Load Balancing in Heterogeneous Systems

Transcription

1 Load Balancing in Heterogeneous Systems P.Neelakantan Department of CSE VNR VJIET, Hyderabad Abstract The grid and cluster computing uses interconnected nodes to solve a problem in parallel in order to improve the response time of the job. Diffusive load balancing algorithms work well when the nodes in the system have the same processing capacity. However, little attention paid in diffusion load balancing techniques, for the nodes with different processing capabilities. In this chapter, a load-balancing algorithm using diffusion technique proposed for distributing the load between the nodes by treating the loads as an integer quantity.the proposed load balancing algorithm distributes apportion ofexcessive workload of a heavily loaded node to a lightly loaded node by considering the node's processing capacities. I. INTRODUCTION Grid and cluster computing consists of many heterogeneous nodes with different processing capacities connected by a communication network. Load balancing must be used when the load among the nodes are not uniform. The nodes in a distributed system must have the uniform distribution of loads after applying the load balancing algorithm. For the migration of the load from the heavily loaded node to lightly loaded node in a heterogeneous distributed system, their processing capacities must be considered. So, the load balancing algorithms designed for homogeneous distributed systems won t work efficiently for the heterogeneous systems. So, there is a need for load balancing algorithm which distributes the load among the nodes in a heterogeneous system by taking into consideration of the processing capacity of the nodes. Distributed systems are geographically spread over different autonomous users and organizations which make them potentially large [4]. Load balancing algorithms are divided into static and dynamic load balancing algorithms. Static load balancing algorithms do not base their decision on the current state of the system, so, their performance will not be optimal when compared to the dynamic load balancing algorithms. In contrast, dynamic load balancing algorithms base their decision on the current state of the system. However, dynamic load balancing algorithms incur more overhead when compared to the static load balancing algorithms. Dynamic policies are further divided into two classes: Centralized and Distributed [1, 2, 3]. In centralized policies one node is held responsible to maintain the information of all the nodes in the system. Collecting the information from all the nodes and then distributing the workload among the nodes will be the difficult job and it imposes much overhead and it won t perform well when the scalability of the system increases. In distributed policies, each node collects the information from all nodes to know whether it is overloaded or under loaded. A dynamic distributed load balancing algorithm has three components: (1) location policy determines suitable nodes for the exchange of the load (2) Information policy is used to collect load information from the nodes in the system (3) transfer policy determines, whether the node is a suitable candidate for transfer of the load. Even this method is not feasible when the number of nodes in the distributed computing is large. The distributed load balancing schemes are further classified into a sender initiated and a receiverinitiated. Iterative load balancing schemes are found to be efficient in balancing the workload among the nodes. Iterative load balancing schemes collect load information from the neighbours which reduce the communication overhead, and achieves faster convergence. In diffusion technique, the node can exchange the information from all its neighbours in a single communication step. In literature, there is no diffusion load-balancing algorithm that deals with heterogeneous systems by considering the load as an 41 P.Neelakantan

2 integer quantity. In this paper, a decentralized loadbalancing algorithm has been devised for heterogeneous systems by using a diffusion technique. II. NOTATIONS,ASSUMPTIONS &MATHEMATICAL MODEL The network is represented by an undirected graph G=(, ), where each node has a processing weight > 0, and a processing capacity > 0 and E is a set of edges represents the communication links connecting the nodes in the heterogeneous distributed system. N: Number of nodes in a distributed system V= {1, 2.N} represents set of nodes in a distributed system L : Load of node i p : processing weights of node i s : processing capacity of node i D : Neighbours of the node i denoted by D ={j j V and (i, j) E} DN : Set consisting of deficient neighbors belonging to the domain of the node i Nless: Set consisting of the nodes having the same minimum load value δ,δ : Indicators of apportion of load sent to the deficient neighbours. Assumptions It has been assumed that a distributed system consists of N heterogeneous nodes interconnected by an underlying arbitrary communication network. Each node i in a system have a processing weight p i>0 and processing capacity >0. The load is defined to be L i= p i/s i. In homogenous case, the value of L i=p i. It has been assumed that jobs arrive at node i according to a Poisson process with rate ( ).A job arrived at node i may be processed locally or migrated through the network to another node j for remote processing. Once the job is migrated it remains there until its completion. It has been assumed, that there is a communication delay incurred when job is transferred from one node to another before the job can be processed in the system. Here, it has been assumed that the bandwidths of the links are same. Each node runs a load balancing algorithms and an infinite buffer to hold jobs. The load-balancing algorithm will balance the arriving jobs at the node such that the response time of the jobs is minimized. In the absence of the load balancer in the node, the job enters the buffer, waits in the queue for processing, leaves the queue and gets processed in the node and then leaves the node. When two nodes are connected by an edge, they are neighbours to each other and the two nodes can exchange their loads (job weight) through that edge. To minimize their loads the nodes transfer their excess load to their neighbours. In general, it is not feasible for a node to collect load information from all the nodes in the network. Therefore, in order to reduce the communication delays, in our model it is assumed that a node knows the load information of its neighbours only. The loads of other nodes are unknown. A network is referred to as a homogeneous network when the processing capacities of all nodes are equal; otherwise it is referred to as a heterogeneous network. In a homogeneous systems, the value of = 1, and therefore =. Mathematical Model Let L i denote the total amount of load originating at node i for processing, L i>=0. Let l i denotes the amount of loads assigned to node i according to a scheduling strategy and x ij denote the amount of load received from node j where j D i. The following equation holds L = + (1.1) The node delay incurs when l i jobs are processed at node i is a linear and increasing function [5]. The node delay for a node i, shall be denoted as ( ) = + (1.2) Where is the time taken to execute a unit load by node i. The objective of the proposed method is to have a load distribution in such way that it reduces the response time of the loads in node i. The response time for each node i depends on the l i and max max t i, t i is the time instance at which the last job is received by node i. If no job is sent to node i by its max neighbour nodes t i =0. Now the total response time is given by = + (1.3) 42 P.Neelakantan

3 The response time of the system is influenced by l i and the amount of the load transferred x ij. Let ( ) is the time instant at which the last job coming from node j received by node i. According to the definition of, the following equation is obtained. ={ ( )}) } (1.4) The objective of the proposed algorithm is to minimize the response time of the system according to equation (1.2). Minimize: P (L, x) = Max{ = + } (1.5) Subject to: L = + where > 0 and 0 III. PROPOSED METHOD The algorithm has been proposed which uses a diffusion technique for sending the loads to deficient neighbours. This algorithm runson each node i. 1.1 Description of the proposed method In a proposed method, a set of deficit neighbours of the domain of node iis given by DN = j D L < L (1.6) After identifying the deficit neighbours of domain of node i, the load average domain of node i is calculated by using the formula L = (1.7) Note in the above equation that the processing capacities of the nodes are taken into account which is true in heterogeneous computing. In general if node i take t itime units to process a job t and node j takes t j time units to process the same job t then the ratio of t i/t j must be equal to one. Otherwise the ratios are different. The former is the case, when the nodes have different processing capacities and the latter refers to when nodes have same processing capacities. After computing the load average of the system, the load difference in the domain of node i is calculated by using the formula LD= ( - ) (1.8) When the value of LD is positive, then it indicates the node i has an excess load, and it is to be sent to the deficit neighbours belong to the set DN i. s i indicates the relative processing capacity of node i when compared to other nodes in the system. If the load difference is a negative value, the node i do not execute the procedure Balance otherwise it calls the procedure Balance. In procedure Balance the input parameters are the load difference and the set of deficit neighbours. The algorithm is terminated when the load difference LD is reached to 1. First, the excess load is sent to the least loaded node in the set deficit neighbourdn i. For this purpose,the nodes in set DN i are sorted in ascending order of their load values which takes the time complexity of O(d log d), where d is the number of neighbour nodes (d= D i ). After sorting the nodes according to their load values in set DN i, the first node is considered for migrating the load and also a check is made to see if any node contains the load value of the first node. If those elements are kept in the set nless by incrementing the counter value and for each element in the set DN i, the following is done: { } ( == ) = { }; = + 1; The Tempsetconsists of the nodes which are present in DN i but not in Nless Tempset = (4.9) The load to be sent is based on Tempset and how much to be sent is determined by use of two indicators and.the minimum of these two values have been used for sending the load to the deficit neighbours until the value of load difference LD reaches zero or 1. = & = Where k Tempset and s Nless (4.10) When the load index of the neighbouring nodes falls below the average load of the domain of i, it is said to be deficient neighbour. The deficient neighbours do not have the excess load to send and they receive apportion of the load sent by the excessive neighbours. If all the loads of the nodes are equal to the average load of the underlying domain, then the domain is in a balanced state. Algorithm () Begin for node i DN = j D L < L Calculate L LD= (L -L )s ; if (LD<0) then exit; Balance (LD, DN ); End Procedure Balance (LD, DN ) Begin While (LD>1) // sort the loads in ascending order in 43 P.Neelakantan

4 set DN lowload = FirstElement in DN ; count = 0 Nless = for k {DN } if (L = L ) Nless = Nless {k}; count = count + 1; endif end for Tempset = DN Nless; δ = ; if( Tempset ) nextloadmin = firstelement in Tempset δ = L L ; for each k Nless if(δ > δ ) Transfer δ to k; LD = LD δ ; end if else Transfer δ tok; LD = LD δ ; end else end for end if else for each k Nless Transfer δ to k; LD = LD δ ; End for End else End while End Algorithm: Balance The above algorithm uses diffusion technique for balancing the load among the nodes in the distributed system. This algorithm receives load information from the neighbours as an input. The idea behind this algorithm is to distribute the load equally among the lightest nodes in until their load value reaches the second lightest node in. This process is repeated until the value of runs out. IV. CONVERGENCE PROOF Lemma 1: Let be the load of node at time t. Let = (,,.. ) be the array of loads in time sorted in ascending order. If there exists at least one node with LD> 0 in time then is lexicographically greater than. Proof: At time some nodes may send their load to the neighbouring nodes and may have their load decreased in time + 1; therefore, these nodes do not send the load at time + 1. This refers to the case when loads of the nodes are distinct. But by using the same idea a similar proof shall be derived where all the loads are not distinct. Let be the set of nodes that send load (i.e., the nodes with LD > 0) in time, and by hypothesis. It shall be observed that a node in S may also receive the load in time t. Let = :. That is, is the node having lowest load value at the time + 1 among the nodes that sent load at time. Let occupies the k-th position of the array.let = (,,.. ) be the array of loads in first k-1 positions of. It has been seen that a node belongs to if its load is among the load values of the array. Nodes that belong to will receive a load (without sending) or remains unchanged in time + 1 depends on choosing the node. Thus, all the load values in are greater than or equal to the corresponding values of. Next, let us consider two cases. i. A set contains a node i that received load in time t. In this case, node i belongs to both and, and its load value is increased in time t + 1. Therefore, there will be at least one load value in a set strictly greater than one value in and therefore is lexicographically greater than. ii. In the complementary case, it has been seen that there are nodes belonging to which do not receive the load in time t. Thus, the load values in set and are equal. Theorem 1 Convergence: In asynchronous heterogeneous networks, if the nodes use the algorithm, the system converges to a balanced state. Proof: If the nodes in the heterogeneous distributed systems are not balanced, it exhibits that there is at least one node in the system heavily loaded, let be the most loaded node. By the choice of, all have. Moreover, as is not balanced, then at least one has. When calculating the nodes such that ( )= have their load value rounded to. But if > and consequently, = - > 0. Thus, the result of Lemma 1 shall be used, which guarantees that the vector of loads sorted in ascending order, in the next time moment, is lexicographically greater than the array of the current step. 44 P.Neelakantan

5 Let us assume algorithm is executed by some of the nodes in the system asynchronously in a given time instant. Let be the set of nodes executed the algorithm and nodes have changed their load value by executing the algorithm in time t. Let be the array of loads sorted in ascending order of load values in time t. It has been shown that is lexicographically greater than. Let be the lightly loaded nodes in a time t. There exists at least one node which is adjacent to node such that >. Now by using the algorithm, node sends a portion of its load to the neighbour node but the node does not send any load in time because it is underloaded when compared to the average load of the domain. In time + 1 the load value of node in \ decrease but its load value never becomes smaller than the load value of h e which is given by >. Thus is lexicographically greater than. Thus it has been proved that the sorted array of load values of nodes in time in are lexicographically greater than the sorted array of load values of nodes in S at time + 1. Lemma 3: A node can only send load to a neighbor if >, ( > ) i.e., where, is the neighbouring nodes (directly connected to the node ) In other words, this rule ensures that if loads of two nodes belong to the same domain differs significantly; the load will be transferred between two nodes belonging to the same domain. Lemma 4: Let be the load of node at time. Then The first constraint says that the load on nodes in a domain of node i does not deviate too much when compared to the load average of the domain of node i and the second one guarantees, the load balancing algorithm must distribute the load to the deficit neighbour in such a way that, the node after transferring its excess load to its neighbour, its load value must remain greater or equal to its deficit neighbour. The second one is a technical constraint because of the job thrashing effect. V. SIMULATION The proposed load-balancing algorithm is compared with two heterogeneous load balancing algorithms: (Maximum Load balance) and (Load balancing using Mobile agents). The simulation framework has been designed with the following setup. Random topology is considered with the system size from 8 to 32 nodes. The connectivity of all the nodes is ensured. The initial load distribution patterns are varying from situations, which exhibit a light unbalance degree to high unbalance situations. The comparison has been carried out in terms of load balancing time, throughput and response time for varying loads. For that purpose, Load Balancing Simulator in Java is developed which allows us to: Test the behaviour of different load balancing algorithms under the same conditions Evaluate the behaviour of the load balancing algorithms for random graph with different sizes of nodes Evaluate the behaviour of the algorithms for different arrival patterns. The load distribution among the nodes is done in four ways for simulation purpose. Entire load is concentrated in 10% of the nodes, while rest of the nodes in the system are start with initial load value equal to 0. 25% of total nodes are having load value initialized to 0. 50% of total nodes are having load value initialized to 0. 75% of total nodes are having load value initialized to 0. System utilization represents the amount of load on the system and is defined as the ratio of the total arrival rate to the aggregate processing rate of the system: Relative processing rate Number of nodes Processing rate(jobs/sec) = Table 1: system configuration 45 P.Neelakantan

6 The simulation has been carried in heterogeneous system consisting of 32 nodes with different processing capacities as mentioned in the table 1. The mean communication time t is assumed sec. Influence of System utilization on Response time of the system Figure-1 to figure-3 reflects response time for different system utilizations when the number of nodes in the system is 32. It is seen that every algorithm provides response time, which is proportional with the system utilization. Moreover, the proposed balancing algorithm performs better than other rival algorithms for each system utilization. It is worth noting that the best response time obtained by the algorithm denote its capacity to coerce the system into a balance load distribution. Notice that the behaviour of all simulated algorithms is very similar under low system utilization, and only a slight increase in response time. Response time ( in Sec) vs. Response time for 25% of idle nodes Fig 1: shows the system utilization vs. response time for node size=32 for 25 % of idle nodes Response time ( in sec) vs. Response time for 50% of idle nodes Fig 2: shows the system utilization vs. response time for node size=32 for 25 % of idle nodes Response time ( in sec) System utilization vs response time for 75% of idle nodes Fig 3: shows the system utilization vs. response time for node size=32 for 25 % of idle nodes Influence of System Size on throughput Figure-4to figure-5 displays the throughput of the system for different system utilization for a varying percentage of idle nodes in the system. It is observed from the simulation that the proposed algorithm provides throughput which is slightly better than the rival algorithms. The throughput is not decreasing with an increase in number of idle nodes in the system. The proposed and rival algorithms exhibit similar throughput. But with the decrease in the percentage of idle nodes in the system, the throughput is decreasing for all the algorithms but the proposed algorithm has done better when compared to the rival algorithms. The throughput obtained by is always greater than that obtained by and. and have the quality of keeping the throughput almost equal for a varying percentage of idle nodes in a system with varying system utilizations. Throughput( jobs/sec) System utilization vs Throughput for 75% of Idle nodes Figure-4 shows the system utilization vs. Throughput for node size=32 for 50% of idle nodes 46 P.Neelakantan

7 Throughput(jobs/sec) vs Throughput for 50% of idle nodes Figure-5 shows the system utilization vs. Throughput for node size=32 for 50% of idle nodes CONCLUSION It has been observed in the literature that little attention was paid in diffusion load balancing algorithms despite of their ability in making the system to converge faster. Another issue that was seen in the literature is only a few algorithms have considered the loads as an integer quantity. Many algorithms assumed that the load could be arbitrarily divided. So keeping in mind these issues, the algorithm using the diffusion technique is designed for distributing the workload in the nodes with different processing capabilities when the load of the nodes is treated as integers. When the loads are distributed to the nodes without considering their processing capacities, it will affect the response time of the system. The proposed algorithm has done well when it is compared to the rival algorithms Maximum Load Balance and Load Balancing using Mobile agent () in terms of response time, throughput and load balancing overhead in a system. In heterogeneous systems, OS Scheduling policies like round robin, priority scheduling must be taken into account while doing load balancing. Another aspect to be considered by the load balancing algorithm includes the deadline of the task, where the task is scheduled in such a way that it has to meet its deadline. These factors are not taken into account while designing proposed algorithms in this thesis as it requires extensive research and different techniques to be incorporated. Much literature is not available on load balancing algorithms that schedule jobs as per deadlines or priorities. Another aspect to be considered in a future work is the size of the queue. In this thesis it has been assumed the queue is of infinite size, where it is not true in distributed systems. The nodes in the distributed systems contain a finite queue size. Without taking consideration of the queue size, when the loads are distributed, the loads are lost as there is no buffer in the receiving end. However, in literature up to today, many authors have assumed the queue size as infinite, as it is an easy approach to attack on the load-balancing problem. Otherwise, the algorithm must be devised in a complex manner. Another aspect to be considered in a distributed system is fault tolerant. As a part of the load balancing process, if jobs sent to the under loaded node and subsequently that node fails after receiving the job, error condition results. So, as a part of future work an efficient algorithm shall be designed by taking into consideration of fault tolerance while doing load balancing among the nodes. Last, but not least, new techniques like Genetic algorithms, Fuzzy logic, neural networks shall be used for a future prediction of loads in the neighboring nodes by a overloaded node to send its excess load. Designing load balancing algorithms with the above techniques will be appropriate and will yield good results. References: [1] X. Tang, S.T. Chanson, Optimizing static job scheduling in a network of heterogeneous computers, in Proceedings of the International Conference on Parallel Processing, August 2000, pp [2] Zhang Y., Kameda H., and Hung S. L. Comparison of dynamic and static load balancing strategies in heterogeneous distributed systems. IEE Proc. Compu. Digit. Tech., 144(2): , 1997 [3] J. Weissman. Scheduling Parallel Computations in a Heterogeneous Environment. PhD thesis, Dept. of Computer Science, Univ. of Virginia, August [4] Wolffe, G. S.; Hosseini, S. H.; Vairavan, K. (1997). An Experimental Study of Workload Indices for Nondedicated, Heterogeneous Systems. In the proceedings of PDPTA 97, v.1, p [5] V. Berten, J. Goossens, and E. Jeannot, On the distribution of sequential jobs in random brokering for heterogeneous computational Grids, IEEE Transactions on Parallel and Distributed Systems, 17 (2) (2006) P.Neelakantan

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