We develop a framework that allows us to address the issues of admission control and routing in high-speed networks under the restriction that once a call is admitted and routed, it has to proceed to completion and no reroutings are allowed. The "no rerouting" restriction appears in all the proposals for future high-speed networks and stems from current hardware limitations, in particular the fact that the bandwidth-delay product of the newly developed optical communication links far exceeds the buffer capacity of the network. In case the goal is to maximize the throughput, our framework yields an on-line O(lognT )- competitive strategy, where n is the number of nodes in the network and T is the maximum call duration. In other words, our strategy results in throughput that is within O(log nT ) factor of the highest possible throughput achievable by an omniscient algorithm that knows all of the requests in advance. Moreover, we show that no on-line strategy can achieve a better competit...

Suppose that n balls are placed into n bins, each ball being placed into a bin chosen independently and uniformly at random. Then, with high probability, the maximum load in any bin is approximately log n log log n . Suppose instead that each ball is placed sequentially into the least full of d bins chosen independently and uniformly at random. It has recently been shown that the maximum load is then only log log n log d +O(1) with high probability. Thus giving each ball two choices instead of just one leads to an exponential improvement in the maximum load. This result demonstrates the power of two choices, and it has several applications to load balancing in distributed systems. In this thesis, we expand upon this result by examining related models and by developing techniques for stu...

Consider the on-line problem where a number of servers are ready to provide service to a set of customers. Each customer's job can be handled by any of a subset of the servers. Customers arrive one-by-one and the problem is to assign each customer to an appropriate server in a manner that will balance the load on the servers. This problem can be modeled in a natural way by a bipartite graph where the vertices of one side (customers) appear one at a time and the vertices of the other side (servers) are known in advance. We derive tight bounds on the competitive ratio in both deterministic and randomized cases. Let n denote the number of servers. In the deterministic case we provide an on-line algorithm that achieves a competitive ratio of k = dlog 2 ne (up to an additive 1) and prove that this is the best competitive ratio that can be achieved by any deterministic on-line algorithm. In a similar way we prove that the competitive ratio for the randomized case is k 0 = ln(n) (up to an a...

Classical routing and admission control strategies achieve provably good performance by relying on an assumption that the virtual circuits arrival pattern can be described by some a priori known probabilistic model. Recently a new online routing framework, based on the notion of competitive analysis, was proposed. This framework is geared towards design of strategies that have provably good performance even in the case where there are no statistical assumptions on the arrival pattern and parameters of the virtual circuits. The online strategies motivated by this framework are quite different from the min-hop and reservation-based strategies. This paper surveys the online routing framework, the proposed routing and admission control strategies, and discusses some of the implementation issues. Research supported by NSF CCR-9304971, ARO DAAH04-95-1-0121, and by Terman Fellowship. E-Mail: plotkin@cs.stanford.edu, URL: http://theory.stanford.edu/people/plotkin/plotkin.html. 1 Introducti...

We study a classical problem in online scheduling. A sequence of jobs must be scheduled on m identical parallel machines. As each job arrives, its processing time is known. The goal is to minimize the makespan. Bartal, Fiat, Karloff and Vohra [3] gave a deterministic online algorithm that is 1.986-competitive. Karger, Phillips and Torng [11] generalized the algorithm and proved an upper bound of 1.945. The best lower bound currently known on the competitive ratio that can be achieved by deterministic online algorithms it equal to 1.837. In this paper we present an improved deterministic online scheduling algorithm that is 1.923-competitive, for all m 2. The algorithm is based on a new scheduling strategy, i.e., it is not a generalization of the approach by Bartal et al. Also, the algorithm has a simple structure. Furthermore, we develop a better lower bound. We prove that, for general m, no deterministic online scheduling algorithm can be better than 1.852-competitive.

In this paper we present a strategy to route unknown duration virtual circuits in a highspeed communication network. Previous work on virtual circuit routing concentrated on the case where the call duration is known in advance. We show that by allowing O(log n) reroutes per call, we can achieve O(log n) competitive ratio with respect to the maximum load (congestion) for the unknown duration case, were n is the number of nodes in the network. This is in contrast to the ( 4p n)lower bound on the competitive ratio for this case if no rerouting is allowed [3]. Our routing algorithm can be also applied in the context of machine load balancing of tasks with unknown duration. We present an algorithm that makes O(log n) reassignments per task and achieves O(log n) competitive ratio with respect to the load, where n is the number of parallel machines. For a special case of unit load tasks we design a constant competitive algorithm. The previously known algorithms that achieve up to polylogarithmic competitive ratio for load balancing of tasks with unknown duration dealt only with special cases of related machines case and unit-load tasks with restricted assignment[4,11].

This paper considers the non-preemptive on-line load balancing problem where tasks have limited duration in time. Upon arrival, each task has to be immediately assigned to one of the machines, increasing the load on this machine for the duration of the task by an amount that depends on both the machine and the task. The goal is to minimize the maximum load. Azar, Broder and Karlin studied the unknown duration case where the duration of a task is not known upon its arrival [4]. They focused on the special case in which for each task there is a subset of machines capable of executing it, and the increase in load due to assigning the task to one of these machines depends only on the task and not on the machine. For this case, they showed an O(n 2=3 )-competitive algorithm, and an \Omega\Gamma p n) lower bound on the competitive ratio, where n is the number of the machines. This paper closes the gap by giving an O( p n)-competitive algorithm. In addition, trying to overco...

In this paper we study the problem of on-line allocation of routes to virtual circuits (both point-topoint and multicast) where the goal is to minimize the required bandwidth. We concentrate on the case of permanent virtual circuits (i.e., once a circuit is established, it exists forever), and describe an algorithm that achieves an O(log n) competitive ratio with respect to maximum congestion, where n is the number of nodes in the network. Informally, our results show that instead of knowing all of the future requests, it is sufficient to increase the bandwidth of the communication links by an O(log n) factor. We also show that this result is tight, i.e. for any on-line algorithm there exists a scenario in which O(log n) increase in bandwidth is necessary. We view virtual circuit routing as a generalization of an on-line load balancing problem, defined as follows: jobs arrive on line and each job must be assigned to one of the machines immediately upon arrival. Assigning a job to a machine increases this machine’s load by an amount that depends both on the job and on the machine. The goal is to minimize the maximum load. For the related machines case, we describe the first algorithm that achieves constant competitive ratio. For the unrelated case (with n machines), we describe a new method that yields O(log n)-competitive

One of the oldest and simplest variants of multiprocessor scheduling is the on-line scheduling problem studied by Graham in 1966. In this problem, the jobs arrive on-line and must be scheduled non-preemptively on m identical machines so as to minimize the makespan. The size of a job is known on arrival. Graham proved that the List Processing Algorithm which assigns each job to the currently least loaded machine has competitive ratio (2 \Gamma 1=m). Recently algorithms with smaller competitive ratios than List Processing have been discovered, culminating in Bartal, Fiat, Karloff, and Vohra's construction of an algorithm with competitive ratio bounded away from 2. Their algorithm has a competitive ratio of at most (2 \Gamma 1=70) 1:986 for all m; hence for m ? 70, their algorithm is provably better than List Processing. We present a more natural algorithm that outperforms List Processing for any m 6 and has a competitive ratio of at most 1:945 for all m, which is significantly closer ...

We investigate balls-into-bins processes allocating m balls into n bins based on the multiple-choice paradigm. In the classical single-choice variant each ball is placed into a bin selected uniformly at random. In a multiple-choice process each ball can be placed into one out of d ≥ 2 randomly selected bins. It is known that in many scenarios having more than one choice for each ball can improve the load balance significantly. Formal analyses of this phenomenon prior to this work considered mostly the lightly loaded case, that is, when m ≈ n. In this paper we present the first tight analysis in the heavily loaded case, that is, when m ≫ n rather than m ≈ n. The best previously known results for the multiple-choice processes in the heavily loaded case were obtained using majorization by the single-choice process. This yields an upper bound of the maximum load of bins of m/n + O ( √ m ln n/n) with high probability. We show, however, that the multiple-choice processes are fundamentally different from the single-choice variant in that they have “short memory. ” The great consequence of this property is that the deviation of the multiple-choice processes from the optimal allocation (that is, the allocation in which each bin has either ⌊m/n ⌋ or ⌈m/n ⌉ balls) does not increase with the number of balls as in the case of the single-choice process. In particular, we investigate the allocation obtained by two different multiple-choice allocation schemes,