Lowest priority waiting time distribution in an accumulating priority Lévy queue

We derive the waiting time distribution of the lowest class in an accumulating priority (AP) queue with positive Lévy input. The priority of an infinitesimal customer (particle) is a function of their class and waiting time in the system, and the particles with the highest AP are the next to be processed. To this end we introduce a new method that relies on the construction of a workload overtaking process and solving a first-passage problem using an appropriate stopping time.     

Scheduling for a processor sharing system with linear slowdown

We consider the problem of scheduling arrivals to a congestion system with a finite number of users having identical deterministic demand sizes. The congestion is of the processor sharing type in the sense that all users in the system at any given time are served simultaneously. However, in contrast to classical processor sharing congestion models, the processing slowdown is proportional to the number of users in the system at any time. That is, the rate of service experienced by all users is linearly decreasing with the number of users. For each user there is an ideal departure time (due date). A centralized scheduling goal is then to select arrival times so as to minimize the total penalty due to deviations from ideal times weighted with sojourn times. Each deviation penalty is assumed quadratic, or more generally convex. But due to the dynamics of the system, the scheduling objective function is non-convex. Specifically, the system objective function is a non-smooth piecewise convex function. Nevertheless, we are able to leverage the structure of the problem to derive an algorithm that finds the global optimum in a (large but) finite number of steps, each involving the solution of a constrained convex program. Further, we put forward several heuristics. The first is the traversal of neighbouring constrained convex programming problems, that is guaranteed to reach a local minimum of the centralized problem. This is a form of a “local search”, where we use the problem structure in a novel manner. The second is a one-coordinate “global search”, used in coordinate pivot iteration. We then merge these two heuristics into a unified “local–global” heuristic, and numerically illustrate the effectiveness of this heuristic.     

Enhanced light‐vapor interactions and all optical switching in a chip scale micro‐ring resonator coupled with atomic vapor

The coupling of atomic and photonic resonances serves as an important tool for enhancing light‐matter interactions and enables the observation of multitude of fascinating and fundamental phenomena. Here, by exploiting the platform of atomic‐cladding wave guides, the resonant coupling of rubidium vapor and an atomic cladding micro ring resonator is experimentally demonstrated. Specifically, cavity‐atom coupling in the form of Fano resonances having a distinct dependency on the relative frequency detuning between the photonic and the atomic resonances is observed. Moreover, significant enhancement of the efficiency of all optical switching in the V‐type pump‐probe scheme is demonstrated. The coupled system of micro‐ring resonator and atomic vapor is a promising building block for a variety of light vapor experiments, as it offers a very small footprint, high degree of integration and extremely strong confinement of light and vapor. As such it may be used for important applications, such as all optical switching, dispersion engineering (e.g. slow and fast light) and metrology, as well as for the observation of important effects such as strong coupling, and Purcell enhancement.

     

Bicriteria problems to minimize maximum tardiness and due date assignment cost in various scheduling environments

We study bicriteria problems of minimizing maximum tardiness and total due date assignment cost in various scheduling environments. We assume that each job can be assigned a different due date without any restriction, and that each due date assignment cost is a non-decreasing function of the quoted due date. We settle the complexity of most of the problems studied by either proving that they are NP-hard or finding a polynomial time solution for them. We also include approximation and non-approximability results for several parallel-machine problems.     

Thermal dehydration and decomposition of copper selenate pentahydrate

Temperature programmed desorption mass spectrometry (TPD-MS) and thermal gravimetry (TG) were utilized for the study of the thermal dehydration and decomposition of copper selenate pentahydrate (CuSeO₄·5H₂O). From the two techniques we suggest that the dehydration is a 3-step process reaching completion at 300 °C. The decomposition process however, is far more complicated consisting of several successive steps occurring between 480 and 900 °C. Initiated with the emission of oxygen, the decomposition of the anhydrous salt continues with SeO₂ emission via several unstable intermediates up to the conversion of the remaining copper monoxide into dicopper monoxide accompanied by oxygen emission.     

Laminar flow in a pipe at low and moderate reynolds numbers

Summary The laminar flow of a homogeneous viscous liquid in the inlet of a pipe is investigated numerically for a range of small and moderate Reynolds numbers where the boundary layer approximation is inapplicable. Velocity profiles and other characteristics of the flow are calculated and the results compared with approximate results obtained by other methods. The limiting case of vanishingly small Reynolds number is also treated analytically.Part of this work was performed while the second author was a summer visitor in the Applied Mathematics Department, Brookhaven National Laboratory, Upton (L.I.), New York.     

Nonlinear problem of an electrical discharge in a flow

The shape of the shoulder of a forming, filling and closing machine is calculated, based on geometrical considerations, first for a circular and subsequently for a rectangular (superelliptic) section. It appears to be possible to formulate a feasible solution for a model represented by a flat triangular region in the centre, which connects two truncated cones; for each cone one half of the intersection of shoulder and cylinder acts as the directrix of the cone. The solutions are formulated in terms which permit direct application in machine construction. Verification of the theory with the help of paper models and with real shoulders on machines shows a very good agreement between theory and practice. In addition practical tests produce excellent results as well.     

Complexity analysis of an assignment problem with controllable assignment costs and its applications in scheduling

We extend the classical linear assignment problem to the case where the cost of assigning agent j to task i is a multiplication of task i’s cost parameter by a cost function of agent j. The cost function of agent j is a linear function of the amount of resource allocated to the agent. A solution for our assignment problem is defined by the assignment of agents to tasks and by a resource allocation to each agent. The quality of a solution is measured by two criteria. The first criterion is the total assignment cost and the second is the total weighted resource consumption. We address these criteria via four different problem variations. We prove that our assignment problem is NP-hard for three of the four variations, even if all the resource consumption weights are equal. However, and somewhat surprisingly, we find that the fourth variation is solvable in polynomial time. In addition, we find that our assignment problem is equivalent to a large set of important scheduling problems whose complexity has been an open question until now, for three of the four variations.