Exact Switching Frequency of the Accumulating-Priority Rule on a Lattice: Ladder-Height and First-Excess Kernels, Random Service and Setup Times

Under the generalized-cμ rule with quadratic delay costs, which is an accumulating-priority rule with relative slope κ=c2μ2/(c1μ1), a server facing two saturated classes alternates between runs of class 1 and runs of class 2. On an integer lattice we show that a class-1 run is a ruin problem for a random walk with moving boundary and a class-2 run is a first-excess problem for a positive-increment walk; the deficit of the first is the starting level of the second, and the overshoot of the second is the starting level of the first. For rational interarrival transforms (geometric, Erlang, hyperexponential) the Wiener–Hopf factorisation gives both laws in closed form from the roots of a Lundberg polynomial. Wald’s identities and the stationary law of the run-start chain then give the long-run switching frequency. The construction covers random service times and class-dependent setup times, which enter as a modified first step of each run. The frequencies agree with an exact lattice computation to three or four digits in all cases tested, for several slopes and arrival shares, and a simulation of the finite queue confirms that the saturated chain is exact when the load including setups is at least one. We also give closed-form undershoot weights, which complete the Gerber–Shiu factorisation of the moving-boundary ruin theory, and for more than two classes we prove a share law (with a correction for starved classes) while no closed form for the switching frequency is available.