WO2000031640A2 - Apparatus for and method of non-linear constraint optimization in storage system configuration - Google Patents
Apparatus for and method of non-linear constraint optimization in storage system configuration Download PDFInfo
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- WO2000031640A2 WO2000031640A2 PCT/US1999/027383 US9927383W WO0031640A2 WO 2000031640 A2 WO2000031640 A2 WO 2000031640A2 US 9927383 W US9927383 W US 9927383W WO 0031640 A2 WO0031640 A2 WO 0031640A2
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- workload
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F9/00—Arrangements for program control, e.g. control units
- G06F9/06—Arrangements for program control, e.g. control units using stored programs, i.e. using an internal store of processing equipment to receive or retain programs
- G06F9/46—Multiprogramming arrangements
- G06F9/50—Allocation of resources, e.g. of the central processing unit [CPU]
- G06F9/5005—Allocation of resources, e.g. of the central processing unit [CPU] to service a request
- G06F9/5011—Allocation of resources, e.g. of the central processing unit [CPU] to service a request the resources being hardware resources other than CPUs, Servers and Terminals
- G06F9/5016—Allocation of resources, e.g. of the central processing unit [CPU] to service a request the resources being hardware resources other than CPUs, Servers and Terminals the resource being the memory
-
- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y10—TECHNICAL SUBJECTS COVERED BY FORMER USPC
- Y10S—TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y10S707/00—Data processing: database and file management or data structures
- Y10S707/99951—File or database maintenance
- Y10S707/99956—File allocation
Definitions
- the present invention relates generally to computer storage systems and pertains more carncuiariy to an apparatus for and a method of non-linear constraint optimization in a storage system configuration.
- Storage systems for computer networks can contain a large number of storage devices having a wide variety of characteristics and a nearly arbitrary interconnection scheme.
- the configuration and management of the storage system is central to the functioning of the computer network.
- the inherent difficulties in configuring and managing the storage system are compounded by the sheer scale of the network. The situation has reached the point where the time needed to configure a new storage system can be several months and the cost of managing the storage system can be several times the purchase cost.
- an objective function may be to ⁇ unimize the cost of the storage system.
- An objective function may be to maximize the performance of the storage system.
- Other objective functions include balancing the load, maximizing the availability. and minimizing the physical footprint.
- a specific piece of data is referred to as a workload unit.
- a set of standards mclude both the workload unit characteristics and the application program access characteristics.
- a standard may be the size of the workload unit.
- a standard may be the access speed or the access frequency of the application program.
- Other standards include request size, request rate, run count, phasing behavior, on time, off " time, and maximum amount of data loss.
- Characteristics include both performance measures and physical descriptions.
- a characteristic may be the quantity of storage available on or the access speed of the storage device.
- a characteristic may be the size or the weight of the storage device.
- the storage allocation problem can be ⁇ iewed on at least two leveis. First. wh ⁇ ther a particular workload unit can be assigned to a particular storage device, that is. whether the constraints are met. Second, whether a particular workload unit should be assigned to a particular storage device given the resulting overall assignment plan, thai is. whether the objective function is optimized.
- Tne objective function is to maximize the value of a set of items placed into the knapsack given the constraint that the sum of the sizes of the set of items in the knapsack must be less then or equal to the capacity of the knapsack.
- the integer knapsack problem there is only one objective function and one constraint.
- the multidimensional Imapsack problem which takes into account multiple capacity dimensions.
- One computer specific optimization problen similar to the storage allocation problem is the standard file allocation problem.
- the objective function is to minimize the file transmission costs of running the tasks given the constr. int that the sum of the sizes of a set of files on a node must be less then or equal to the capacity of the node.
- the standard file allocation problem there is only one objective function. Altematively, there are a number of variants of the standard file allocation pioblem. Even so, all of the file allocation problems differ from the storage allocation problem in that they assume a fixed number of nodes.
- an objective function to choose the best set of storage devices may require that storage devices be added or that storage devices remain unused. As a result, solutions to the file allocation problems cannot be used directly to solve the storage allocation problem.
- the objective function for a storage system is determined, the workload units are selected and their standards are determined, and the storage devices are selected and their characteristics are determined. Tnese selections and determinations are then used by a constraint based solver through non-linear constraint integer optimization to generate an assignment plan for the workload units to the storage devices.
- FIG. i is a block diagram of a computer storage constraint optimization system.
- FIG. 2 is a flow diagram of a computer storage constraint optimization method.
- Tne purpose of the present invention is to provide a formalization of the storage allocation problem and a collection of a range of solutions.
- the storage allocation problem is referred to as being NP-hard. which means that there is currently no known way of finding a provably optimal solution without checking every possible solution. Generally this is not practical or necessary.
- the present invention discloses an apparatus and a method that reach a good, generally acceptable, approximate solution in a relatively short amount of time.
- the present invention formalizes the storage allocation problem as a non-linear constraint integer programming problem. In this way one can apply results from a wide body of research on integer programming and optimization. This body of research gives one insight into such problems as well as a large body of heuristics that can give good, though not provably optimal, solutions to the problem.
- a constraint C is satisfied for a set of workload units and a storage device d if the function C ,(W.c ' . is true. It may be said that an assignment of a subset of workload units d to a storage dt vice d is feasible if. for all constraints C, in the set of constraints C. the function C.(W c . ⁇ ) is true.
- t he optimaiity of the resulting overall assignment plan is evaluated based on an objective function, such as Of .w.d). that represents the value of assigning a workload unit to a storage device given that a subset of the workload units is already assigned to the storage device. At least initially, the subset of workload units may be a null set. Typically, a bigger objective function value implies a better assignment. Given this fo ⁇ nalization. r is now possible to design a system to solve the storage allocation problem.
- FIG. 1 a block diagram of a computer storage constraint optimization system 10. according to the present invention, is shown.
- S/stem 10 includes a constraint based solver 12.
- Solver 12 takes as an input both workload unit standards 14 from a plurality of workload units (not shown) and storage device charac teristics 16 from a plurality of storage devices (not shown) (the standards 14 and the cha ⁇ cteristics 16 are referred to collectively as constraints) and generates as an output an asshmment plan 18.
- Solver 12 accomplishes this through the operation of a solver algo ⁇ thm 20 and a performance model 22 both of which are governed by at least one objectr 'e function 24.
- Solver algorithm 20 serves the functions of determining where to start, where to proceed with, and where to stop the assignment process.
- Performance model 2 serves the functions of determining if an assignment can be made and what resources are left if such an assignment is made.
- Objective function 24 guides the assignment process _nd evaluates the result. Recall from above that there may be multiple and competing objective functions
- system 10 shown may be implemented using software, hardware, or a dedicated signal processor (DSP).
- DSP dedicated signal processor
- FIG. 2. a flow diagram of a computer storage constraint optimization method is shown.
- the method assigns a plurality of workload units to a plurality of storage devices and begins with step 30.
- At step 30. at least one objective function is determined.
- Next, at step 32 one or more of the plurality of workload units is selected.
- Tnen. at step 34 the standards of the selected workload units are determined.
- At ⁇ tcD 36. one or more of the plurality of storage devices is selected.
- Tnen. at step 38 the characteristics of the selected storage devices are determined. .An assignment plan is generated for the workload units to the storage devices, in step 40.
- the assignment plan is evaluated either against some criteria such as the objective function or against some alternative plan.
- the assignment plan is implemented if a:, proved.
- One of ordinary skill in the art will realize that additional steps may be added to the method or that one or more of the steps may be repeated as necessary to generate an approved assignment plan.
- a range of solutions is possible depending on what optimization heuristic is used to generate the assignment plan at step 40.
- a collection of heuristics will be presented below. Tney are generally organized into two groups. First, a set of well known heuristics are adapted to the storage allocation problem as formalized above. These are referred to here as the first-fit heuristic and its variants. One of ordinary skill in the art will readily identify other well known heuristics, such as simulated annealing, genetic algorithms, and linear relaxation, that could be adapted without departing from the spirit of the invention. In the interest of brevity, the first-fit heuristics are presented as examples of the plethora of well known heuristics that could be used. Second, a set of new heuristics, referred to here as the gradient based heuristic and its variants, will be disclosed. No one optimization heuristic is preferred above all the others. The choice depends on what constraints and objective functions are being considered.
- the first collection of heuristics is based on the first-fit heuristic.
- the essence of the "simple" heuristic is to systematically assign each workload unit to the first storage device for which all of the constraints are satisfied, that is. the first device that it "fits" on.
- n workload units ⁇ w,. w ; ....w tone ⁇ .
- D ⁇ d ,. d ,....d m ⁇ .
- workload unit w ⁇ w,. d ,....d m ⁇ .
- a second variant to the simple first-fit heuristic is to son both the workload units and the storage devices before performing the heuristic. Carefully sorting the order of both the workload units and the storage devices increases the odds that a good assignment plan will be found. For example, sorting the workload units from largest size to smallest size tends to increase the chances that later workload units will fit into the space left over in the storage devices by the earlier assignments.
- a third variant to the simple first-fit heuristic is to perform the heuristic using a more round robin approach to which of the storage devices is checked first. Instead of always first checking the first storage device d, with each new workload unit, the storage device that follows the storage device that received the previous assignment is checked first the next time. For example, assume that in the usual manner the first workload unit w, is assigned to a storage device d .. Then, for the next workload unit w ,. one first checks the next storage device d._, and works back to d by wrapping around from the last storage devic ⁇ d. to the first d ,. Again, this process is repeated until all of the workload demonstr s have been addressed or all of the storage devices are full. Implicit in this variant is that it acts to evenly distribute the workload units over the storage devices used in the assig. lment plan. This third variant could also be used in combination with either of the previous two variants.
- Tne second collection of heuristics originates from a gradient based heuristic.
- Tne first-fit heuristics produce assignment plans that satisfy" the basic constraints, but thev do not take into account the interplay between the different dimensions represented by _he constraints.
- the gradient based heuristics combine the inequality constraints in different dimensions to formulate the assignment plan.
- Such a constraint can be said to be consumable because one can determine how much the addition of a workload unit to a storage device. with some workload units already assigned to i will consume of that resource.
- the calculation of how much resource is consumed by the addition of workload unit w is trivial for a linear constraint, where the amount consumed is equal to a,w. That is, the amount consumed can be determined by looking at workload unit w in isolation.
- the amount of resources consumed by the addition of workload unit w will depend on what subset of workload units W are already assigned to the storage device.
- one begins by calculating a set of first gradient functions for all of the workload units and all of the storage devices. Then, one determines which first gradient function has the greatest value and assigns the corresponding workload unit to the corresponding storage device. Next, one calculates a set of new gradient functions for the remaining workload units and all of the storage devices. Then, one determines which new gradient function has the greatest value and assigns the corresponding workload unit to the corresponding storage device. This process is repeated until ail of the workload units have been addressed or all of the storage devices are full.
- the key to the gradient based heuristic is a non-negative gradient function G(W.w.d). that represents the value of assigning a workload unit v to a storage device d given that a subset of the workload units W is already assigned to the storage device. At least initially, the subset of workload units may be a null set.
- Tne gradient function is equal to zero if the assignment is infeasible. Typically, a bigger gradient function value implies a better assignment A number of gradient functions are available.
- a first gradient function according to the present invention is referred to as the cosine-gradient function.
- the theory is to represent the storage device as a point in vector space with dimensions thai correspond to the remaining capacity of each of its consumable constraints. With respect to a particular storage device and the corresponding set of workload units assigned to it a new workload unit can also be represented as a point in this vector space by plotting the amount of each consumable constraint that the workload unit would consume if it were to be assigned to the storage device. Ideally, one would like to find a set of workload units that fully utilizes all of the consumable constraints. One way that one can accomplish this is to assign workload units with opposite needs.
- the gradient function to be the cosine of the angle of incidence between the vector representing how much of each consumable constraint the workload unit would use if it were placed on the storage device. and the vector representing how much of each consumable constraint the storage device has available, then one obtains a gradient between zero and one such that larger gradient values. corresponding to smaller angles, imply be ⁇ er fits. So. the cosine-gradient function can be written as
- a second gradient function according to the present invention is referred to as the normalized-cosine-gradient function.
- the normalized-cosine-gradient function can be written as
- a third gradient function is an adaptation of the gradient function presented by Toyoda but extended to multiple "knapsacks" or storage devices and to consumable constraints.
- Each element b, of the vector B is the amount of a consumable resource / used by the subset of workload units W previously assigned to a particular storage device d divided by the total amount of that resource that that storage device started with.
- Each element f, of the vector F is the amount of a consumable resource / used by assigning a workload unit w to a particular storage device d divided by the total amount of that resource that that storage device started with. So. the Toyoda gradient function can be written as - 1 >-
- an improved range of solutions is possible through the application of one or more obit-tive functions.
- Any particular goal can be achieved by the computer storage constraint optimization method through the application of an objective function. O(W.w.d ..
- the objective function might be the expect ⁇ d fraction of unused bandwidth if w ikload units W and w were assigned to storage device d.
- the composite gradient function can be written as
- vvh ⁇ r _ G( W.w.d) is either the cosin ⁇ -gradient function or the normalized-cosine-gradient funct'on outlined above.
- any objective function can be composited with the gradient functions to weight them for a particular goal as desired.
- Toyoda gradient function An extension of the Toyoda gradient function is to composite it with the objective function of supporting the greatest number of workload units on the cheapest set of storage devices. Tnen. the composite gradient function can be written as
- wher ⁇ maxcost is the maximum cost of any storage device in the set of storage devices. In most cases, all of the workload units have an equal weight so for all w. ben ⁇ fit(w) is equal to one. This gradient function has produced quite good assignment plans of workloads onto a low cost set of devices.
- This gradient function has also given good assignment plans, especially in the case where the workload units have quite disparate sizes.
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EP99963927A EP1145127A2 (en) | 1998-11-20 | 1999-11-18 | Apparatus for and method of non-linear constraint optimization in storage system configuration |
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US09/197,114 | 1998-11-20 | ||
US09/197,114 US6366931B1 (en) | 1998-11-20 | 1998-11-20 | Apparatus for and method of non-linear constraint optimization in storage system configuration |
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WO2000031640A3 WO2000031640A3 (en) | 2001-05-10 |
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EP1370950A4 (en) * | 2001-02-13 | 2011-01-05 | Candera Inc | System and method for policy based storage provisioning and management |
US7664858B2 (en) | 2003-11-21 | 2010-02-16 | Sap Ag | Method for balancing load between processors in a multi-processor environment |
WO2008142005A1 (en) * | 2007-05-17 | 2008-11-27 | International Business Machines Corporation | Scalable performance-based volume allocation in large storage controller collections |
US7917705B2 (en) | 2007-05-17 | 2011-03-29 | International Business Machines Corporation | Scalable performance-based volume allocation in large storage controller collections |
US8412890B2 (en) | 2007-05-17 | 2013-04-02 | International Business Machines Corporation | Scalable performance-based volume allocation in large storage controller collections |
WO2014151928A2 (en) * | 2013-03-14 | 2014-09-25 | California Institute Of Technology | Distributed storage allocation for heterogeneous systems |
WO2014151928A3 (en) * | 2013-03-14 | 2014-11-13 | California Institute Of Technology | Distributed storage allocation for heterogeneous systems |
US9632829B2 (en) | 2013-03-14 | 2017-04-25 | California Institute Of Technology | Distributed storage allocation for heterogeneous systems |
Also Published As
Publication number | Publication date |
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EP1145127A2 (en) | 2001-10-17 |
US6366931B1 (en) | 2002-04-02 |
US20020046316A1 (en) | 2002-04-18 |
WO2000031640A3 (en) | 2001-05-10 |
WO2000031640A9 (en) | 2001-07-19 |
US6526420B2 (en) | 2003-02-25 |
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