The PDE Method for the Analysis of Randomized Load Balancing Networks

We introduce a new framework for the analysis of large-scale load balancing networks with general service time distributions, motivated by applications in server farms, distributed memory machines, cloud computing and communication systems. For a parallel server network using the so-called SQ(d) load balancing routing policy, we use a novel representation for the state of the system and identify its fluid limit, when the number of servers goes to infinity and the arrival rate per server tends to a constant. The fluid limit is characterized as the unique solution to a countable system of coupled partial differential equations (PDE), which serve to approximate transient Quality of Service parameters such as the expected virtual waiting time and queue length distribution. In the special case when the service time distribution is exponential, our method recovers the well-known ordinary differential equation characterization of the fluid limit.

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