2019/03/16 by Rami Atar, Amarjit Budhiraja, Atar, Rami +5 · 1 citation
Business, Management and Accounting · Decision Sciences · Economics, Econometrics and Finance · #60F10 #60J27 #60K25 #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1903.06870
openalex publication_date 2019/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For the M/M/1+M model at the law-of-large-numbers scale, the long run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: The decay rate of the probability that the long run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.