Correctness within a constant of an optimal buffer allocation rule of thumb

The problem is to allocate a fixed number of buffers among the nodes of an open network of exponential servers with Bernoulli routing and Poisson arrivals so as to optimize some performance criterion associated with the time to buffer overflow, such as maximizing its mean or maximizing the probability that it exceeds some value. In earlier work, the authors used pathwise probabilistic arguments to derive a simple rule of thumb for this problem: allocate the buffers in inverse proportion to the logarithms of the effective service rates at the nodes. Effective service rate denotes the ratio of the service rate to the stationary arrival rate in the network with infinite buffers. They showed that this rule of thumb is accurate to within a known constant times the logarithm of the number of buffers as the number of buffers to be allocated becomes large. In the present paper, the authors use time reversal and Poisson clumping arguments to show that their rule of thumb is, in fact, much better than previously demonstrated. They show that the optimal buffer allocation is within a constant of the rule of thumb as the number of buffers to be allocated becomes large, although now they cannot estimate the constant. In numerical terms, the earlier result reduced the search space for the optimal buffer allocation from O(N^J-1) to O((log N)^J-1), where J denotes the number of nodes and N the number of buffers to be allocated. The improvement reduces the search space to O(1)