Engineering solution of a basic call-center model

An algorithm is developed to rapidly compute approximations for all the standard steady-state performance measures in the basic call-center queueing model M/GI/s/r + GI, which has a Poisson arrival process, independent and identically distributed (IID) service times with a general distribution, s servers, r extra waiting spaces and IID customer abandonment times with a general distribution. Empirical studies of call centers indicate that the service-time and abandon-time distributions often are not nearly exponential, so that it is important to go beyond the Markovian M/M/s/r + M special case, but the general service-time and abandon-time distributions make the realistic model very difficult to analyze directly. The proposed algorithm is based on an approximation by an appropriate Markovian M/M/s/r + M(n) queueing model, where M(n) denotes state-dependent abandonment rates. After making an additional approximation, steady-state waiting-time distributions are characterized via their Laplace transforms. Then the approximate distributions are computed by numerically inverting the transforms. Simulation experiments show that the approximation is quite accurate. The overall algorithm can be applied to determine desired staffing levels, e.g., the minimum number of servers needed to guarantee that, first, the abandonment rate is below any specified target value and, second, that the conditional probability that an arriving customer will be served within a specified deadline, given that the customer eventually will be served, is at least a specified target value.