patterns without road toll and parking fee. In the regime of duration dependent parking fee, the social surplus decreases with the increase of duration dependent parking fee rate.

ENDOGENOUS SCHEDULING PREFERENCES AND CONGESTION

We consider the timing of activities through a dynamic model of commuting with congestion, in which workers care solely about leisure and consumption. Implicit preferences for the timing of the commute form endogenously due to temporal agglomeration economies. Equilibrium exists uniquely and is indistinguishable from that of a generalized version of the classical Vickrey bottleneck model, based on exogenous trip‐timing preferences, but optimal policies differ: the Vickrey model will misstate the benefits of a capacity increase, it will underpredict the benefits of congestion pricing, and pricing may make people better off even without considering the use of revenues.

Existence of simultaneous route and departure choice dynamic user equilibrium

This paper is concerned with the existence of the simultaneous route-and-departure choice dynamic user equilibrium (SRDC-DUE) in continuous time, first formulated as an infinite-dimensional variational inequality in Friesz et al. (1993). In deriving our existence result, we employ the generalized Vickrey model (GVM) introduced in Han et al. (2013a,b) to formulate the underlying network loading problem. As we explain, the GVM corresponds to a path delay operator that is provably strongly continuous on the Hilbert space of interest. Finally, we provide the desired SRDC-DUE existence result for general constraints relating path flows to a table of fixed trip volumes without invocation of a priori bounds on the path flows.

A partial differential equation formulation of Vickrey’s bottleneck model, part II: Numerical analysis and computation

The Vickrey model, originally introduced in Vickrey (1969), is one of the most widely used link-based models in the current literature in dynamic traffic assignment (DTA). One popular formulation of this model is an ordinary differential equation (ODE) that is discontinuous with respect to its state variable. As explained in Ban et al. (2011) and Han et al. (2013), such an irregularity induces difficulties in both continuous-time analysis and discrete-time computation. In Han et al. (2013), the authors proposed a reformulation of the Vickrey model as a partial differential equation (PDE) and derived a closed-form solution to the aforementioned ODE. This reformulation enables us to rigorously prove analytical properties of the Vickrey model and related DTA models.In this paper, we present the second of a two-part exploration regarding the PDE formulation of the Vickrey model. As proposed by Han et al. (2013), we continue research on the generalized Vickrey model (GVM) in a discrete-time framework and in the context of DTA by presenting a highly computable solution methodology. Our new computational scheme for the GVM is based on the closed-form solution mentioned above. Unlike finite-difference discretization schemes which could yield non-physical solutions (Ban et al., 2011), the proposed numerical scheme guarantees non-negativity of the queue size and the exit flow as well as first-in-first-out (FIFO). Numerical errors and convergence of the computed solutions are investigated in full mathematical rigor. As an application of the GVM, a class of network system optimal dynamic traffic assignment (SO-DTA) problems is analyzed. We show existence of a continuous-time optimal solution and propose a discrete-time mixed integer linear program (MILP) as an approximation to the original SO-DTA. We also provide convergence results for the proposed MILP approximation.

A partial differential equation formulation of Vickrey’s bottleneck model, part I: Methodology and theoretical analysis

This paper is concerned with the continuous-time Vickrey model, which was first introduced in Vickrey (1969). This model can be described by an ordinary differential equation (ODE) with a right-hand side which is discontinuous in the unknown variable. Such a formulation induces difficulties with both theoretical analysis and numerical computation. Moreover it is widely suspected that an explicit solution to this ODE does not exist. In this paper, we advance the knowledge and understanding of the continuous-time Vickrey model by reformulating it as a partial differential equation (PDE) and by applying a variational method to obtain an explicit solution representation. Such an explicit solution is then shown to be the strong solution to the ODE in full mathematical rigor. Our methodology also leads to the notion of generalized Vickrey model (GVM), which allows the flow to be a distribution, instead of an integrable function. As explained by Han et al. (in press), this feature of traffic modeling is desirable in the context of analytical dynamic traffic assignment (DTA). The proposed PDE formulation provides new insights into the physics of The Vickrey model, which leads to a number of modeling extensions as well as connection with first-order traffic models such as the Lighthill–Whitham–Richards (LWR) model. The explicit solution representation also leads to a new computational method, which will be discussed in an accompanying paper, Han et al. (in press).

An Integrated Dynamic Model of Road Traffic Congestion Based on Simple Car-Following Theory: Exploring Hypercongestion

This paper presents a dynamic version of the standard static economic model of road traffic congestion, based on car-following theory. It is proven formally and illustrated numerically that the “hypercongested” equilibria found in the standard model are dynamically unstable. For arrival rates of users below the road's maximum capacity, the model reproduces the non-hypercongested stationary state outcomes found in the standard model. When the arrival rate exceeds this maximum capacity, however, the model produces outcomes consistent with Vickrey's model of bottleneck congestion. The model thus offers an integration and a generalization of these two archetype models.