513 research outputs found
Block-Structured Supermarket Models
Supermarket models are a class of parallel queueing networks with an adaptive
control scheme that play a key role in the study of resource management of,
such as, computer networks, manufacturing systems and transportation networks.
When the arrival processes are non-Poisson and the service times are
non-exponential, analysis of such a supermarket model is always limited,
interesting, and challenging.
This paper describes a supermarket model with non-Poisson inputs: Markovian
Arrival Processes (MAPs) and with non-exponential service times: Phase-type
(PH) distributions, and provides a generalized matrix-analytic method which is
first combined with the operator semigroup and the mean-field limit. When
discussing such a more general supermarket model, this paper makes some new
results and advances as follows: (1) Providing a detailed probability analysis
for setting up an infinite-dimensional system of differential vector equations
satisfied by the expected fraction vector, where "the invariance of environment
factors" is given as an important result. (2) Introducing the phase-type
structure to the operator semigroup and to the mean-field limit, and a
Lipschitz condition can be obtained by means of a unified matrix-differential
algorithm. (3) The matrix-analytic method is used to compute the fixed point
which leads to performance computation of this system. Finally, we use some
numerical examples to illustrate how the performance measures of this
supermarket model depend on the non-Poisson inputs and on the non-exponential
service times. Thus the results of this paper give new highlight on
understanding influence of non-Poisson inputs and of non-exponential service
times on performance measures of more general supermarket models.Comment: 65 pages; 7 figure
Validity of heavy traffic steady-state approximations in generalized Jackson Networks
We consider a single class open queueing network, also known as a generalized
Jackson network (GJN). A classical result in heavy-traffic theory asserts that
the sequence of normalized queue length processes of the GJN converge weakly to
a reflected Brownian motion (RBM) in the orthant, as the traffic intensity
approaches unity. However, barring simple instances, it is still not known
whether the stationary distribution of RBM provides a valid approximation for
the steady-state of the original network. In this paper we resolve this open
problem by proving that the re-scaled stationary distribution of the GJN
converges to the stationary distribution of the RBM, thus validating a
so-called ``interchange-of-limits'' for this class of networks. Our method of
proof involves a combination of Lyapunov function techniques, strong
approximations and tail probability bounds that yield tightness of the sequence
of stationary distributions of the GJN.Comment: Published at http://dx.doi.org/10.1214/105051605000000638 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Delay analysis of two batch-service queueing models with batch arrivals: Geo(X)/Geo(c)/1
In this paper, we compute the probability generating functions (PGF's) of the customer delay for two batch-service queueing models with batch arrivals. In the first model, the available server starts a new service whenever the system is not empty (without waiting to fill the capacity), while the server waits until he can serve at full capacity in the second model. Moments can then be obtained from these PGF's, through which we study and compare both systems. We pay special attention to the influence of the distribution of the arrival batch sizes. The main observation is that the difference between the two policies depends highly on this distribution. Another conclusion is that the results are considerably different as compared to Bernoulli (single) arrivals, which are frequently considered in the literature. This demonstrates the necessity of modeling the arrivals as batches
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