A ϕ\phi-Competitive Algorithm for Scheduling Packets with Deadlines

In the online packet scheduling problem with deadlines (PacketScheduling, for short), the goal is to schedule transmissions of packets that arrive over time in a network switch and need to be sent across a link. Each packet has a deadline, representing its urgency, and a non-negative weight, that represents its priority. Only one packet can be transmitted in any time slot, so, if the system is overloaded, some packets will inevitably miss their deadlines and be dropped. In this scenario, the natural objective is to compute a transmission schedule that maximizes the total weight of packets which are successfully transmitted. The problem is inherently online, with the scheduling decisions made without the knowledge of future packet arrivals. The central problem concerning PacketScheduling, that has been a subject of intensive study since 2001, is to determine the optimal competitive ratio of online algorithms, namely the worst-case ratio between the optimum total weight of a schedule (computed by an offline algorithm) and the weight of a schedule computed by a (deterministic) online algorithm. We solve this open problem by presenting a $\phi$-competitive online algorithm for PacketScheduling (where $\phi\approx 1.618$ is the golden ratio), matching the previously established lower bound.Comment: Major revision of the analysis and some other parts of the paper. Another revision will follo

Better Approximation Bounds for the Joint Replenishment Problem

The Joint Replenishment Problem (JRP) deals with optimizing shipments of goods from a supplier to retailers through a shared warehouse. Each shipment involves transporting goods from the supplier to the warehouse, at a fixed cost C, followed by a redistribution of these goods from the warehouse to the retailers that ordered them, where transporting goods to a retailer $\rho$ has a fixed cost $c_\rho$. In addition, retailers incur waiting costs for each order. The objective is to minimize the overall cost of satisfying all orders, namely the sum of all shipping and waiting costs. JRP has been well studied in Operations Research and, more recently, in the area of approximation algorithms. For arbitrary waiting cost functions, the best known approximation ratio is 1.8. This ratio can be reduced to 1.574 for the JRP-D model, where there is no cost for waiting but orders have deadlines. As for hardness results, it is known that the problem is APX-hard and that the natural linear program for JRP has integrality gap at least 1.245. Both results hold even for JRP-D. In the online scenario, the best lower and upper bounds on the competitive ratio are 2.64 and 3, respectively. The lower bound of 2.64 applies even to the restricted version of JRP, denoted JRP-L, where the waiting cost function is linear. We provide several new approximation results for JRP. In the offline case, we give an algorithm with ratio 1.791, breaking the barrier of 1.8. In the online case, we show a lower bound of 2.754 on the competitive ratio for JRP-L (and thus JRP as well), improving the previous bound of 2.64. We also study the online version of JRP-D, for which we prove that the optimal competitive ratio is 2

A φ-competitive algorithm for scheduling packets with deadlines

In the online packet scheduling problem with deadlines (PacketScheduling, for short), the goal is to schedule transmissions of packets that arrive over time in a network switch and need to be sent across a link. Each packet has a deadline, representing its urgency, and a non-negative weight, that represents its priority. Only one packet can be transmitted in any time slot, so, if the system is overloaded, some packets will inevitably miss their deadlines and be dropped. In this scenario, the natural objective is to compute a transmission schedule that maximizes the total weight of packets which are successfully transmitted. The problem is inherently online, with the scheduling decisions made without the knowledge of future packet arrivals. The central problem concerning PacketScheduling, that has been a subject of intensive study since 2001, is to determine the optimal competitive ratio of online algorithms, namely the worst-case ratio between the optimum total weight of a schedule (computed by an offline algorithm) and the weight of a schedule computed by a (deterministic) online algorithm. We solve this open problem by presenting a ϕ-competitive online algorithm for PacketScheduling (where ϕ ≈ 1.618 is the golden ratio), matching the previously established lower bound

Online packet scheduling with bounded delay and lookahead

We study the online bounded-delay packet scheduling problem (Packet Scheduling), where packets of unit size arrive at a router over time and need to be transmitted over a network link. Each packet has two attributes: a non-negative weight and a deadline for its transmission. The objective is to maximize the total weight of the transmitted packets. This problem has been well studied in the literature; yet currently the best published upper bound is 1.828 [8],still quite far from the best lower bound ofφ≈1.618 [11, 2, 6].In the variant of Packet Scheduling with s-bounded instances, each packet can be scheduled in at most s consecutive slots, starting at its release time. The lower bound of φ applies even to the special case of 2-bounded instances, and a φ-competitive algorithm for 3-boundedinstances was given in [5]. Improving that result, and addressing a question posed by Goldwasser [9], we present a φ-competitive algorithm for 4-boundedinstances. We also study a variant of Packet Scheduling where an online algorithm has the additional power of1-lookahead, knowing at time t which packets will arrive at time t+ 1. For Packet Scheduling with 1-lookahead restricted to 2-bounded instances, we present an online algorithm with competitive ratio12(√13−1)≈1.303 and we prove a nearly tight lower boundof14(1 +√17)≈1.281. In fact, our lower bound result is more general: using only 2-boundedinstances, for any integer`≥0 we prove a lower bound of12(`+1)(1 +√5 + 8`+ 4`2) for online algorithms with`-look ahead, i.e., algorithms that at time t can see all packets arriving by time t+`. Finally, for non-restricted instances we show a lower bound of 1.25 for randomized algorithms with`-lookahead, for any`≥0

Online Algorithms for Multi-Level Aggregation

In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes of an edge-weighted tree T, and have to be served eventually. A service is defined as a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the TCP Acknowledgment Problem, while for trees of depth 2, it is equivalent to the Joint Replenishment Problem. Aggregation problem for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and in supply-chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests. Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant competitive online algorithm for networks of arbitrary (fixed) number of levels. The competitive ratio is O(D^4 2^D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines. We also show several additional lower and upper bound results for some special cases of MLAP, including the Single-Phase variant and the case when the tree is a path

Epidemiology of chronic kidney disease after renal transplantation according to different methods of glomerular filtration rate estimation

Wzory obecnie stosowane przy obliczaniu szacowanego wskaźnika filtracji kłębuszkowej (eGFR) są oparte na standaryzacji przeprowadzonej w populacji pacjentów z przewlekłą chorobą nerek. Stosowanie tych wzorów w populacji biorców nerki przeszczepionej może wiązać się ze zmniejszoną precyzją tych oznaczeń. W niniejszym artykule przedstawiono aktualny stan wiedzy na temat metod obliczania eGFR u biorców nerki przeszczepionej na podstawie danych z literatury oraz badań własnych. Opisano czynniki potencjalnie wpływające na zmienność wyników obliczania eGFR w tej grupie pacjentów. Artykuł zawiera dyskusję na temat implikacji klinicznych wynikających ze zmiany stosowanego wzoru do obliczania eGFR. Opisano także potencjalny wpływ wyboru metody oznaczania eGFR na zmienność częstości występowania poszczególnych stadiów niewydolności nerki przeszczepionej. Ponadto zasugerowano wybór metody oznaczania filtracji kłębuszkowej u biorców nerki przeszczepionej.Formulas used currently for estimation of glomerular filtration rate (GFR) have been standardized in population of patients with chronic kidney disease. Application of these formulas in population of renal transplant recipients brings a risk of diminished precision of these equations. We present current state of knowledge on methods of estimation of GFR in renal transplant recipients, based on contemporary literature data and our own analyses. Factors that have influence on variability of results are also described in this article. We discuss consequences of choice of estimation formulas on clinical decision making process. Beyond that we analyze a potential influence of estimation formulas choice on post transplant chronic kidney disease epidemiology. The article contains also suggestion of the best method of GFR estimation in population of renal transplant recipients

Effect of changing P/Ge and Mn/Fe ratios on the magnetocaloric effect and structural transition in the (Mn,Fe)2 (P,Ge) intermetallic compounds

The magnetocaloric effect in the MnxFe2-xP1-yGey intermetallic compounds with the amount of Mn in the range of x = 1.05 to 1.17 and amount of Ge in the range of y = 0.19 to 0.22 has been studied. It was found that a higher Ge/P ratio causes an increase in Curie temperature, magnetocaloric effect at low field (up to 1 T), activation energy of structural transition and a decrease in thermal hysteresis, as well as transition enthalpy. Contrary to this observation, higher Mn/Fe ratio causes a decrease in Curie temperature, slight decrease of magnetocaloric effect at low magnetic field, and an increase in thermal hysteresis. Simultaneous increase of both ratios may be very advantageous, as the thermal hysteresis can be lowered and magnetocaloric effect can be enhanced without changing the Curie temperature. Some hints about optimization of the composition for applications at low magnetic fields (0.5 T to 2 T) have been presented

Structure and Properties of Copper Pyrophosphate by First-Principle Calculations

Investigated the structural, electronic, and magnetic properties of copper pyrophosphate dihydrate (CuPPD) by the first-principle calculations based on the density functional theory (DFT). Simulations were performed with the generalized gradient approximation (GGA) of the exchangecorrelation functional (Exc) supplemented by an on-site Coulomb self-interaction (U–Hubbard term). It was confirmed that the GGA method did not provide a satisfactory result in predicting the electronic energy band gap width (Eg) of the CuPPD crystals. Simultaneously, we measured the Eg of CuPPD nanocrystal placed inside mesoporous silica using the ultraviolet–visible spectroscopy (UV–VIS) technique. The proposed Hubbard correction for Cu-3d and O-2p states at U = 4.64 eV reproduces the experimental value of Eg = 2.34 eV. The electronic properties presented in this study and the results of UV–VIS investigations likely identify the semiconductor character of CuPPD crystal, which raises the prospect of using it as a component determining functional properties of nanomaterials, including quantum dots