Loss rates in large loss systems with subexponential demands

Abstract

The analysis of stochastic loss networks has long been of interest in telephone engineering, and is becoming important in the areas of service and information systems, workforce management and inventory control. In more traditional settings, where requests for fixed amount of resources arrive as Poisson process and require them for some almost surely finite time, it is well known that the request loss rate can be explicitly computed using a generalization of the well known Erlang formula (see [8], [5]). In the cases of an increasing number of request types and their higher arrival rates, larger resource capacities are required in order to avoid large losses. However, computing loss rates for these systems becomes intractable, since the state space increases exponentially fast. Using compound point processes to capture stochastic variability in the request process, we introduce a new class of models in this framework that allows us to compute simple asymptotic expressions for loss rates in a quite general setting. In particular, in this paper we analyze loss systems in the presence of random resource requirements. Assuming that requests arrive according to a renewal process, requiring resources for some random time with finite mean, we compute an explicit asymptotic formula for the request loss rate as resource capacity grows large. In addition, we extend our model to incorporate requests for different resource types, as well as shifting the start times of engagements by some random time (advance reservations). Although asymptotic, our experiments show excellent match between derived formulas and simulation results even for relatively small resource capacities and relatively large values of loss rates.