528 research outputs found
Human activity modeling and Barabasi's queueing systems
It has been shown by A.-L. Barabasi that the priority based scheduling rules
in single stage queuing systems (QS) generates fat tail behavior for the tasks
waiting time distributions (WTD). Such fat tails are due to the waiting times
of very low priority tasks which stay unserved almost forever as the task
priority indices (PI) are "frozen in time" (i.e. a task priority is assigned
once for all to each incoming task). Relaxing the "frozen in time" assumption,
this paper studies the new dynamic behavior expected when the priority of each
incoming tasks is time-dependent (i.e. "aging mechanisms" are allowed). For two
class of models, namely 1) a population type model with an age structure and 2)
a QS with deadlines assigned to the incoming tasks which is operated under the
"earliest-deadline-first" policy, we are able to analytically extract some
relevant characteristics of the the tasks waiting time distribution. As the
aging mechanism ultimately assign high priority to any long waiting tasks, fat
tails in the WTD cannot find their origin in the scheduling rule alone thus
showing a fundamental difference between the present and the A.-L. Barabasi's
class of models.Comment: 16 pages, 2 figure
Erlang arrivals joining the shorter queue
We consider a system in which customers join upon arrival the shortest of two single-server queues. The interarrival times between customers are Erlang distributed and the service times of both servers are exponentially distributed. Under these assumptions, this system gives rise to a Markov chain on a multi-layered quarter plane. For this Markov chain we derive the equilibrium distribution using the compensation approach. The obtained expression for the equilibrium distribution matches and re??nes heavy-traffic approximations and tail asymptotics obtained earlier in the literature. Keywords: random walks in the quarter plane, compensation approach, join the shorter queue, tail asymptotic
Heavy-tailed Distributions In Stochastic Dynamical Models
Heavy-tailed distributions are found throughout many naturally occurring
phenomena. We have reviewed the models of stochastic dynamics that lead to
heavy-tailed distributions (and power law distributions, in particular)
including the multiplicative noise models, the models subjected to the
Degree-Mass-Action principle (the generalized preferential attachment
principle), the intermittent behavior occurring in complex physical systems
near a bifurcation point, queuing systems, and the models of Self-organized
criticality. Heavy-tailed distributions appear in them as the emergent
phenomena sensitive for coupling rules essential for the entire dynamics
EUROPEAN CONFERENCE ON QUEUEING THEORY 2016
International audienceThis booklet contains the proceedings of the second European Conference in Queueing Theory (ECQT) that was held from the 18th to the 20th of July 2016 at the engineering school ENSEEIHT, Toulouse, France. ECQT is a biannual event where scientists and technicians in queueing theory and related areas get together to promote research, encourage interaction and exchange ideas. The spirit of the conference is to be a queueing event organized from within Europe, but open to participants from all over the world. The technical program of the 2016 edition consisted of 112 presentations organized in 29 sessions covering all trends in queueing theory, including the development of the theory, methodology advances, computational aspects and applications. Another exciting feature of ECQT2016 was the institution of the Takács Award for outstanding PhD thesis on "Queueing Theory and its Applications"
Exponential Tail Bounds on Queues: A Confluence of Non-Asymptotic Heavy Traffic and Large Deviations
In general, obtaining the exact steady-state distribution of queue lengths is
not feasible. Therefore, we establish bounds for the tail probabilities of
queue lengths. Specifically, we examine queueing systems under Heavy-Traffic
(HT) conditions and provide exponentially decaying bounds for the probability
, where is the HT parameter denoting how
far the load is from the maximum allowed load. Our bounds are not limited to
asymptotic cases and are applicable even for finite values of , and
they get sharper as . Consequently, we derive non-asymptotic
convergence rates for the tail probabilities. Unlike other approaches such as
moment bounds based on drift arguments and bounds on Wasserstein distance using
Stein's method, our method yields sharper tail bounds. Furthermore, our results
offer bounds on the exponential rate of decay of the tail, given by
for any finite value of .
These can be interpreted as non-asymptotic versions of Large Deviation (LD)
results.
We demonstrate our approach by presenting tail bounds for: (i) a continuous
time Join-the-shortest queue (JSQ) load balancing system, (ii) a discrete time
single-server queue and (iii) an queue. We not only bridge the gap
between classical-HT and LD regimes but also explore the large system HT
regimes for JSQ and systems. In these regimes, both the system size and
the system load increase simultaneously. Our results also close a gap in the
existing literature on the limiting distribution of JSQ in the super-NDS
(a.k.a. super slowdown) regime. This contribution is of an independent
interest. Here, a key ingredient is a more refined characterization of state
space collapse for JSQ system, achieved by using an exponential Lyapunov
function designed to approximate the norm.Comment: 37 pages, 1 figur
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