Some new results about an old model: The M/G/1 queue


Stochastic networks are an integral part of everyone’s daily life. For example, traffic jams, waiting lines in a supermarket, communication systems and etc. are all examples of such networks. One classic model in this direction is the  M/G/1 queue, i.e., assume that there is a single server with customers who arrive according to a Poisson process. There is an incredible amount of existing

about this queue, but still there are some open questions that deserve answers: 1. Externalities: Assume that the service discipline is first-comes, first-served and at time t = 0, the existing workload in the system is v ≥ 0 minutes. What would be the impact (on the waiting times of the other customers) of an additional customer who arrives at time t = 0 and requests x > 0 minutes of service? It turns out that the aggregate impact yields a stochastic process which we call the externalities process. We discover its properties. Furthermore, we give a similar analysis also for the analogue

process when the service discipline is last-comes, first-served preemptiveresume. 2. Moments of functionals of the M/G/1 workload: There are many functionals of the workload that appear in literature, e.g., the length of a busy period, the number of customers who arrive during a busy period, the total waiting-time of customers who arrive during a busy period, the area

below the workload process during a busy period. In this work, we observe that all of these functionals are special cases of a random measure which is determined by the workload process. Then, this fact is exploited in order to derive a general recursive formula for the high-order joint moments of these functionals. Furthermore, there is an interesting relation to certain

Poisson’s equation in a corresponding Markov process. 

Supervisors Michel Mandjes (UvA)

Royi Jacobovic


University of Amsterdam (UvA)




This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement Grant Agreement No 101034253.