M. Mitzenmacher, "The power of two choices in randomized load balancing," IEEE Transactions on Parallel and Distributed Systems, vol. 12, no. 10, pp. 1094--1104, October 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....sizes and in adapting previous arguments to this more general setting. In addition, we provide simulation results demonstrating the load balance that results as the system size scales into the millions. 1 Introduction A well known paradigm for balancing load is the power of two choices [1, 9, 10, 15], whereby an item is stored at the less loaded of two (or more) random alternatives, which we refer to variously as bins and servers. These methods are used in standard hashing with chaining to reduce the maximum number of items, or load, in a bin with high probability. Using these methods, two or ....

MITZENMACHER, M. The power of two choices in randomized load balancing. Ph.D. thesis, U.C. Berkeley, 1996.

....The main idea is simply stated: Basing decisions upon a few random samples of a large state space is often a good surrogate for making decisions with complete knowledge of the state. See [18] for a general exposition of randomized algorithms, 24] 8] for application to switching, and [17], 20] for other applications to networking. Organization of the paper The rest of the paper exploits the above observations and proposes some new algorithms and proof techniques. The results are divided into two parts: Section II deals with throughput and Section III deals with delay. Section ....

Mitzenmacher M., "The power of two choices in randomized load balancing ", Ph.D. thesis, University of California at Berkeley, 1996

....high probability and the virtual node scheme suggested is not applicable since it requires an additional factor of 52 n) connectivity. To remedy this, schemes such as load balancing by allowing two or more hashed locations can bring the maximum load down to (log log n) with high probability [10]. The first outgoing edge of each node is to its closest neighbor on the circle when moving clock wise along the circle (increasing hashes in our implementation) These edges form a ring of all the nodes of the network. One should note that one cannot do any better using only one outgoing edge ....

MITZENMACHER, M. The Power of Two Choices in Randomized Load Balancing. IEEE Transactions on Parallel and Distributed Systems 12, 10 (2001), 1094--1104.

....with high probability and the virtual node scheme suggested is not applicable since it requires an additional factor of## ## connectivity. To remedy this, schemes such as load balancing by allowing two or more hashed locations can bring the maximum load down to ##### ### ## with high probability [10]. The first outgoing edge of each node is to its closest neighbor on the circle when moving clock wise along the circle (increasing hashes in our implementation) These edges form a ring of all the nodes of the network. One should note that one cannot do any better using only one outgoing edge ....

MITZENMACHER, M. The Power of Two Choices in Randomized Load Balancing. IEEE Transactions on Parallel and Distributed Systems 12, 10 (2001), 1094--1104.

....system to determine to which server to allocate a request. Many studies have focused on the strategy of using a subset of the load information available. This involves first randomly choosing a small number, k, of homogeneous servers and then choosing the least loaded server from within that set [Mit96] ELZ86] VDK96] ABKU94] KLH92] In particular, for homogeneous systems, Mitzenmacher [Mit96] studies the tradeoffs of various choices of k and various degrees of staleness of load information reported. As the degree of staleness increases, smaller values of k are preferable. Genova et al. ....

....of using a subset of the load information available. This involves first randomly choosing a small number, k, of homogeneous servers and then choosing the least loaded server from within that set [Mit96] ELZ86] VDK96] ABKU94] KLH92] In particular, for homogeneous systems, Mitzenmacher [Mit96] studies the tradeoffs of various choices of k and various degrees of staleness of load information reported. As the degree of staleness increases, smaller values of k are preferable. Genova et al. GC00] propose an algorithm, which we call Inv Load that first randomly selects k servers. The ....

M. Mitzenmacher. The Power of Two Choices in Randomized Load Balancing. PhD thesis, UC Berkeley, September 1996.

....[1] also examined several related problems, including a closed dynamic model where at each step a random ball is deleted and re inserted into the system. This result was generalized to natural queueing theoretic models independently by Vvedenskaya, Dobrushin, and Karpelevich [14] and Mitzenmacher [9, 10]. See also the work by Eager, Lazowska, and Zahorjan [2] The standard model is as follows. Suppose that tasks arrive at a bank of n First In First Out processors as a Poisson process of rate n, where 1. Note the arrival rate per processor is a xed constant. Tasks require an ....

.... That these di erential equations accurately describe the behavior of the process follows from the framework established by Kurtz [3, 4, 5] Here we will just assume that the intuitive di erential equations are the proper uid limit; further details and similar results can be found in for example [9, 12, 14]. For the static load balancing problem, we are most interested in the case of n balls and n bins. For the uid limit model, this corresponds to the time t = 1. As the w i are increasing, we can bound their behavior using simple manipulation: w i 1 (t) w i (1) w i 1 (t) ....

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M. Mitzenmacher, The Power of Two Choices in Randomized Load Balancing. Ph.D. thesis, University of California, Berkeley, September 1996.

....while three is not too much di erent from two. Besides improving the maximum load, using two hash functions in this way leads to a more equal distribution of the load across buckets. A numerical analysis of this hashing process is given in [11] and extensions to queueing models are presented in [10, 9, 16]. The hashing scheme we examine here is a variation of the d random scheme, with better performance and characteristics that make it more suitable for the IP lookup problem. It was rst introduced and analyzed theoretically by V ocking [15] a simpler analysis more relevant to our discussion was ....

....can a ect its probability of being deleted, can be handled using the analysis of [15] although this approach does not give the numerical answers we desire here. The model where a random bucket is chosen and an item is deleted from that bucket can also be handled using these techniques, however [9]. We modify the equation (1) to account for deletions by noting that the total number of balls is i 0 i(x 2i x 2i 1 ) and the number of balls that can be deleted that cause a reduction in x i is b c(x i x i 2 ) Hence the equations that describe the behavior of the system are given by ....

M. Mitzenmacher. The Power of Two Choices in Randomized Load Balancing. Ph.D. thesis, University of California, Berkeley, September 1996.

....parallelize and could therefore even be used for very large systems with thousands of disks. A well known technique, shortest queue, maintains a FIFO queue of committed requests. A newly arrived request fi; jg that could be served on two disks i and j is committed to the disk with shortest queue [6, 31, 23]. We show that executing the requests locally in FIFO order is indeed optimal. However, immediately committing a request when it arrives is unnecessary. Perhaps the simplest more flexible strategy, lazy, puts a request fi; jg in the queues of both disks i and j. When a disk i finishes a request, ....

....performs much better than mirroring yet is still significantly worse than RDA which chooses both destinations randomly. Even the simplest scheduling heuristics for RDA are quite difficult to treat analytically for asynchronous request arrivals and small e. Vvedenskaya et al. 31] and Mitzenmacher [23] analyze the shortest queue heuristics as D for Poisson arrivals with fixed arrival rate l = D= 1 e) and exponentially distributed service times. They give different kinds of evidence that this limit result transfers to the finite case. However, experiments indicate [23] that the agreement ....

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M. Mitzenmacher. The Power of Two Choices in Randomized Load Balancing. PhD thesis, UC Berkeley, 1996.

....n) m = n log n O( m n ) m = O(n log n) 8) 2.1 Two bins at once Suppose we put balls into bins by the following steps: 1. Choose 2 bins at random. 2. Put the ball into the less full bin. Then the max load is O(log log n) for n balls and n bins and m n O(log log n) for m balls and n bins [1]. Moreover, if we choose d bins at random rather than 2 bins, the max load is O( log log n log d ) if we choose d bins at random and always go left when there is a tie, then the max load is O( log log n d ) 1] Theorem 2 Throw n 8 balls into n bins by choosing 2 bins uniformly at ....

.... log n) for n balls and n bins and m n O(log log n) for m balls and n bins [1] Moreover, if we choose d bins at random rather than 2 bins, the max load is O( log log n log d ) if we choose d bins at random and always go left when there is a tie, then the max load is O( log log n d ) [1]. Theorem 2 Throw n 8 balls into n bins by choosing 2 bins uniformly at random and picking the less loaded bin. The max load is O(log log n) We can formulate the lling process as a graph G = V; E) where E = fe i = u; v) ball i chooses bins u and vg and V = f1; ng. Claim 3 ....

M.D. Mitzenmacher, \The Power of Two Choices in Randomized Load Balancing", Ph.D. Thesis, EECS Department, UC Berkeley, 1996.

....is to avoid the excessive amount of coordination required in a centralized system, some coordination is obviously necessary to ensure that load is properly balanced. To illustrate some of the considerations that go into designing a distributed system, we introduce the supermarket model used in [11, 29], etc. Suppose we have a collection of n servers all processing customers at the same rate (e.g. no express lanes for customers with few items) Each server has its own queue that customers must wait in before they are serviced and can leave the checkout area. Furthermore, assume that customers ....

....tasks are running up waiting time while only one gets serviced. If service times are less variable, then there are fewer big tasks which hold up all the tasks behind them. We address this issue in more detail in Chapter 2. The distribution on which queuing research predominantly focuses on (e.g. [11, 25, 26, 28, 29, 33]) is the exponential distribution. More recently, research has shown that many real service time distributions actually have much larger variance than the exponential distribution. In particular, many observed service distributions have been showed to fit a heavy tailed distribution. 4 In ....

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M. Mitzenmacher. The Power of Two Choices in Randomized Load Balancing. PhD thesis, University of California at Berkeley, September 1996.

....nodes. The mapping of this problem to our parallel disk subsystem is clear. Disk requests are the tasks and random disk block allocation with random replication is equivalent to randomly probing for the least loaded server. The most recent work related to this problem is that of Mitzenmacher [11] which has a detailed analysis of several variations of a multi server queuing model in which customers can randomly probe d servers and join the one with the shortest queue. The results show that an exponential improvement in the mean waiting time of customers is obtained in the system when ....

....only have two choices. Our experimental results confirm that replicating data on multiple disks can significantly reduce the variation of individual disks loads, providing a more balanced and efficient system. However there are significant differences between our system and the models analyzed in [11] with respect to service time distributions, arrival processes, etc. Since we require accuracy on the tail of the response time distribution we resorted to simulation and measurement. Models, however, are useful for configuration planning and a first order approximation of the delay bounds that ....

M.D. Mitzenmacher, "The Power of Two Choices in Randomized Load Balancing", PhD Dissertation, University of California at Berkeley, Computer Science Department, 1996.

....given the same random distribution of blocks from Figure 1. # Figure 4: The same distribution of Figure 1 but with 20 inter node replication The problem of routing requests to the least loaded node for load balancing has been studied in the context of distributed systems [12, 13] In [14], it is shown that most of the improvement in load balancing occurs when there are exactly two choices. Although our system provides two types of block replication, a block is only replicated using either intra node or inter node replication, and the percentages of total replicated blocks of each ....

M. D. Mitzenmacher, "The Power of Two Choices in Randomized Load Balancing", Ph.D. Dissertation, University of California at Berkeley, Computer Science Department, 1996.

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M. Mitzenmacher. The power of two choices in randomized load balancing. Ph. D. thesis, U.C. Berkeley, 1996.

....sizes and in adapting previous arguments to this more general setting. In addition, we provide simulation results demonstrating the load balance that results as the system size scales into the millions. 1 Introduction A well known paradigm for balancing load is the power of two choices [1, 9, 10, 15], whereby an item is stored at the less loaded of two (or more) random alternatives, which we refer to variously as bins and servers. These methods are used in standard hashing with chaining to reduce the maximum number of items, or load, in a bin with high probability. Using these methods, two or ....

....At the boundary of the theoretical and the practical, it would be an improvement if the theory could be used to accurately predict the resulting load distribution. In the case of uniform bin sizes, this can be done quite well using methods based on di erential equations pioneered by Mitzenmacher [9]. While not as accurate as di erential equaitons, the witness tree approach, as demonstrated by V ocking [15] gives a somewhat tighter analysis than the original argument of Azar et al. It is not clear whether either of these methods can be made to apply to this setting, but perhaps some other ....

Mitzenmacher, M. The power of two choices in randomized load balancing. Ph.D. thesis, U.C. Berkeley, 1996.

....d left policy. Note that when d = 2, the maximum load for the 2 left policy is #( matching our policy with one random choice and one choice from memory. The 2 left policy, however, requires more randomness resources, as well as an a priori agreement on the grouping of b ns. Mitzenmacher [7] develops an approach using the theory of large deviations for studying these loadb alancing prob lemsb y deriving di#erential equations for the limitingb ehavior as n grows large. This methodology is also useful for studying dynamic, queueing versions of the prob lem, where theb alls correspond ....

....to queues. Arrivals occur according to a rate n# (# 1) Poisson process at ab ank of n independent rate i exponential servers. We will assume that i = n, so that the net service rate is larger than the net arrival rate. We refer to this general queueing setup as the model, following [7]. As in theb alls andb ins prob lem, for each arrival d queues are chosen independently and uniformly at random, and m queues are stored in memory. The arrival is placed in the least loaded of the d m queues, and the m least loaded of the d m queues are kept in memory. The case of no memory (m ....

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M. Mitzenmacher. The Power of Two Choices in Randomized Load Balancing, PhD thesis. University of California, Berkeley, 996. Journal article appears as: The Power of Two Choices in

....also examined several related problems, including a closed dynamic model where at each step a random ball is deleted and re inserted into the system. This result was generalized to natural queueing theoretic models independently by by Vvedenskaya, Dobrushin, and Karpelevich [14] and Mitzenmacher [9, 10]. See also the work by Eager, Lazowska, and Zahorjan [2] The standard model is as follows. Suppose that tasks arrive at a bank of n First In First Out processors as a Poisson process of rate Harvard University, Computer Science Department. 29 Oxford St. Cambridge, MA 02138. email: ....

.... these differential equations accurately describe the behavior of the process follows from the framework established by Kurtz [3, 4, 5] Here we will just assume that the intuitive differential equations are the proper fluid limit; further details and similar results can be found in for example [9, 12, 14]. For the static load balancing problem, we are most interested in the case of n balls and n bins. For the fluid limit model, this corresponds to the time t = 1. As the w i are increasing, we can bound their behavior using simple manipulation: w i Gamma1 (t) w i (1) w i Gamma1 ....

[Article contains additional citation context not shown here]

M. Mitzenmacher, "The Power of Two Choices in Randomized Load Balancing", Ph.D. thesis, University of California, Berkeley, September 1996.

No context found.

M. Mitzenmacher, "The power of two choices in randomized load balancing," IEEE Transactions on Parallel and Distributed Systems, vol. 12, no. 10, pp. 1094--1104, October 2001.

No context found.

M. Mitzenmacher, The power of two choices in randomized load balancing, Ph.D. Thesis, University of California, Berkeley, August 1996.

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M. Mitzenmacher. The power of two choices in randomized load balancing. IEEE Transactions on Parallel and Distributed Systems, 12(10), 2001.

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M. Mitzenmacher. The power of two choices in randomized load balancing. IEEE Transactions on Parallel and Distributed Systems, 12(10), 2001.

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M. Mitzenmacher. The power of two choices in randomized load balancing. IEEE Transactions on Parallel and Distributed Systems, 12(10), 2001.

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M. Mitzenmacher. The Power of Two Choices in Randomized Load Balancing. PhD thesis, University of California, Berkeley, August 1996.

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M. Mitzenmacher, "The Power of Two Choices in Randomized Load Balancing," Ph.D. thesis, 1996.

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M. Mitzenmacher. The Power of Two Choices in Randomized Load Balancing. PhD thesis, University of California, Berkeley, August 1996.

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M. Mitzenmacher. The Power of Two Choices in Randomized Load Balancing. PhD thesis, University of California, Berkeley, August 1996.

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