# Dynamic Multi-User Load Balancing in Distributed Systems

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6 5. Determine the lightest node (, ) for the user job as follows: i. Calculate where is given by Proposition 4.2. ii. Calculate. iii. If,then node is the lightest loaded node for the user job. iv. If,where is the number of transfers the job has already had, send the user job to node.gotostep1. v. Add the user job to the local queue. Go to Step 1. NCOOPC tries to balance the of each user ( ) at all the nodes ( ) statically whereas DNCOOPC tries to balance the of each user at all the nodes ( ) dynamically. For a user job arriving at node that is eligible to transfer, each potential destination node ( ) is compared with node. Definition 4.2 If, then node is said to be more heavily loaded than node for a user job. We use the following proposition to determine a light node relativeto node forauser job, in the DNCOOPC algorithm discussed below. Proposition 4.4 If node is more heavily loaded than node for a user job, then,,where The goal of DNCOOPC is to balance the workload among the nodes dynamically in order to obtain a useroptimal solution ( to minimize the expected response time of the individual users). We base the derivation of DNCOOPC on the static NCOOPC of Section 3. We use the following proposition to express in equation (14) in terms of the mean service time at node ( ) and the mean number of jobs of user at node (, ). Proposition 4.3 Equation (14) can be expressed in terms of and ( )as Proof: Similar to Proposition 4.1. (22) Rewriting equation (15) in terms of and, we have (23) Let denote the utilization of the network at a given instant as in DGOS and denote the utilization of the network at a given instant excluding user traffic. We next express of Proposition 4.3 and from equation (23), which use the mean estimates of the system parameters, in terms of instantaneous variables. (24) (25) (26) Proof: Similar to Proposition 4.2. The description of the components of dynamic NCOOPC (information policy, transfer policy and location policy) is similar to that of the components of dynamic GOS. To calculate the threshold s in DNCOOPC, all the nodes execute NCOOPC every time units to determine the optimal loads ( ). A node with the lightest load for a user job is determined using Proposition 4.4. We next describe the dynamic NCOOPC (DNCOOPC) algorithm. DNCOOPC algorithm: For each node ( ): Steps 1., 2. are similar to DGOS. 3. Steps (i), (iii) are similar to DGOS. ii. Use the NCOOPC algorithm to calculate. When a job of user ( ) arrives with a mean interarrival time of : 4. If, then add the job to the local job queue. Go to Step Determine the lightest node (, ) for the user job as follows: i. Calculate where is given by Proposition 4.4. Steps (ii) - (v) are similar to DGOS. s,

9 Expected Response Time (sec) Expected Response Time (sec) DNCOOPC increases and for a high bias (e.g. =1),the performance of DNCOOPC is similar to that of GOS. The effect of bias on DGOS is similar GOS bias 0.2 bias 0.6 bias 0.8 bias System Utilization (%) Figure 6. Variation of expected response time of DNCOOPC with system utilization for various biases (OV = 5%) In Figure 7, we present the variation of expected response time of DNCOOPC with exchange period of system state information. The system utilization is fixed at 80%. The other parameters are fixed as in Figure 2. From Figure 7, it can be observed that the expected response time of DNCOOPC increases with an increase in exchange period. This is because, for high values of P, outdated state information is exchanged between the nodes and optimal load balancing will not be done. The effect of exchange period on DGOS is similar GOS OV = 0% GOS OV = 5% GOS OV = 10% DNCOOPC OV = 0% DNCOOPC OV = 5% DNCOOPC OV = 10% 6. Conclusions and Future Work In this paper we proposed two dynamic load balancing schemes (DGOS and DNCOOPC) for multi-user jobs in heterogeneous distributed systems. DGOS tries to minimize the expected response time of the entire system, whereas DNCOOPC tries to minimize the expected response time of the individual usres. These dynamic schemes use the number of jobs of the users in the queue at each node as state information. We used simulations to compare the performance of the dynamic schemes with that of the static schemes (GOS and NCOOPC). It was observed that, at low communication overheads, both DGOS and DNCOOPC show superior performance over GOS and NCOOPC. Also, the performance of DNCOOPC which tries to minimize the expected response time of the individual users is very close to that of DGOS which tries to minimize the expected response time of the entire system. Furthermore, DNCOOPC provides almost equal expected response times for all the users and so is a fair load balancing scheme. It was also observed that, as the bias, exchange period and the overheads for communication increases, both DGOS and DNCOOPC yield similar performance to that of the static schemes. In future work, we plan to propose dynamic load balancing schemes where the jobs arriving from various users to a node not only differ in their arrival rates but also differ in their execution times. Acknowledgements Some of the reviewers comments which helped improve the quality of the paper are gracefully acknowledged. Appendix In this section we present the proofs of the results used in the paper. Proof of Proposition 4.1 The expected response time of a user jobprocessedat computer in equation (1) can be expressed in terms of as Exchange Period (sec) Using Little s law ( above equation can be written in terms of, as )[8],the Figure 7. Variation of expected response time with exchange period ( = 80%) (28) Remark: Note that we assume the jobs from various users have the same execution time at a node.

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