Using queuing theory to describe steady‐state runoff‐runon phenomena and connectivity under spatially variable conditions

Abstract

Runoff‐runon occurs when spatially variable infiltration capacities result in runoff generated in one location potentially infiltrating downslope in an area with higher infiltration capacity. The runoff‐runon process is invoked to explain field observations of runoff ratios that decline with plot length and steady‐state infiltration rates that increase gradually with rainfall intensity. To illustrate the influence of spatial variability and runoff‐runon on net infiltration and runoff generation, we use both (i) field rainfall simulation on soil with a high infiltration capacity and (ii) numerical rainfall‐runoff‐runon simulations over a spatially variable area. Numerical simulations have shown that the spatial variability of soil infiltration properties affects surface runoff generation; however, it has proven difficult to represent analytically the associated runoff‐runon process. We argue that given some simplifying assumptions, the runoff‐runon phenomenon can be represented by queuing theory, well known in the literature on stochastic processes. Using this approach we report simple analytic expressions derived from the queuing literature that quantify the total runoff under steady‐state conditions from a spatially variable tilted 2‐D plane, and the runoff produced by the area connected with the lower boundary of the plane under these conditions. These quantities are shown to be equal to the waiting time and the “sampled” busy period, respectively, of a first‐in first‐out queue with exponentially distributed arrivals and service times. The queuing theory model is shown to be consistent with field observations of runoff; however, the approach requires some simplifying assumptions and restrictions that may negate the benefits of the reported analytic solutions.