Optimal buffer allocation in tandems of last come first served queues

A large fixed number of buffer spaces is given. We consider the problem of allocating these spaces among the nodes of a tandem of last-come-first-served queues with general service time distributions and Poisson external 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. Consider the following rule of thumb: allocate the buffer spaces in inverse proportion to the logarithms of the effective service rates at the nodes. Here effective service rate denotes the ratio of the service rate to the stationary arrival rate. We prove that this rule of thumb achieves a nearly optimal buffer allocation under the assumption that the service time distributions satisfy an exponential tail condition. This problem has been studied earlier in the context of Jackson networks, where it was shown that the same rule of thumb achieves an allocation that is close to optimal. The technique of proof here is similar, but there are important differences. Both Jackson networks and the LCFS tandems considered here are product form networks (with infinite buffers). Optimism should lead us to expect that the near optimality of this rule of thumb holds much more generally for product-form networks, but this remains a conjecture at present.