1. Aspects of E-Science, Mathematics and Theoretical Computer Science Professor Iain Stewart Department of Computer Science University of Durham March 2003

2. E-Science, Mathematics and Theoretical Computer ScienceSlide 2 of 10 A Holy Grail of Grid Computing “Every user connected to the Grid should have access to all resources available anywhere on the Grid, with the user presented with an illusion of a single centrally-organized computer.” The challenge is to design the protocols, algorithms, paradigms and systems so that the aggregation of the physical computers, communication links and other available resources achieves this illusion. Ideally, such systems would optimally allocate available resources such as –computational resources (CPU time, file servers) –information resources (databases, video) –communication resources (links, quality of service) –services (help-desk, access to specialized algorithms) –hardware (printers, cameras) …

3. E-Science, Mathematics and Theoretical Computer ScienceSlide 3 of 10 Different Aspects of this Challenge There are clearly many different aspects to this challenge. One such is the basic technological interoperability, for which the paradigm of distributed object services has become the fundamental framework. This paradigm is the basis of most modern commercial distributed platforms, e.g., –CORBA –Microsoft’s DCOM –Java’s RMI Standards such as XML and protocols such as SOAP result in a web-like infrastructure for distributed processing, following this paradigm. However, let us focus on another aspect: the sharing of resources.

4. E-Science, Mathematics and Theoretical Computer ScienceSlide 4 of 10 Resource Sharing A major difficulty as regards the efficient sharing of resources is that the different resources generally belong to different organizations. A system for resource sharing should provide for “motivation” for resource sharing and an evaluation and application of different “costs”. What emerges is some kind of economic system involving, in its simplest form, “payments” for services. A very basic architecture is as follows. –Each seller providing a service may attach a price-tag for this service. –When a buyer wishes to use a particular type of service, it calls a central “service market place” that functions as the seller-request broker. –The market performs a “reverse auction” and the seller charging the lowest price wins and gets to service the request at an agreed price. Of course, the meanings of terms like “price” and “cost” can be wide- ranging.

5. E-Science, Mathematics and Theoretical Computer ScienceSlide 5 of 10 A Very Basic Example Scenario Printers provide the service of printing and each printer charges a price for this service. Customers supply their requests to the “printing market” and receive the opportunity to print on a printer as returned from the “printing market”. In reality, many parameters distinguish one print job from another: number of pages, page size, printer’s location, … The system needs to be aware of these parameters (and there is an issue relating to how these parameters are specified). This results in a parameter space for (our one) service-type where each printer can print for some subset of the parameter space at some given cost. Service requests will in general involve a subset of the parameter space, e.g., “A4 or Letter is fine, so long as the resolution is high”.

6. E-Science, Mathematics and Theoretical Computer ScienceSlide 6 of 10 A Resulting Economic System Our economic system is based on a common currency in which all participants can express their economic preferences. Each service provider - printer – has a certain cost for supplying the service at each point of the parameter space provided by the provider, e.g., –4p for A4 single-sided printing, high resolution –3p for A4 single-sided printing, low resolution –2p for A4 double-sided printing, low resolution Each service requester – buyer – has a certain economic benefit from receiving the service, again parameter-dependent. Let S be the parameter space of our service-type. Our economic model is given by: –provider p’s cost function c p : S  R + –buyer b’s utility function u b : S  R +

7. E-Science, Mathematics and Theoretical Computer ScienceSlide 7 of 10 Allocation of Resources Each provider sends to the market a function corresponding to the provider’s cost function – the quote function q p : S  R +. The quote function provides a catalogue of the services provided by the provider and amount of money demanded. Note that q p may be different from c p, depending on the provider’s inclination to make a profit; and this inclination may change. A buyer sends to the market a representation of the utility function u b. Of course, there is an issue as to how one supplies these functions to market.

8. E-Science, Mathematics and Theoretical Computer ScienceSlide 8 of 10 Allocation of Resources When the market receives a buyer request, the market attempts to match this request to the best provider and choosing the best parameter values. For parameters s, the optimisation criteria is surplus b,p (s) = u b (s) – q p (s). For a given provider, the parameters are chosen to maximize this surplus. Of course, there can be significant complexity-theoretic issues here (depending upon function representation, the size of the parameter space, etc.). Once the optimal provider p* and parameter choice s* have been found, the market can fix any payment d in the range q s* (p*)  d  u b (p*). There can be economic issues as to which specific payment d is chosen by the market.

9. E-Science, Mathematics and Theoretical Computer ScienceSlide 9 of 10 Decentralized Load Balancing In our system above, we obtain an optimal allocation of each request to the service provided that is best for it. However, we have assumed no conflict of requests. The whole point of allocation in distributed systems is that requests should be partitioned between the providers. This gives rise to the situation where users’ utilities depend upon the time until their request is serviced; so, in effect, service-time is a parameter in the parameter space of the service-type. Note that the scheduling of one buyer’s request to a provider can have a dramatic impact on the the costs of scheduling a subsequent request. Furthermore, optimal global allocation is NP-hard and cannot be done by scheduling requests in an online fashion (as they come).

10. E-Science, Mathematics and Theoretical Computer ScienceSlide 10 of 10 Verdict An the heart of e-Science lies a considerable amount of mathematics and theoretical computer science, specifically: –Decentralized scheduling –Combinatorial auctions –Algorithms and complexity with significant overlaps with –operations research –artificial intelligence –mathematical economics –discrete mathematics. For more details, see Naom Nisan’s pages on economic mechanisms in computation: http://www.cs.huji.ac.il/~noam/mkts.html