On generalized processor sharing and objective functions: analytical framework

Today, telecommunication networks host a wide range of heterogeneous services. Some demand strict delay minima, while others only need a best-effort kind of service. To achieve service differentiation, network traffic is partitioned in several classes which is then transmitted according to a flexible and fair scheduling mechanism. Telecommunication networks can, for instance, use an implementation of Generalized Processor Sharing (GPS) in its internal nodes to supply an adequate Quality of Service to each class. GPS is flexible and fair, but also notoriously hard to study analytically. As a result, one has to resort to simulation or approximation techniques to optimize GPS for some given objective function. In this paper, we set up an analytical framework for two-class discrete-time probabilistic GPS which allows to optimize the scheduling for a generic objective function in terms of the mean unfinished work of both classes without the need for exact results or estimations/approximations for these performance characteristics. This framework is based on results of strict priority scheduling, which can be regarded as a special case of GPS, and some specific unfinished-work properties in two-class GPS. We also apply our framework on a popular type of objective functions, i.e., convex combinations of functions of the mean unfinished work. Lastly, we incorporate the framework in an algorithm to yield a faster and less computation-intensive result for the optimum of an objective function

Sojourn times in a processor sharing queue with multiple vacations

We study an M/G/1 processor sharing queue with multiple vacations. The server only takes a vacation when the system has become empty. If he finds the system still empty upon return, he takes another vacation, and so on. Successive vacations are identically distributed, with a general distribution. When the service requirements are exponentially distributed we determine the sojourn time distribution of an arbitrary customer. We also show how the same approach can be used to determine the sojourn time distribution in an M/M/1-PS queue of a polling model, under the following constraints: the service discipline at that queue is exhaustive service, the service discipline at each of the other queues satisfies a so-called branching property, and the arrival processes at the various queues are independent Poisson processes. For a general service requirement distribution we investigate both the vacation queue and the polling model, restricting ourselves to the mean sojourn time

Sojourn times in a processor sharing queue with multiple vacations

We study an M/G/1 processor sharing queue with multiple vacations. The server only takes a vacation when the system has become empty. If he finds the system still empty upon return, he takes another vacation, and so on. Successive vacations are identically distributed, with a general distribution. When the service requirements are exponentially distributed we determine the sojourn time distribution of an arbitrary customer. We also show how the same approach can be used to determine the sojourn time distribution in an M/M/1-PS queue of a polling model, under the following constraints: the service discipline at that queue is exhaustive service, the service discipline at each of the other queues satisfies a so-called branching property, and the arrival processes at the various queues are independent Poisson processes. For a general service requirement distribution we investigate both the vacation queue and the polling model, restricting ourselves to the mean sojourn time

Markov-modulated M/G/1 type queue in heavy traffic and its application to time-sharing disciplines

International audienceThis paper deals with a single-server queue with modulated arrivals, service requirements and service capacity. In our first result, we derive the mean of the total workload assuming generally distributed service requirements and any service discipline which does not depend on the modulating environment. We then show that the workload is exponentially distributed under heavy-traffic scaling. In our second result, we focus on the discriminatory processor sharing (DPS) discipline. Assuming exponential, class-dependent service requirements, we show that the joint distribution of the queue lengths of different customer classes under DPS undergoes a state-space collapse when subject to heavy-traffic scaling. That is, the limiting distribution of the queue-length vector is shown to be exponential, times a deterministic vector. The distribution of the scaled workload, as derived for general service disciplines, is a key quantity in the proof of the state-space collapse

Oral Contraceptive Use and Breast Cancer Risk: Retrospective and Prospective Analyses From a BRCA1 and BRCA2 Mutation Carrier Cohort Study

Background: For BRCA1 and BRCA2 mutation carriers, the association between oral contraceptive preparation (OCP) use and breast cancer (BC) risk is still unclear. Methods: Breast camcer risk associations were estimated from OCP data on 6030 BRCA1 and 3809 BRCA2 mutation carriers using age-dependent Cox regression, stratified by study and birth cohort. Prospective, left-truncated retrospective and full-cohort retrospective analyses were performed. Results: For BRCA1 mutation carriers, OCP use was not associated with BC risk in prospective analyses (hazard ratio [HR] = 1.08, 95% confidence interval [CI] = 0.75 to 1.56), but in the left-truncated and full-cohort retrospective analyses, risks were increased by 26% (95% CI = 6% to 51%) and 39% (95% CI = 23% to 58%), respectively. For BRCA2 mutation carriers, OCP use was associated with BC risk in prospective analyses (HR = 1.75, 95% CI = 1.03 to 2.97), but retrospective analyses were inconsistent (left-truncated: HR = 1.06, 95% CI = 0.85 to 1.33; full cohort: HR = 1.52, 95% CI = 1.28 to 1.81). There was evidence of increasing risk with duration of use, especially before the first full-term pregnancy (BRCA1: both retrospective analyses, P < .001 and P = .001, respectively; BRCA2: full retrospective analysis, P = .002). Conclusions: Prospective analyses did not show that past use of OCP is associated with an increased BC risk for BRCA1 mutation carriers in young middle-aged women (40-50 years). For BRCA2 mutation carriers, a causal association is also not likely at those ages. Findings between retrospective and prospective analyses were inconsistent and could be due to survival bias or a true association for younger women who were underrepresented in the prospective cohort. Given the uncertain safety of long-term OCP use for BRCA1/2 mutation carriers, indications other than contraception should be avoided and nonhormonal contraceptive methods should be discussed