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

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

Characteristics of carbohydrate metabolism in the surgical treatment of morbid obesity and type 2 diabetes mellitus using various modifications of biliopancreatic diversion with duodenal switch

BACKGROUND: Biliopancreatic diversion (BPD), and its modifications, is the most effective surgical bariatric treatment of morbid obesity and associated metabolic disturbances. However, at present comparative studies of the dynamics of carbohydrate metabolism after various modifications of the BPD are lacking.AIM: comparative assessment for the effectiveness of biliopancreatic diversion with duodenal switch (BPD-DS) in the HessMarceau and single-anastomosis duodeno-ileal bypass with sleeve gastrectomy (SADI-S) modifications for correcting carbohydrate metabolism disorders and achieving remission of Diabetes mellitus type 2 (DM2) within a period of up to five years after both operations.MATERIALS AND METHODS: within the framework of a prospective study, 200 patients with morbid obesity were operated on using the BPD-DS (group 1, n = 100) and SADI (group 2, n = 100) methods, the dynamics of clinical and laboratory parameters was analyzed in groups depending on the presence, or absence, of DM2, with an emphasis on the analysis of indicators of carbohydrate metabolism. DM2 was diagnosed in 35 (35.0%) patients in group 1 (BPD - DS) and 45 (45.0%) in group 2 (SADI). All patients underwent a standard set of clinical, laboratory and instrumental examination methods before, immediately after and at 3, 6, 9, 12, 18, 24, 36, 48 and 60 months after the operation.RESULTS: 5 years after the operation, complete remission of DM2 was achieved in 38 (84.4%) and 32 (91.4%) patients from the SADI and BPD-DS groups, respectively, and 7 (15.6%) and 3 (8.6 %) of patients achieved partial remission. The level of C-peptide, which also decreased after both modifications of BPS, was higher in patients after BPS in the SADI modification, both in patients with DM2 36 months (p<0.05), and in patients without DM2 at 3, 12 and 24 months after surgery (p<0.05).CONCLUSION: The frequency of achieving stable remission of DM2 is comparable in both groups. The carbohydrate profile of patients after SADI is characterized by higher levels of glucose and C-peptide compared to BPD-DS at different periods of follow-up over five years

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

Transform-domain analysis of packet delay in network nodes with QoS-aware scheduling

In order to differentiate the perceived QoS between traffic classes in heterogeneous packet networks, equipment discriminates incoming packets based on their class, particularly in the way queued packets are scheduled for further transmission. We review a common stochastic modelling framework in which scheduling mechanisms can be evaluated, especially with regard to the resulting per-class delay distribution. For this, a discrete-time single-server queue is considered with two classes of packet arrivals, either delay-sensitive (1) or delay-tolerant (2). The steady-state analysis relies on the use of well-chosen supplementary variables and is mainly done in the transform domain. Secondly, we propose and analyse a new type of scheduling mechanism that allows precise control over the amount of delay differentiation between the classes. The idea is to introduce N reserved places in the queue, intended for future arrivals of class 1

Enchanced preoperative rehabilitation as part of the complex perioperative rehabilitation of patients with super-obesity and obstructive sleep apnea syndrome (clinical outlook)

Obesity is a global social and economic problem. The bariatric surgery is a most effective treatment for obesity. The presented clinical case demonstrates the usage of principles of enchanced perioperative rehabilitation for the preoperative preparation of a patient with super obesity and with severe obstructive sleep apnea and alveolar hypoventilation syndrome.A 54-year-old patient was hospitalized with complaints of obesity, impossibility of persistent weight loss conservatively, severe daytime sleepiness, frequent nocturnal awakenings (up to 8 times per night). The patient’s weight was 230 kg with a height of 157 cm (BMI 93.5 kg / m2). The examination revealed a syndrome of sleep apnea of mixed genesis of extremely severe degree, chronic night hypoxemia of an extremely severe degree. Preoperative preparation was performed in accordance with the program of enchanced perioperative rehabilitation. The duration of preoperative preparation was 19 days; weight loss — 40 kg (%WL -17,4), compensation of comorbidities was achieved as well. After that the patient underwent a laparoscopic sleeve gastrectomy. There were no complications in the postoperative period. Length of postoperative hospital stay was 6 days. At follow-up examination one year after surgery, body weight dropped from 230 to 153 kg (% WL-33.5), a significant improvement of the quality of life was achieved.The enchanced perioperative rehabilitation program can be successfully used as an effective method for preoperartive preparation of the patients with morbid obesity in combination with severe obstructive sleep apnea syndrome.and obesity hypoventilation. It can be a reasonable alternative to the standard program with preoperative intragastric balloon treatment. The use of this technique allows to increase the effectiveness of treatment of these high-risk patients, as well as to reduce the risk of perioperative complications

Moment-Generating Algorithm for Response Time in Processor Sharing Queueing Systems

Response times are arguably the most representative and important metric for measuring the performance of modern computer systems. Further, service level agreements (SLAs), ranging from data centres to smartphone users, demand quick and, equally important, predictable response times. Hence, it is necessary to calculate moments, at least, and ideally response time distributions, which is not straightforward. A new moment-generating algorithm for calculating response times analytically is obtained, based on M/M/1 processor sharing (PS) queueing models. This algorithm is compared against existing work on response times in M/M/1-PS queues and extended to M/M/1 discriminatory PS queues. Two real-world case studies are evaluated