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The minimumbacklog problem
 IN PROC. 5TH CANAD. CONF. COMPUT. GEOM
, 1993
"... We introduce and study the minimumbacklog problem (MBP). The MBP arises in sensor networks and is related to the classic kserver problem. It can be understood as a 2person game played on a graph G = (V, E). The “player ” moves along the edges of the graph; the opponent is the “adversary. ” The ..."
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We introduce and study the minimumbacklog problem (MBP). The MBP arises in sensor networks and is related to the classic kserver problem. It can be understood as a 2person game played on a graph G = (V, E). The “player ” moves along the edges of the graph; the opponent is the “adversary. ” The game proceeds in timesteps. In each timestep the adversary pours a total of one unit of water into “cups ” that are located on the vertices of the graph, arbitrarily distributing the water among the cups. The player then moves from her current vertex to an adjacent vertex and empties the cup at that vertex. The player’s objective is to minimize the maximum amount of water (the backlog) in any cup at any time. We show that the competitive ratio of any algorithm for the MBP has a lower bound of Ω(∆), where ∆ is the diameter of the graph. Thus, we focus on determining a strategy for the player that guarantees a uniform upper bound on the backlog. In general graphs, the deamortization analysis of Dietz and Sleator gives a bound of O( ∆ ln V ). Our main result is that in geometric settings (e.g., sensor fields), one can obtain substantially better bounds on the maximum backlog. In particular, for a 2dimensional nbyn grid, we achieve a backlog of O(n √ ln ln n), improving the O(n ln n) upper bound for general graphs, and coming close to the naive Ω(n) lower bound. Then, in a model of continuous motion of the player and continuous pouring by the adversary, for cups placed at m points in the plane we show that the backlog can be bounded by O(D √ ln ln m), where D is the diameter of the point set. Our methods apply also to higher (fixed) dimensions. We study also the variant of the MBP in which the adversary has a location within the graph and must act locally (filling cups) with respect to his position, just as the player acts locally (emptying cups) with respect to her position. We prove that deciding the value of this game is PSPACEhard.
Maximizing throughput in multiqueue switches
 Algorithmica
, 2004
"... We study a basic problem in MultiQueue switches. A switch connects m input ports to a single output port. Each input port is equipped with an incoming FIFO queue with bounded capacity B. A switch serves its input queues by transmitting packets arriving at these queues, one packet per time unit. Sin ..."
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Cited by 18 (5 self)
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We study a basic problem in MultiQueue switches. A switch connects m input ports to a single output port. Each input port is equipped with an incoming FIFO queue with bounded capacity B. A switch serves its input queues by transmitting packets arriving at these queues, one packet per time unit. Since the arrival rate can be higher than the transmission rate and each queue has limited capacity, packet loss may occur as a result of insufficient queue space. The goal is to maximize the number of transmitted packets. This general scenario models most current networks (e.g., IP networks) which only support a “best effort ” service in which all packet streams are treated equally. A 2competitive algorithm for this problem was designed in [5] for arbitrary B. Recently, a 17 ≈ 1.89competitive algorithm was presented for B> 1 in 9 [3]. Our main result in this paper shows that for B which is not too small our algorithm can do
Scheduling algorithms for providing flexible, ratebased, quality of service guarantees for packetswitching in Banyan networks
 IN PROCEEDINGS OF THE 38TH ANNUAL CONFERENCE ON INFORMATION SCIENCES AND SYSTEMS (CISS
, 2004
"... We consider the problem of providing flexible, ratebased, quality of service guarantees for a particular class of multistage switch networks that includes Banyan networks. We focus on solving a type of online, traffic scheduling problem, whose input at each time step is a set of desired traffic r ..."
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Cited by 11 (0 self)
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We consider the problem of providing flexible, ratebased, quality of service guarantees for a particular class of multistage switch networks that includes Banyan networks. We focus on solving a type of online, traffic scheduling problem, whose input at each time step is a set of desired traffic rates through the switch network. These traffic rates in general cannot be exactly achieved since they treat the incoming data as fluid, that is, they assume arbitrarily small fractions of packets can be transmitted at each time step. The goal of the traffic scheduling problem is to closely approximate the given sequence of traffic rates by a sequence of switch uses throughout the network in which only whole packets are sent. The focus of this paper is bounding the costs incurred in using such an approximation, in terms of the additional buffer size, called backlog, required. Our contributions in this paper apply to a class of multistage
Universal Bounds on Buffer Size for Packetizing Fluid Policies in Input Queued, Crossbar Switches
, 2004
"... In this paper, we consider a type of online, traffic scheduling problem in input queued, crossbar switches. The input to a problem, at each time step, is a set of desired traffic rates. These traffic rates in general cannot be exactly achieved since they assume arbitrarily small fractions of packets ..."
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Cited by 3 (2 self)
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In this paper, we consider a type of online, traffic scheduling problem in input queued, crossbar switches. The input to a problem, at each time step, is a set of desired traffic rates. These traffic rates in general cannot be exactly achieved since they assume arbitrarily small fractions of packets can be transmitted at each time step. The goal of the traffic scheduling problem is to closely approximate the given sequence of traffic rates by a sequence of switch uses in which only whole packets are sent. The focus of this paper is bounding the costs incurred in using such an approximation, in terms of the additional buffer size required.
On centralized smooth scheduling
, 2005
"... Abstract This paper studies evenly distributed sets of natural numbers and their applications to scheduling in a distributed environment. Such sets, called smooth sets, have the property that their quantity within each interval is proportional to the size of the interval, up to a bounded additive d ..."
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Abstract This paper studies evenly distributed sets of natural numbers and their applications to scheduling in a distributed environment. Such sets, called smooth sets, have the property that their quantity within each interval is proportional to the size of the interval, up to a bounded additive deviation; namely, for ρ, ∆ ∈ R a set A of natural numbers is (ρ, ∆)smooth if abs(I · ρ − I ∩ A) < ∆ for any interval I ⊂ N. The current paper studies scheduling persistent clients on a single slotoriented resource in a flexible, predictable and distributed manner. Each client γ has a given rate ρ γ that defines the share of the resource he is entitled to receive and the goal is a smooth schedule in which, for some predefined ∆, each client γ is served in a (ρ γ , ∆)smooth set of slots (natural numbers). The paper focuses on a distributed environment where each client by itself (without any interclient communication) resolves (computes), slot after slot, whether or not it owns this slot. The paper presents extremely efficient schedules under which a client resolves each slot in a constant time. The paper considers two scheduling frameworks. The first one, the Flat Scheduling Framework, is the common problem where the rates of the clients are given a priori. In the second and novel framework, the OpenMarket Scheduling Framework, fractions of the resource are bought and sold by dealers. Each dealer, upon receiving his set of slots, may choose either to become a client and use his share, or to remain a dealer and sell fractions of his share to other dealers. In this framework, the allocation process is highly distributed; moreover, fractions of several resources can be combined into a single virtual resource of new capabilities. The paper presents two scheduling techniques. Both techniques, in both frameworks, produce smooth schedules with highly efficient distributed resolutions a client resolves each slot in O(1) time on a RAM with a moderate number of memory words, all of a small size. Each technique has its pros and cons. For example, one technique utilizes 100% of the resource but its resolution algorithm requires a number of words which is linear in the number of clients; the other technique utilizes only 99% of the resource but its resolution algorithm requires just O(1) words. One of these techniques yields a solution to Tijdeman's Hierarchial Chairman Assignment Problem which outperforms prior solutions. The other technique naturally extends to the problem of scheduling multiple resources, under the restriction that a client may be served concurrently by at most one resource. The extension yields the first solution to this problem having efficient distributed resolution. Prior solutions produce a special type of smooth scheduling called Pfair scheduling, are centralized, and are less efficient than ours.
Optimal Backlog in the Plane
, 2008
"... Suppose that a cup is installed at every point of a planar set P, and that somebody pours water into the cups. The total rate at which the water flows into the cups is 1. A player moves in the plane with unit speed, emptying the cups. At any time, the player sees how much water there is in every cu ..."
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Suppose that a cup is installed at every point of a planar set P, and that somebody pours water into the cups. The total rate at which the water flows into the cups is 1. A player moves in the plane with unit speed, emptying the cups. At any time, the player sees how much water there is in every cup. The player has no information on how the water will be poured into the cups in the future; in particular, the pouring may depend on the player’s motion. The backlog of the player is the maximum amount of water in any cup at any time, and the player’s objective is to minimise the backlog. Let D be the diameter of P. If the water is poured at the rate of 1/2 into the cups at the ends of a diameter, the backlog is Ω(D). We show that there is a strategy for a player that guarantees the backlog of O(D), matching the lower bound up to a multiplicative constant. Note that our guarantee is independent of the number of the cups.
Approximating fluid schedules in packetswitched networks
, 2004
"... Doctor of Philosophy We consider a problem motivated by the desire to provide
exible, ratebased, quality of service guarantees for packets sent over switches and switch networks. Our focus is solving a type of online, trac scheduling problem, whose input at each time step is a set of desired trac ..."
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Doctor of Philosophy We consider a problem motivated by the desire to provide
exible, ratebased, quality of service guarantees for packets sent over switches and switch networks. Our focus is solving a type of online, trac scheduling problem, whose input at each time step is a set of desired trac rates through the switch network. These trac rates in general cannot be exactly achieved since they treat the incoming data as
uid, that is, they assume arbitrarily small fractions of packets can be transmitted at each time step. The goal of the trac scheduling problem is to closely approximate the given sequence of trac rates by a sequence of switch uses throughout the network in which
On Centralized Smooth Scheduling
, 2005
"... This paper studies evenly distributed sets of natural numbers and their applications to scheduling in a centralized environment. Such sets, called smooth sets, have the property that their quantity within each interval is proportional to the size of the interval, up to a bounded additive deviation; ..."
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This paper studies evenly distributed sets of natural numbers and their applications to scheduling in a centralized environment. Such sets, called smooth sets, have the property that their quantity within each interval is proportional to the size of the interval, up to a bounded additive deviation; namely, for ρ, ∆ ∈ R a set A of natural numbers is (ρ,∆)smooth if abs(I  · ρ − I ∩A) < ∆ for any interval I ⊂ N. The current paper studies scheduling persistent clients on a single slotoriented resource in a flexible and predictable manner. Each client γ has a given rate ργ that defines the share of the resource he is entitled to receive and the goal is a smooth schedule in which, for some predefined ∆, each client γ is served in a (ργ,∆)smooth set of slots (natural numbers). The paper considers a centralized environment where a single algorithm computes the user of the current slot. (An accompanying paper studies a distributed environment in which each client by itself computes whether or not it owns the current slot.) An important contribution of this paper is the construction of a smooth schedule with an extremely efficient algorithm that computes the user of each slot in O(log log q) time and O(n) space, where n is the number of clients and
On Centralized Smooth Scheduling Dedicated to the memory of Professor Shimon Even for his inspiration and encouragement
"... Abstract This paper studies evenly distributed sets of natural numbers and their applications to schedulingin a centralized environment. Such sets, called smooth sets, have the property that their quantity within each interval is proportional to the size of the interval, up to a bounded additive dev ..."
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Abstract This paper studies evenly distributed sets of natural numbers and their applications to schedulingin a centralized environment. Such sets, called smooth sets, have the property that their quantity within each interval is proportional to the size of the interval, up to a bounded additive deviation;namely, for!, \Delta 2 R a set A of natural numbers is (!, \Delta)smooth if abs(I  * ! I &quot; A) < \Delta for anyinterval I! N.The current paper studies scheduling persistent clients on a single slotoriented resource in a
Approximating Fluid Schedules in Crossbar PacketSwitches and Banyan Networks
, 2006
"... We consider a problem motivated by the desire to provide flexible, ratebased, quality of service guarantees for packets sent over input queued switches and switch networks. Our focus is solving a type of online traffic scheduling problem, whose input at each time step is a set of desired traffic ra ..."
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We consider a problem motivated by the desire to provide flexible, ratebased, quality of service guarantees for packets sent over input queued switches and switch networks. Our focus is solving a type of online traffic scheduling problem, whose input at each time step is a set of desired traffic rates through the switch network. These traffic rates in general cannot be exactly achieved since they assume arbitrarily small fractions of packets can be transmitted at each time step. The goal of the traffic scheduling problem is to closely approximate the given sequence of traffic rates by a sequence of transmissions in which only whole packets are sent. We prove worstcase bounds on the additional buffer use, which we call backlog, that results from using such an approximation. We first consider the, input queued, crossbar switch. Our main result is an online packetscheduling algorithm using no speedup that guarantees backlog at most @ CIA P R packets at each input port and each output port. Upper bounds on worstcase backlog have been proved for the case of constant fluid schedules, such as the P P CP bound of Chang, Chen, and Huang (INFOCOM, 2000). Our main result for the crossbar switch is the first, to our knowledge, to bound backlog in terms of switch size for arbitrary, timevarying fluid schedules, without using speedup. Our main result for Banyan networks is an exact characterization of the speedup required to maintain bounded backlog, in terms of polytopes derived from the network topology.