Stochastic order results and equilibrium joining rules for the Bernoulli Feedback Queue

We consider customer joining behaviour for a system that consists of a FCFS queue with Bernoulli feedback. A consequence of the feedback characteristic is that the sojourn time of a customer already in the system depends on the joining decisions taken by future arrivals to the system. By establishing stochastic order results for coupled versions of the system, we establish the existence of homogeneous Nash equilibrium joining policies for both single and multiple customer types which are distinguished through distinct quality of service preference parameters. Further, it is shown that for a single customer type, the homogeneous policy is unique

Approximate expected delay costs for call and contact centre models under light traffic regimes

This paper studies the form of certain expected delay costs as a function of the arrival rate for customers who pass through a service facility that allows for reneging and retrials. We show that, under certain light traffic conditions, these costs are continuously increasing and convex functions of the arrival rate (within a finite interval). This result is first explored for the processor sharing system, in which a penalty cost is incurred for reneging from the service facility for good without ever receiving service, and then we consider a system with a more general structure governing the output processes and costs incurred per unit time, but without the penalty cost. A suggested application for these results, in which game theoretic considerations are utilized for gauging customer behaviour within a decentralized context, is briefly discussed

Volumetric Uncertainty Bounds and optimal configurations for Converging Beam Triple LIDAR

We consider the problem of quantifying uncertainty for converging beam triple LIDAR when the input uncertainty follows a uniform distribution. We determine expressions for the range (i.e. set of reachable points) for the reconstructed velocity vector as a function of any particular setting of the nominal input parameters and determine an explicit lower (and upper) bound on the (averaged) volume (with respect to Lebesgue measure), in R 3 , of that range. We show that the size of any such bound is inversely proportional to the absolute value of the triple scalar product of the unit vectors characterizing the Doppler measurement directions (optimized over the uncertainty region) in R 6 associated with the nominal angle settings under consideration. This leads to the conclusion that the nominal LIDAR configurations that minimize output uncertainty ought to be those in which the value of the triple scalar product of the Doppler unit vectors is at its largest

On the Nash Equilibria for the FCFS Queuing System with load-increasing service rate

We consider a service system (QS) that operates according to the FCFS discipline, and in which the service rate is an increasing function of the queue length. Customers arrive sequentially to the system and decide whether or not to join, using decision rules based upon the queue length on arrival to QS. Each customer is interested in selecting a rule that meets a certain optimality criterion with regards to their expected sojourn time in the system; as a consequence, the decision rules of other customers need to be taken into account. Within a particular class of decision rules for an associated infinite player game, the structure of the Nash equilibrium routing policies is characterized. We prove that within this class, there exist a finite number of Nash equilibria, and that at least one of these is non-randomized. Finally, we explore the extent to which the Nash equilibria are characteristic of customer joining behaviour under a learning rule based on system-wide data with the aid of simulation experiments

Error analysis for a Static Convergent Beam Triple LIDAR

In this paper we consider the problem of uncertainty propagation and quantification for the converging triple-beam LIDAR technology, proposed for measuring wind velocity passing through a fixed point in space. Converging triple-beam LIDAR employs the use of three non-parallel, noncoplanar, laser beams which are directed from a fixed platform, typically at ground level, that extend to meet at the point at which measurement of velocity is sought. Coordinate values of the velocity are ascertained with respect to unit vectors along the lines of sight of the laser beams (Doppler vectors), which are then resolved in order determine the velocity in terms of Cartesian coordinates (i.e. with respect to the standard basis). However, if there is any discrepancy between the recorded values of the coordinates with respect to the Doppler unit vectors and/or the perceived angle settings for such vectors with what they really should be, however small, then this will lead to errors in the reconstructed Cartesian coordinates. The aim of this paper to quantify the potential size of this error by consideration of the variance-covariance matrix of the reconstructed Cartesian coordinates

The Bernoulli Feedback Queue with Balking: stochastic order results and equilibrium joining rules

We consider customer joining behaviour for a system that consists of a FCFS queue with Bernoulli feedback. A consequence of the feedback characteristic is that the sojourn time of a customer already in the system depends on the joining decisions taken by future arrivals to the system. By establishing stochastic order results for coupled versions of the system, we prove the existence, and uniqueness, of Nash equilibrium joining policies, and show that these are characterized by (possibly randomized) threshold rules. We contrast the Nash rule with the socially optimizing joining rule that minimizes the long-term expected average sojourn time (or cost) per customer. The latter rule is characterized by a nonrandomized threshold, and we show that the Nash rule admits at least as many customers into the system as the socially optimizing one

Error propagation analysis for a Static Convergent Beam Triple LIDAR

We consider the issue of uncertainty propagation and quantification for the converging triplebeam LIDAR technology used for measuring wind velocity passing through a fixed point in space. Converging triple-beam LIDAR employs the use of three non-parallel, non-coplanar, laser beams which are directed from a fixed platform, typically at ground level, that extend to meet at the point at which measurement of velocity is sought. Coordinate values of the velocity are ascertained with respect to unit vectors along the lines of sight of the laser beams (Doppler vectors), which are then resolved in order determine the velocity in terms of Cartesian coordinates (i.e. with respect to the standard basis). However, if there is any discrepancy between the recorded values of the coordinates with respect to the Doppler unit vectors and/or the perceived angle settings for such vectors with what they really should be, however small, then this will lead to errors in the reconstructed Cartesian coordinates. The aim of this paper is to quantify the potential size of this error by consideration of its variance within each component of the reconstructed velocity vector through the use of the error propagation formula

On the Nash equilibria for the FCFS queueing system with load-increasing service rate

We consider a service system (QS) that operates according to the first-come-first-served (FCFS) discipline, and in which the service rate is an increasing function of the queue length. Customers arrive sequentially at the system, and decide whether or not to join using decision rules based upon the queue length on arrival. Each customer is interested in selecting a rule that meets a certain optimality criterion with regard to their expected sojourn time in the system; as a consequence, the decision rules of other customers must be taken into account. Within a particular class of decision rules for an associated infinite-player game, the structure of the Nash equilibrium routeing policies is characterized. We prove that, within this class, there exist a finite number of Nash equilibria, and that at least one of these is nonrandomized. Finally, with the aid of simulation experiments, we explore the extent to which the Nash equilibria are characteristic of customer joining behaviour under a learning rule based on system-wide data