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We analyze randomized dynamic load balancing schemes for multi-serverprocessor sharing systems when the number of servers in the system is large andthe servers have heterogeneous service rates. In particular, we focus on theclassical power-of-two load balancing scheme and a variant of it in which anewly arrived job is assigned to the server having the least instantaneousLagrange shadow cost among two randomly chosen servers. The instantaneousLagrange shadow cost at a server is given by the ratio of the number ofunfinished jobs at the server to the capacity of the server. Two differentapproaches of analysis are presented for each scheme. For exponential joblength distribution, the analysis is done using the mean field approach and formore general job length distributions the analysis is carried out assuming anasymptotic independence property. Analytical expressions to compute meansojourn time of jobs are found for both schemes. Asymptotic insensitivity ofthe schemes to the type of job length distribution is established. Numericalresults are presented to validate the theoretical results and to show that,unlike the homogeneous scenario, the power-of-two type schemes considered inthis paper may not always result in better behaviour in terms of the meansojourn time of jobs.

Analysis of Load Balancing in Large Heterogeneous Processor Sharing Systems