3 research outputs found
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
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