6 research outputs found
Convergence to Equilibrium States for Fluid Models of Many-server Queues with Abandonment
Fluid models have become an important tool for the study of many-server
queues with general service and patience time distributions. The equilibrium
state of a fluid model has been revealed by Whitt (2006) and shown to yield
reasonable approximations to the steady state of the original stochastic
systems. However, it remains an open question whether the solution to a fluid
model converges to the equilibrium state and under what condition. We show in
this paper that the convergence holds under a mild condition. Our method builds
on the framework of measure-valued processes developed in Zhang (2013), which
keeps track of the remaining patience and service times
The impact of reneging in processor sharing queues
We investigate an overloaded processor sharing queue with renewal arrivals and generally distributed service times. Impatient customers may abandon the queue, or renege, before completing service. The random time representing a customer’s patience has a general distribution and may be dependent on his initial service time requirement. We propose a scaling procedure that gives rise to a fluid model, with nontrivial yet tractable steady state behavior. This fluid model captures many essential features of the underlying stochastic model, and we use it to analyze the impact of impatience in processor sharing queues. We show that this impact can be substantial compared with FCFS, and we propose a simple admission control policy to overcome these negative impacts
Heavy traffic limit for a processor sharing queue with soft deadlines
This paper considers a GI/GI/1 processor sharing queue in which jobs have
soft deadlines. At each point in time, the collection of residual service times
and deadlines is modeled using a random counting measure on the right
half-plane. The limit of this measure valued process is obtained under
diffusion scaling and heavy traffic conditions and is characterized as a
deterministic function of the limiting queue length process. As special cases,
one obtains diffusion approximations for the lead time profile and the profile
of times in queue. One also obtains a snapshot principle for sojourn times.Comment: Published at http://dx.doi.org/10.1214/105051607000000014 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Diffusion approximation for a processor sharing queue in heavy traffic
Consider a single server queue with renewal arrivals and i.i.d. service times
in which the server operates under a processor sharing service discipline. To
describe the evolution of this system, we use a measure valued process that
keeps track of the residual service times of all jobs in the system at any
given time. From this measure valued process, one can recover the traditional
performance processes, including queue length and workload. We show that under
mild assumptions, including standard heavy traffic assumptions, the (suitably
rescaled) measure valued processes corresponding to a sequence of processor
sharing queues converge in distribution to a measure valued diffusion process.
The limiting process is characterized as the image under an appropriate lifting
map, of a one-dimensional reflected Brownian motion. As an immediate
consequence, one obtains a diffusion approximation for the queue length process
of a processor sharing queue