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

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 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

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 include several additional results in the paper. We show that a standard lower-bound technique for MLAP, based on so-called Single-Phase instances, cannot give super-constant lower bounds (as a function of the tree depth). This result is established by giving an online algorithm with optimal competitive ratio 4 for such instances on arbitrary trees. We also study the MLAP variant when the tree is a path, for which we give a lower bound of 4 on the competitive ratio, improving the lower bound known for general MLAP. We complement this with a matching upper bound for the deadline setting

Online algorithms for multi-level aggregation

In the multilevel 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 the root of T. 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 Transmission Control Protocol acknowledgment problem, whereas for trees of depth 2, it is equivalent to the joint replenishment problem. Aggregation problems for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and 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 trees of arbitrary (fixed) depth. The competitive ratio is O(D42D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines

New results on multi-level aggregation

International audienceIn the Multi-Level Aggregation Problem (MLAP ), requests for service arrive at the nodes of an edge-weighted rooted tree . Each service is represented by a subtree X of 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 a waiting cost between its arrival and service time. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. The currently best online algorithms for the MLAP achieve competitive ratios polynomial in the tree depth, while the best lower bound is only 3.618. In this paper, we report some progress towards closing this gap, by improving this lower bound and providing several tight bounds for restricted variants of MLAP: (1) We first study a Single-Phase variant of MLAP where all requests are released at the beginning and expire at some unknown time θ, for which we provide an online algorithm with optimal competitive ratio of 4. (2) We prove a lower bound of 4 on the competitive ratio for MLAP, even when the tree is a path. We complement this with a matching upper bound for the deadline variant of MLAP on paths. Additionally, we provide two results for the offline case: (3) We prove that the Single-Phase variant can be solved optimally in polynomial time, and (4) we give a simple 2-approximation algorithm for offline MLAP with deadlines

CD73 in Autoimmune Arthritis

Adenosine is a potent anti-inflammatory molecule that plays an important role in many diseases. Extracellular levels of adenosine are determined by a combination of membrane transporters and ecto-nucleotidases such as CD73. Therapeutic targeting of the adenosinergic pathway, such as administration of adenosine receptor agonists, could be a valuable approach in the treatment of rheumatoid arthritis(RA). Until recently, the role of CD73 in RA pathogenesis had not been established. Using CD73-deficient gene-targeted mice, we demonstrated that CD73 plays a critical protective role in collagen-induced arthritis (CIA) in mice. Our findings, together with the results of recently published human studies, thus suggests that enhancement of CD73 activity may be a novel therapeutic approach in RA

Effect of cloudiness on the production of electricity by photovoltaic panels

The paper deals with the influence of different types of cloud on the production of electricity by photovoltaic panels. In the introductory part of the paper, processes in the atmosphere are described, giving rise to various types of clouds and, consequently, to the formation of cloud. In the next section of the paper, the system is described on which the research and the methodology of acquisition and data evaluation were carried out. The last part of the paper summarizes the overall results of the research