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Optimal Tree Structures for Large-Scale Grids J. Palmer I. Mitrani School of Computing Science University of Newcastle NE1 7RU [email protected] [email protected] Grid Performability, Modelling and Measurement AHM’04 Outline Introduction The model Computation of the optimal tree structure A simple heuristic Results Conclusions and future work Grid Performability, Modelling and Measurement AHM’04 2 Introduction In the provision of a Grid Within such a provision, it service, a provider may have will be desirable that the heterogeneous clusters of clusters are hosted in a resources offering a variety of cost effective manner services Job arrivals Potential bottle-neck Master Node ... Server Server Server Server Server Grid Performability, Modelling and Measurement AHM’04 3 Job arrivals additional decision-making process Master Node additional transfer delays Master Node Master Node ... Server Server ... Server Server Server Server The problem of load-balancing considers how best to distribute incoming jobs across a fixed tree structure Instead, our approach considers the dynamic reconfiguration of the underlying tree structure as load changes Grid Performability, Modelling and Measurement AHM’04 4 Job arrivals dynamic network reconfiguration Master Node Job arrivals Master Node Master Node Master Node ... Server Server ... Server Server Server Server Master Node ... ... ... Server Server Master Node Master Node Server Server Server Server Server Server Grid Performability, Modelling and Measurement AHM’04 Server 5 The model What value of k minimizes the overall average response time of the system? ck level 1 master node transfer delay T1 ... c k) level 2 ... ... ... k master nodes k sub-clusters of N/k service nodes Grid Performability, Modelling and Measurement AHM’04 6 Job distribution policies Different job distribution policies have been considered: level i ici ki transfer delay Ti ... level i+1 1. Each dependent has a separate queue; the master places new jobs into i. those queues in random order ii. the queue which is currently shortest iii. those queues in cyclic order 2. Dependents at the final service cluster level have a joint queue Grid Performability, Modelling and Measurement AHM’04 7 Computation of the optimal tree structure The average response time at each level i master node is given by: 1 Wi i (1 i ) where i ci , i number of dependents i At the final service level, approximated by an M/M/n queue: 1 (n n ) W final 2 j 1 ( j 1)! (n 1)!(n ) j 0 j! (n 1)!(n ) n 1 j where n 2 n 1 j n n number of servers in each cluster , Grid Performability, Modelling and Measurement AHM’04 8 1 Computation of the optimal tree structure For a flat structure ( c1>N for stability): W W1 W final For a two level tree structure: W W1 T1 W2 W final The objective is to minimise the latter with respect to k Grid Performability, Modelling and Measurement AHM’04 9 Computation of the optimal tree structure At each master node we require i 1 So, for a given parameter set, k has upper and lower bounds so that no master node becomes saturated: N c2 k c1 Average response times for each value of k within this range have been evaluated and compared to find the minimum Hence, the optimal value of k has been determined numerically This gives the optimal network configuration with a single layer of master nodes Grid Performability, Modelling and Measurement AHM’04 10 A simple heuristic Consider the total offered load at the level 1 master node and one of the level 2 master nodes: k N f (k ) 2 c1 c2 k N This total load can be minimized with respect to k to find an initial value for k given N, c1 and c2: 2 Nc1 k 3 c2 Grid Performability, Modelling and Measurement AHM’04 11 Results Average response time as k varies Parameters: N 100, c1 c2 100, T1 0.001, 8, 0.1 Load is 80%, flat structure not feasible heuristic predicts k = 6 optimal k = 4 Grid Performability, Modelling and Measurement AHM’04 12 Results Optimal number of clusters as load increases Parameters: N 100, c1 c2 100, T1 0.001, 0.1 Grid Performability, Modelling and Measurement AHM’04 13 Conclusions and Future Work Encouraging results suggest dynamic network configuration will reduce long-term average response times A simple heuristic is available for initial network configuration Future work includes: 1. extension to include further tiers of master nodes 2. different modelling assumptions for how a master node makes a routing decision - shortest queue - cyclic order Grid Performability, Modelling and Measurement AHM’04 14 Acknowledgment This work was carried out as part of the collaborative project GridSHED, funded by North-East Regional e-Science Centre and BT This project also aims to develop Grid middleware to demonstrate the legitimacy of our models, providing a basis for the development of commercially viable Grid hosting environments Project web page: http://www.neresc.ac.uk/projects/GridSHED/ Grid Performability, Modelling and Measurement AHM’04 15