470 research outputs found
On Sojourn Times in the Finite Capacity Queue with Processor Sharing
We consider a processor shared queue that can accommodate at most a
finite number of customers. We give an exact expression for the sojourn
time distribution in the finite capacity model, in terms of a Laplace
transform. We then give the tail behavior, for the limit , by
locating the dominant singularity of the Laplace transform.Comment: 10 page
Decomposing the queue length distribution of processor-sharing models into queue lengths of permanent customer queues
We obtain a decomposition result for the steady state queue length distribution in egalitarian processor-sharing (PS) models. In particular, for an egalitarian PS queue with customer classes, we show that the marginal queue length distribution for class factorizes over the number of other customer types. The factorizing coefficients equal the queue length probabilities of a PS queue for type in isolation, in which the customers of the other types reside \textit{ permanently} in the system. Similarly, the (conditional) mean sojourn time for class can be obtained by conditioning on the number of permanent customers of the other types. The decomposition result implies linear relations between the marginal queue length probabilities, which also hold for other PS models such as the egalitarian processor-sharing models with state-dependent system capacity that only depends on the total number of customers in the system. Based on the exact decomposition result for egalitarian PS queues, we propose a similar decomposition for discriminatory processor-sharing (DPS) models, and numerically show that the approximation is accurate for moderate differences in service weights. \u
On Spectral Properties of Finite Population Processor Shared Queues
We consider sojourn or response times in processor-shared queues that have a
finite population of potential users. Computing the response time of a tagged
customer involves solving a finite system of linear ODEs. Writing the system in
matrix form, we study the eigenvectors and eigenvalues in the limit as the size
of the matrix becomes large. This corresponds to finite population models where
the total population is . Using asymptotic methods we reduce the
eigenvalue problem to that of a standard differential equation, such as the
Hermite equation. The dominant eigenvalue leads to the tail of a customer's
sojourn time distribution.Comment: 28 pages, 7 figures and 5 table
Randomized Assignment of Jobs to Servers in Heterogeneous Clusters of Shared Servers for Low Delay
We consider the job assignment problem in a multi-server system consisting of
parallel processor sharing servers, categorized into ()
different types according to their processing capacity or speed. Jobs of random
sizes arrive at the system according to a Poisson process with rate . Upon each arrival, a small number of servers from each type is
sampled uniformly at random. The job is then assigned to one of the sampled
servers based on a selection rule. We propose two schemes, each corresponding
to a specific selection rule that aims at reducing the mean sojourn time of
jobs in the system.
We first show that both methods achieve the maximal stability region. We then
analyze the system operating under the proposed schemes as which
corresponds to the mean field. Our results show that asymptotic independence
among servers holds even when is finite and exchangeability holds only
within servers of the same type. We further establish the existence and
uniqueness of stationary solution of the mean field and show that the tail
distribution of server occupancy decays doubly exponentially for each server
type. When the estimates of arrival rates are not available, the proposed
schemes offer simpler alternatives to achieving lower mean sojourn time of
jobs, as shown by our numerical studies
On the Sojourn Time Distribution in a Finite Population Markovian Processor Sharing Queue
We consider a finite population processor-sharing (PS) queue, with Markovian
arrivals and an exponential server. Such a queue can model an interactive
computer system consisting of a bank of terminals in series with a central
processing unit (CPU). For systems with a large population and a
commensurately rapid service rate, or infrequent arrivals, we obtain various
asymptotic results. We analyze the conditional sojourn time distribution of a
tagged customer, conditioned on the number of others in the system at the
tagged customer's arrival instant, and also the unconditional distribution. The
asymptotics are obtained by a combination of singular perturbation methods and
spectral methods. We consider several space/time scales and parameter ranges,
which lead to different asymptotic behaviors. We also identify precisely when
the finite population model can be approximated by the standard infinite
population -PS queue.Comment: 60 pages and 3 figure
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