Approximation for Scheduling on Parallel Machines with Fixed Jobs or Unavailability Periods

We survey results that address the problem of non-preemptive scheduling on parallel machines with fixed jobs or unavailability periods with the purpose of minimizing the maximum completion time. We consider both identical and uniform processors, and also address the special case of scheduling on nonsimultaneous parallel machines, which may start processing at different times. The discussed results include polynomial-time approximation algorithms that achieve the best possible worst-case approximation bound of 1.5 in the class of polynomial algorithms unless P = NP for scheduling on identical processors with at most one fixed job on each machine and on uniform machines with at most one fixed job on each machine. The presented heuristics have similarities with the LPT algorithm or the MULTIFIT algorithm and they are fast and easy to implement. For scheduling on nonsimultaneous machines, experiments suggest that they would perform well in practice. We also include references to the relevant work in this area that contains more complex algorithms. We then discuss the main methods of argument used in the approximation bound proofs for the simple heuristics, and comment upon current challenges in this area by describing aspects of related practical problems from the automotive industry

Multiprocessor Scheduling with Availability Constraints

We consider the problem of scheduling a given set of tasks on multiple pro- cessors with predefined periods of unavailability, with the aim of minimizing the maximum completion time. Since this problem is strongly NP-hard, polynomial ap- proximation algorithms are being studied for its solution. Among these, the best known are LPT (largest processing time first) and Multifit with their variants. We give a Multifit-based algorithm, FFDL Multifit, which has an optimal worst- case performance in the class of polynomial algorithms for same-speed processors with at most two downtimes on each machine, and for uniform processors with at most one downtime on each machine, assuming that P 6= NP. Our algorithm finishes within 3/2 the maximum between the end of the last downtime and the end of the optimal schedule. This bound is asymptotically tight in the class of polynomial algorithms assuming that P 6= NP. For same-speed processors with at most k downtimes on each machine our algorithm finishes within ( 3 2 + 1 2k ) the end of the last downtime or the end of the optimal schedule. For problems where the optimal schedule ends after the last downtime, and when the downtimes represent fixed jobs, the maximum completion time of FFDL Multifit is within 3 2 or ( 3 2+ 1 2k ) of the optimal maximum completion time. We also give an LPT-based algorithm, LPTX, which matches the performance of FFDL Multifit for same-speed processors with at most one downtime on each machine, and is thus optimal in the class of polynomial algorithms for this case. LPTX differs from LPT in that it uses a specific order of processors to assign tasks if two processors become available at the same time. For a similar problem, when there is at most one downtime on each machine and no more than half of the machines are shut down at the same time, we show that a bound of 2 obtained in a previous work for LPT is asymptotically tight in the class of polynomial algorithms assuming that P 6= NP

Scheduling on uniform nonsimultaneous parallel machines

Abstract We consider the problem of scheduling on uniform processors which may not start processing at the same time with the purpose of minimizing the maximum completion time. We give a variant of the Multifit algorithm which generates schedules which end within 1.382 times the optimal maximum completion time for the general problem, and within √ 6/2 times the optimal maximum completion time for problem instances with at most two processors. This results from properties of a variant of the Multifit algorithm for scheduling on uniform processors with simultaneous start times. We also show that if a better approximation bound of the Multifit variant for scheduling on uniform processors will be found in the future, this bound will also apply to our Multifit variant for scheduling on nonsimultaneous uniform processors

Availability of a Hybrid FSO/RF Link While Using the Link’s Diversity for Packet Scheduling

Hybrid-free space optical and radio frequency wireless links are a way of providing reliable transport of real-time traffic in outdoor wireless environments. We consider a link layer protocol that assigns packets to each physical channel of such a hybrid link, which first attempts to send each packet over one of the links (the main link) and, if unsuccessful, sends the packet over the other link (the backup link). The hybrid link processes high-priority traffic by using the link layer protocol and additional (background) traffic at low priority over the backup link. In this setting, high-priority traffic can be transmitted at a rate as high as the maximum capacity of the main link, assuming that the backup link can compensate for main link capacity deterioration, with no need for reconfigurations aimed at adapting to changes in weather conditions, which is an advantage over other approaches. From the perspective of link availability for high-priority traffic, we compare our approach to using another protocol that does not require reconfigurations, which could be employed if the backup link is expected to have a constant transmission rate during the time interval of interest. For situations where both links can be represented by finite-state Markov models with states corresponding to channel bit error rates, as has been done in previous literature for radio frequency links and for free space optical links affected by strong atmospheric turbulence and Gaussian noise, we give a way to provide probabilistic quality of service guarantees for background traffic assuming that the high-priority traffic is insured to not exceed a given constant rate

Transformation Systems Semantics and Compositionality of Algebraic Petri Nets

In the rst part of the paper, place-transition nets and algebraic highlevel nets are translated into algebra transformation systems, thus de ning the net semantics. Both sequential and concurrent nets are considered. Some results concerning the relation between the net semantics resulting from a rule-based system corresponding to the net, and a transformation system built directly from the net are presented

PERIODONTAL DISEASE AND COMPLICATIONS IN THE CONTEXT OF THE COVID-19 PANDEMIC. INACTIVATION OF PATHOGENS AT THE LEVEL OF A DENTAL OFFICE IN THE CONTEXT OF THE PANDEMIC. REVIEW.

When a disease is new to humans, only very few people are immune to it. In such cases, there is no vaccine and the result can be a rapid increase in the incidence of the disease. How dangerous or deadly that disease is – it all depends on the specific virus in question and the general state of health of each individual patient. If a disease does not only spread regionally but ends up affecting entire countries and continents, then experts speak of a pandemic. Periodontal disease is a bacterial infection that causes inflammation of the supporting tissues of the tooth. Experts say that there is a dangerous link between this and the COVID-19 infection, which leads to hospitalization or even admission to intensive care, periodontal disease can lead to complications of Covid-19. The association between coronavirus disease 2019 infection and periodontitis has also been investigated in the retrospective studies. In the current context we have solutions to eradicate already known pathogens, but new pathogens or changes to existing ones occur that require time and research to find a solution