25 research outputs found
Modulated Branching Processes, Origins of Power Laws and Queueing Duality
Power law distributions have been repeatedly observed in a wide variety of
socioeconomic, biological and technological areas. In many of the observations,
e.g., city populations and sizes of living organisms, the objects of interest
evolve due to the replication of their many independent components, e.g.,
births-deaths of individuals and replications of cells. Furthermore, the rates
of the replication are often controlled by exogenous parameters causing periods
of expansion and contraction, e.g., baby booms and busts, economic booms and
recessions, etc. In addition, the sizes of these objects often have reflective
lower boundaries, e.g., cities do not fall bellow a certain size, low income
individuals are subsidized by the government, companies are protected by
bankruptcy laws, etc.
Hence, it is natural to propose reflected modulated branching processes as
generic models for many of the preceding observations. Indeed, our main results
show that the proposed mathematical models result in power law distributions
under quite general polynomial Gartner-Ellis conditions, the generality of
which could explain the ubiquitous nature of power law distributions. In
addition, on a logarithmic scale, we establish an asymptotic equivalence
between the reflected branching processes and the corresponding multiplicative
ones. The latter, as recognized by Goldie (1991), is known to be dual to
queueing/additive processes. We emphasize this duality further in the
generality of stationary and ergodic processes.Comment: 36 pages, 2 figures; added references; a new theorem in Subsection
4.
Characterizing Heavy-Tailed Distributions Induced by Retransmissions
Consider a generic data unit of random size L that needs to be transmitted
over a channel of unit capacity. The channel availability dynamics is modeled
as an i.i.d. sequence {A, A_i},i>0 that is independent of L. During each period
of time that the channel becomes available, say A_i, we attempt to transmit the
data unit. If L<A_i, the transmission was considered successful; otherwise, we
wait for the next available period and attempt to retransmit the data from the
beginning. We investigate the asymptotic properties of the number of
retransmissions N and the total transmission time T until the data is
successfully transmitted. In the context of studying the completion times in
systems with failures where jobs restart from the beginning, it was shown that
this model results in power law and, in general, heavy-tailed delays. The main
objective of this paper is to uncover the detailed structure of this class of
heavy-tailed distributions induced by retransmissions. More precisely, we study
how the functional dependence between P[L>x] and P[A>x] impacts the
distributions of N and T. In particular, we discover several functional
criticality points that separate classes of different functional behavior of
the distribution of N. We also discuss the engineering implications of our
results on communication networks since retransmission strategy is a
fundamental component of the existing network protocols on all communication
layers, from the physical to the application one.Comment: 39 pages, 2 figure
Information Ranking and Power Laws on Trees
We study the situations when the solution to a weighted stochastic recursion
has a power law tail. To this end, we develop two complementary approaches, the
first one extends Goldie's (1991) implicit renewal theorem to cover recursions
on trees; and the second one is based on a direct sample path large deviations
analysis of weighted recursive random sums. We believe that these methods may
be of independent interest in the analysis of more general weighted branching
processes as well as in the analysis of algorithms
Heavy-Tailed Limits for Medium Size Jobs and Comparison Scheduling
We study the conditional sojourn time distributions of processor sharing
(PS), foreground background processor sharing (FBPS) and shortest remaining
processing time first (SRPT) scheduling disciplines on an event where the job
size of a customer arriving in stationarity is smaller than exactly k>=0 out of
the preceding m>=k arrivals. Then, conditioning on the preceding event, the
sojourn time distribution of this newly arriving customer behaves
asymptotically the same as if the customer were served in isolation with a
server of rate (1-\rho)/(k+1) for PS/FBPS, and (1-\rho) for SRPT, respectively,
where \rho is the traffic intensity. Hence, the introduced notion of
conditional limits allows us to distinguish the asymptotic performance of the
studied schedulers by showing that SRPT exhibits considerably better asymptotic
behavior for relatively smaller jobs than PS/FBPS.
Inspired by the preceding results, we propose an approximation to the SRPT
discipline based on a novel adaptive job grouping mechanism that uses relative
size comparison of a newly arriving job to the preceding m arrivals.
Specifically, if the newly arriving job is smaller than k and larger than m-k
of the previous m jobs, it is routed into class k. Then, the classes of smaller
jobs are served with higher priorities using the static priority scheduling.
The good performance of this mechanism, even for a small number of classes m+1,
is demonstrated using the asymptotic queueing analysis under the heavy-tailed
job requirements. We also discuss refinements of the comparison grouping
mechanism that improve the accuracy of job classification at the expense of a
small additional complexity.Comment: 26 pages, 2 figure
Proceedings of the Sixth International Workshop on Web Caching and Content Distribution
OVERVIEW
The International Web Content Caching and Distribution Workshop (WCW) is a premiere technical meeting for researchers and practitioners interested in all aspects of content caching, distribution and delivery on the Internet. The 2001 WCW meeting was held on the Boston University Campus. Building on the successes of the five previous WCW meetings, WCW01 featured a strong technical program and record participation from leading researchers and practitioners in the field. This report includes all the technical papers presented at WCW'01.
Note: Proceedings of WCW'01 are published by Elsevier. Hardcopies of these proceedings can be purchased through the workshop organizers. As a service to the community, electronic copies of all WCW'01 papers are accessible through Technical Report BUCSāTRā2001ā017, available from the Boston University Computer Science Technical Report Archives at http://www.cs.bu.edu/techreps. [Ed.note: URL outdated. Use http://www.bu.edu/cs/research/technical-reports/ or http://hdl.handle.net/2144/1455 in this repository to access the reports.]Cisco Systems; InfoLibria; Measurement Factory Inc; Voler