On the M(n)/M(n)/s queue with impatient calls

The paper is concerned with the analysis of an s-server queueing system wherein the calls may leave the system due to impatience. The individual maximal waiting times are assumed to be i.i.d. and arbitrarily distributed. The arrival and cumulative service rates may depend on the number n of calls in the system, but the service rate is assumed to be constant for n>s. For this system, denoted by M(n)/M(n)/s+GI, we derive a system of integral equations for the vector of the residual maximal waiting times of the waiting calls and their original maximal waiting times. By solving these equations explicitly we obtain the stability condition and for the steady state of the system, the occupancy distribution and various waiting time distributions. The results are also new for special cases analyzed in earlier papers. As an application of the M(n)/M(n)/s+GI system we give a performance analysis of an automatic call distributor system (ACD system) of finite capacity with outbound and impatient inbound calls; numerical results are given for the case of maximal waiting times as the minimum of constant and exponentially distributed times.