The Weibull distribution is a popular and widely used distribution in reliability and in lifetime data analysis. Since 1958, the Weibull distribution has been modified by many researchers to allow for non-monotonic hazard functions. Many modifications of the Weibull distribution have achieved the above purpose. On the other hand, the number of parameters has increased, the forms of the survival and hazard functions have become more complicated and the estimation problems have risen. This thesis provides an extensive review of some discrete and continuous versions of the modifications of the Weibull distribution, which could serve as an important reference and encourage further modifications of the Weibull distribution. Four different modifications of the Weibull distribution are proposed to address some of the above problems using different techniques. First model, with five parameters, is constructed by considering a two-component serial system with one component following a Weibull distribution and another following a modified Weibull distribution. A new method has been proposed to reduce the number of parameters of the new modified Weibull distribution from five to three parameters to simplify the distribution and address the estimation problems. The reduced version has the same desirable properties of the original distribution in spite of having two less parameters. It can be an alternative distribution for all modifications of the Weibull distribution with bathtub shaped hazard rate functions. To deal with unimodal shaped hazard rates, the third model with four parameters, named as the exponentiated reduced modified Weibull distribution is introduced. This model is flexible, has a nice physical interpretation and has the ability to capture monotonically increasing, unimodal and bathtub shaped hazard rates. It is a generalization of the reduced modified Weibull distribution. The proposed distribution gives the best fit comparing to other modifications of the Weibull distribution including those having similar properties. A three-parameter discrete distribution is introduced based on the reduced distribution. It is one of only three discrete distributions allowing for bathtub shaped hazard rate functions. Four real data sets have applied to this distribution. The new distribution is shown to outperform at least three other models including the ones allowing for bathtub shaped hazard rate functions. The new models show flexibility and can be used to model different kinds of real data sets better than other modified versions of Weibull distribution including those having the same number of parameters. The mathematical properties and statistical inferences of the new models are studied. Based on a simulation study the performances of the MLEs of each model are assessed with respect to sample size n. We find no evidence that the generalized modified Weibull distribution can provide a better fit than the exponentiated Weibull distributionfor data sets exhibiting the modified unimodal hazard function.

Background. Lower-level graphics programming interfaces such as Direct3D 12 re-quire synchronization and data hazards between dependent workloads to be resolvedmanually. A barrier is a primitive used to resolve synchronization and data hazardsin a manner to achieve correct behavior by allowing developers to define waits be-tween workloads. However, due to its coarse-grained interface, workloads may beredundantly blocked, and data hazards resolved conservatively leading to excessiveGPU cache flushes even with correct usage. Objectives. To evaluate if the novel and more fine-grained enhanced barriers APIin Direct3D 12 can provide any improvements over Direct3D 12 resource (legacy)barriers in AMD FidelityFX applications using Direct3D 12. Methods. An experiment as a research method was carried out to investigate theeffects of enhanced barriers in existing Direct3D 12 applications. Frame time and thenumber of GPU cache flushes and invalidations occurring per frame were the primarymetrics measured. This was carried out by replacing legacy barriers with enhancedbarriers in three of AMD’s open-source, state-of-the-art image quality toolkits; Fideli-tyFX Super Resolution (FSR), FidelityFX Super Resolution 2 (FSR2), and Stochas-tic Screen Space Reflections (SSSR). Results. The use of enhanced barriers in FSR, FSR2, and SSSR showed that nosignificant differences were found in terms of frame time and the number of cacheflushes and invalidations occurring within a frame when compared to using resourcebarriers. Configurations of enhanced barriers that may reduce pipeline stall times re-main theoretical and could not be verified due to minuscule differences. These includecompute-only workload synchronization and non-blocking barrier layout transitions. Conclusions. Replacing legacy barriers with enhanced barriers in FSR, FSR2, andSSSR proved to be feasible, but lacks performance benefits for it to be desirable.However, the use of barriers can vary depending on the application, therefore, differ-ent results can arise given other synchronization scenarios. For existing Direct3D 12applications using resource barriers, it may be advisable to not upgrade to enhancedbarriers. / Bakgrund. Grafikprogrammeringsgränssnitt på lägre nivå, som till exempel Di-rect3D 12, kräver att synkronisering och datahinder mellan beroende arbetsbelast-ningar löses manuellt. En barriär är en primitiv som används för att lösa synkroniser-ing och datahinder som tillåter korrekt beteende genom att låta utvecklare definieraväntetider mellan arbeten. Men på grund av sitt högnivå-gränssnitt kan arbetsbelast-ningar onödigt blockeras och datahinder löses konservativt med överdrivet mycketGPU-cache-flushningar även vid korrekt användning. Syfte. För att utvärdera om det nya och mer detaljerade enhanced barriers API iDirect3D 12 kan ge några förbättringar jämfört med Direct3D 12 Resource (Legacy)Barriers i AMD FidelityFX-applikationer som använder Direct3D 12. Metod. Ett experiment som en forskningsmetod genomfördes för att undersöka ef-fekterna av enhanced barriers i befintliga Direct3D 12-applikationer. Bildtid och an-talet GPU-cache-flushar och ogiltigförklaringar som inträffar per bild var de primäramätvärdena. Detta genomfördes genom att ersätta legacy barriers med enhanced bar-riers i tre av AMD:s öppen källkodsbaserade verktyg för bildkvalitet; FidelityFX Su-per Resolution (FSR), FidelityFX Super Resolution 2 (FSR2) och Stochastic ScreenSpace Reflections (SSSR). Resultat. Användningen av enhanced barriers i FSR, FSR2 och SSSR visade attinga betydande skillnader kunde hittas när det gäller bildtid och antalet cache-flusharoch ogiltigförklaringar som inträffar inom en bild jämfört med användning av resourcebarriers. Konfigurationer av enhanced barrier som kan minska pipeline-stopp tiderförblir teoretiska och kunde inte verifieras på grund av minimala skillnader. Dessainkluderar synkronisering av endast compute-shader arbete och icke-blockerandeövergångar av barriär-layout. Slutsatser. Att ersätta legacy barriers med enhanced barriers i FSR, FSR2 ochSSSR visade sig vara genomförbart, men saknar prestandafördelar för att vara ön-skvärt. Dock kan användningen av barriärer variera beroende på applikationen, ochdärför kan olika resultat uppstå med olika synkroniseringsscenarier. För befintligaDirect3D 12-applikationer som använder resursbarriärer kan det vara lämpligt attinte uppgradera till förbättrade barriärer.