A Stochastic Analysis of a Network with Two Levels of Service

In this paper a stochastic model of a call center with a two-level architecture is analyzed. A first-level pool of operators answers calls, identifies, and handles non-urgent calls. A call classified as urgent has to be transferred to specialized operators at the second level. When the operators of the second level are all busy, the operator of first level handling the urgent call is blocked until an operator at the second level is available. Under a scaling assumption, the evolution of the number of urgent calls blocked at level 1 is investigated. It is shown that if the ratio of the number of operators at level 2 and 1 is greater than some threshold, then, essentially, the system operates without congestion, with probability close to 1, no urgent call is blocked after some finite time. Otherwise, we prove that a positive fraction of the operators of the first level are blocked due to the congestion of the second level. Stochastic calculus with Poisson processes, coupling arguments and formulations in terms of Skorokhod problems are the main mathematical tools to establish these convergence results.