Consider a Jackson type closed queueing network in which each queue has a single exponential server. Assume that N customers are moving among k queues. We propose a candidate procedure which yields a lower bound of the network throughput which is sharper than those which are currently available: Let (ρ1, …, ρk) be the loading vector, let x be a real number with 0≤x≤N, and let y(x) denote that y is a function of x and be the unique positive solution of the equation Σki=1 y(x)ρi/(N–y(x)xρi) = 1. Whitt has shown that y(N) is a lower bound for the throughput. In this paper, we present evidence that y(N–1) is also a lower bound. In doing so, we are led to formulate a rather general conjecture on ‘Migrating Critical Points’ (MCP). The MCP conjecture asserts that zeros of successive derivatives of certain rational functions migrate at an accelerating rate. We provide a proof of MCP in the polynomial case and some other special cases, including that in which the rational function has exactly two real poles and fewer than three real zeros.
