Randomized load balancing greatly improves the sharing of resources in a number of applications while being simple to 
implement.  One model that has been extensively studied is the supermarket model with exponential service time 
distribution. The result due to Vvedenskaya et. al. (1996) showed that when each arriving job is assigned to the 
shortest of d > 1 randomly chosen queues, the equilibrium queue sizes decay doubly exponentially in the limit as 
n goes to infinity.  This is a substantial improvement over the case d=1, where queue sizes decay exponentially.

It is desirable to study load balancing models with more general, especially heavy-tailed, service time distributions 
since such service times occur widely in practice. However, the method of analysis for exponential service time 
distributions does not easily generalize. We describe a program for treating general service time distributions. 
The program depends on an ansatz that asserts the uniqueness and asymptotic independence of the limiting equilibrium 
distribution. Assuming the ansatz, we obtain that the limiting distribution is a fixed point for an object named queue 
at the cavity, and that the queue size distribution is insensitive to the service distribution with PS and LIFO-PR 
service disciplines. We establish the ansatz and charaterize the tail behavior of the queue size distribution for the 
FIFO service discipline and the service time distribution with a decreasing hazard rate.