Heavy-Traffic Approximations for Service Systems With Blocking

Abstract

This paper develops approximations for the blocking probability and related congestion measures in service systems with s servers, r extra waiting spaces, blocked customers lost, and independent and identically distributed service times that are independent of a general stationary arrival process (the G/GI/s/r model). The approximations are expressed in terms of the normal distribution and the peakedness of the arrival process. They are obtained by applying previous heavy-traffic limit theorems and a conditioning heuristic. There are interesting connections to Hayward's approximation, generalized peakedness, asymptotic expansions for the Erlang loss function, the normal-distribution method, and bounds for the blocking probability. For the case of no extra waiting space, a renewal arrival process and an exponential service-time distribution (the GI/M/s/0 model), a heavy-traffic local limit theorem by A. A. Borovkov implies that the blocking depends on the arrival process only through the first two moments of the renewal interval as the offered load increases. Moreover, in this situation Hayward's approximation is asymptotically correct.