Second-order properties of the throughput of a closed queueing network

The authors show that the throughput of a single-class closed queueing network (CQN) of Jackson type, as a function of the job population, is nondecreasing concave [respectively, convex, anti-starshaped, starshaped, subadditive or superadditive] if the service rate at each node, as a function of the local queue length, has the same property. The key to the proofs is the concept of ‘equilibrium rate’. For a discrete positive random variable Y the equilibrium rate is defined as a function: r(0)=0, r(n)=P[Y=n-1]/P[Y=n], n≥1. It turns out that the equilibrium rate r(n) is nondecreasing in n if and only if the pmf of Y is a Polya frequency function of order two; and it is known that the said property (hence, the nondecreasing property of equilibrium rates) is preserved under convolution. Here, the authors represent the CQN throughput function as the equilibrium rate of a sum of independent random variables, each corresponding to a node in the network, with the service rate of the node as its equilibrium rate. The second order properties of the CQN throughput are then established by proving that the second-order properties of equilibrium rates are also preserved under convolution.