Prof. Haya Kaspi (Israeli Institute of Technology, Haifa, Israel)

Title: Measure Valued Processes in the Asymptotic Approximation of Many Servers Queues

Abstract: The lecture focuses on queueing systems with many servers serving in parallel, where the arrival process into the system is a quite general counting process, the service times of various customers are i.i.d. random variables with general distribution and are independent of the arrival process, and the number of servers N is large. A primary motivation for studying such systems is that they arise as models for telephone call centers. While most research to date on such systems assumes that the service time is exponentially distributed, a fact which makes the number of customers in the system a Markov process, statistical analysis of large service stations performed recently have shown that the service times are typically non exponential but rather Lognormal or Weibul distributed. An extension of the exponentially distributed service times to phase type service distribution by Puhalski and Reiman, lead to a Markov process with a finite dimensional state descriptor. The general service time assumption lead us to represent the Markovian dynamics of the system in terms of a process that describes the total number of customers in the system, as well as a measured valued process that keeps track of the ”ages” (the time in service) of the various customers in service. In the call center application, it is natural to consider an asymptotic approximation in the limit, as the number of servers and the arrival rate go to infinity and the mean traffic intensity increases to 1, in such a way that the limiting probability of a positive queue is strictly between 0 and 1. This asymptotic regime is often referred to as the QED (Quality and Efficiency Driven) regime that was introduced in the seminal paper by Halfin and Whitt in 1981, and dealt with such systems with exponentially distributed service time. Fluid (first order)and diffusion (second order) approximations of the pair consisting of the number of customers in the system and the measure valued process described above, in heavy traffic as N ! 1 will be discussed in this lecture.