Asymptotic Expansions for the Sojourn Time Distribution in the M/G/1M/G/1-PS Queue

We consider the $M/G/1$ queue with a processor sharing server. We study the conditional sojourn time distribution, conditioned on the customer's service requirement, as well as the unconditional distribution, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. Our results demonstrate the possible tail behaviors of the unconditional distribution, which was previously known in the cases $G=M$ and $G=D$ (where it is purely exponential). We assume that the service density decays at least exponentially fast. We use various methods for the asymptotic expansion of integrals, such as the Laplace and saddle point methods.Comment: 45 page

Asymptotic Expansions for the Conditional Sojourn Time Distribution in the M/M/1M/M/1-PS Queue

We consider the $M/M/1$ queue with processor sharing. We study the conditional sojourn time distribution, conditioned on the customer's service requirement, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison \cite{MO} and Flatto \cite{FL}.Comment: 30 pages, 3 figures and 1 tabl

Decomposing the queue length distribution of processor-sharing models into queue lengths of permanent customer queues

We obtain a decomposition result for the steady state queue length distribution in egalitarian processor-sharing (PS) models. In particular, for an egalitarian PS queue with $K$ customer classes, we show that the marginal queue length distribution for class $k$ factorizes over the number of other customer types. The factorizing coefficients equal the queue length probabilities of a PS queue for type $k$ in isolation, in which the customers of the other types reside \textit{ permanently} in the system. Similarly, the (conditional) mean sojourn time for class $k$ can be obtained by conditioning on the number of permanent customers of the other types. The decomposition result implies linear relations between the marginal queue length probabilities, which also hold for other PS models such as the egalitarian processor-sharing models with state-dependent system capacity that only depends on the total number of customers in the system. Based on the exact decomposition result for egalitarian PS queues, we propose a similar decomposition for discriminatory processor-sharing (DPS) models, and numerically show that the approximation is accurate for moderate differences in service weights. \u

The mean-field behavior of processor sharing systems with general job lengths under the SQ(d) policy

This paper addresses the mean-field behavior of large-scale systems of parallel servers with a processor sharing service discipline when arrivals are Poisson and jobs have general service time distributions when an SQ() routing policy is used. Under this policy, an arrival is routed to the server with the least number of progressing jobs among randomly chosen servers. The limit of the empirical distribution is then used to study the statistical properties of the system. In particular, this shows that in the limit as grows, individual servers are statistically independent of others (propagation of chaos) and more importantly, the equilibrium point of the mean-field is insensitive to the job length distributions that has important engineering relevance for the robustness of such routing policies used in web server farms. We use a framework of measure-valued processes and martingale techniques to obtain our results. We also provide numerical results to support our analysis

Unsolved issues related to multivitamin and multimineral support for patients following bariatric surgery

In the review article the authors discuss unsolved issues regarding vitamin and mineral support of patients undergoing bariatric/metabolic operations. Many patients refer for the surgery already having deficiencies of macro- and micronutrients and need preoperative preparation for improving of vitamin-mineral status. Vitamin-mineral support is indicated after each bariatric operation, however the volume of nutritive support and the choice and doses of vitamins and minerals can be varied depending on type of surgery. Hypoabsorptive procedures such as duodenal switch or SADI-S (Single anastomosis duodeno–ileal bypass with sleeve gastrectomy) usually demand more significant protein, calcium and iron supplementations together with administration of multivitamin complexes containing fat-soluble vitamins. Existing standard vitamin-mineral complexes may not contain necessary doses of iron, calcium, fat-soluble and other vitamins as well as some microelements like cooper, zink, selenium etc. Thus, sufficient vitamin-mineral support may need up to 10 tablets or pills daily depending on kind of surgery. That can influence on patient’s compliance, lead to refusal of supplementations and thus to undesirable metabolic consequences after surgery. The authors have concluded about necessity of working out of cost-effective domestic series of vitamin-mineral combinations with high bio-availability specially adapted to every class of bariatric/metabolic operation. To author’s opinion this can lead to better patient’s compliance and can prevent undesirable side effects after metabolic surgery

A stochastic network with mobile users in heavy traffic

We consider a stochastic network with mobile users in a heavy-traffic regime. We derive the scaling limit of the multi-dimensional queue length process and prove a form of spatial state space collapse. The proof exploits a recent result by Lambert and Simatos which provides a general principle to establish scaling limits of regenerative processes based on the convergence of their excursions. We also prove weak convergence of the sequences of stationary joint queue length distributions and stationary sojourn times.Comment: Final version accepted for publication in Queueing Systems, Theory and Application

High power femtosecond chirped pulse amplification in large mode area photonic bandgap Bragg fibers

International audienceWe report on high power amplification of femtosecond pulses in 40-μm core diameter Yb-doped photonic bandgap Bragg fibers. The robustness to bending and transverse spatial behavior of these fibers is analyzed through simulations. The fibers are used in both stages of a moderately stretched (150 ps) femtosecond chirped pulsed amplification (CPA) system. A compressed average power of 6.3W is obtained using a low-index polymer-coated Bragg fiber with excellent beam quality and high efficiency, in agreement with numerical simulations. The use of an air-clad Bragg fiber allows us to scale the output power to 47 W at a repetition rate of 35MHz. This experiment demonstrates the great potential of Bragg fibers to increase the mode area and the power of practical bending-tolerant femtosecond fiber systems

405-nm pumped Ce3+-doped silica fiber for broadband fluorescence from cyan to red

A pure Ce-doped silica fiber is fabricated using modified chemical vapor deposition (MCVD) technique. Fluorescence characteristics of a Ce-doped silica fiber are experimentally investigated with continuous wave pumping from 440 nm to 405 nm. Best pump absorption and broad fluorescence spectrum is observed for ~ 405 nm laser. Next, the detailed analysis of spectral response as a function of pump power and fiber length is performed. It is observed that a -10dB spectral width of ~ 280 mn can be easily achieved with different combinations of the fiber length and pump power. Lastly, we present, for the first time to the best of our knowledge, a broadband fluorescence spectrum with -10dB spectral width of 301 nm, spanning from ~ 517.36 nm to ~ 818 nm, from such fibers with non-UV pump lasers