912 research outputs found
Analysis of loss networks with routing
This paper analyzes stochastic networks consisting of finite capacity nodes
with different classes of requests which move according to some routing policy.
The Markov processes describing these networks do not, in general, have
reversibility properties, so the explicit expression of their invariant
distribution is not known. Kelly's limiting regime is considered: the arrival
rates of calls as well as the capacities of the nodes are proportional to a
factor going to infinity. It is proved that, in limit, the associated rescaled
Markov process converges to a deterministic dynamical system with a unique
equilibrium point characterized by a nonstandard fixed point equation.Comment: Published at http://dx.doi.org/10.1214/105051606000000466 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Exponential penalty function control of loss networks
We introduce penalty-function-based admission control policies to
approximately maximize the expected reward rate in a loss network. These
control policies are easy to implement and perform well both in the transient
period as well as in steady state. A major advantage of the penalty approach is
that it avoids solving the associated dynamic program. However, a disadvantage
of this approach is that it requires the capacity requested by individual
requests to be sufficiently small compared to total available capacity. We
first solve a related deterministic linear program (LP) and then translate an
optimal solution of the LP into an admission control policy for the loss
network via an exponential penalty function. We show that the penalty policy is
a target-tracking policy--it performs well because the optimal solution of the
LP is a good target. We demonstrate that the penalty approach can be extended
to track arbitrarily defined target sets. Results from preliminary simulation
studies are included.Comment: Published at http://dx.doi.org/10.1214/105051604000000936 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Efficient estimation of blocking probabilities in non-stationary loss networks
This paper considers estimation of blocking probabilities in a nonstationary loss network. Invoking the so called MOL (Modified Offered Load) approximation, the problem is transformed into one requiring the solution of blocking probabilities in a sequence of stationary loss networks with time varying loads. To estimate the blocking probabilities Monte Carlo simulation is used and to increase the efficiency of the simulation, we develop a likelihood ratio method that enables samples drawn at a one time point to be used at later time points. This reduces the need to draw new samples every time independently as a new time point is considered, thus giving substantial savings in the computational effort of evaluating time dependent blocking probabilities. The accuracy of the method is analyzed by using Taylor series approximations of the variance indicating the direct dependence of the accuracy on the rate of change of the actual load. Finally, three practical applications of the method are provided along with numerical examples to demonstrate the efficiency of the method
Perfect simulation for interacting point processes, loss networks and Ising models
We present a perfect simulation algorithm for measures that are absolutely
continuous with respect to some Poisson process and can be obtained as
invariant measures of birth-and-death processes. Examples include area- and
perimeter-interacting point processes (with stochastic grains), invariant
measures of loss networks, and the Ising contour and random cluster models. The
algorithm does not involve couplings of the process with different initial
conditions and it is not tied up to monotonicity requirements. Furthermore, it
directly provides perfect samples of finite windows of the infinite-volume
measure, subjected to time and space ``user-impatience bias''. The algorithm is
based on a two-step procedure: (i) a perfect-simulation scheme for a (finite
and random) relevant portion of a (space-time) marked Poisson processes (free
birth-and-death process, free loss networks), and (ii) a ``cleaning'' algorithm
that trims out this process according to the interaction rules of the target
process. The first step involves the perfect generation of ``ancestors'' of a
given object, that is of predecessors that may have an influence on the
birth-rate under the target process. The second step, and hence the whole
procedure, is feasible if these ``ancestors'' form a finite set with
probability one. We present a sufficiency criteria for this condition, based on
the absence of infinite clusters for an associated (backwards) oriented
percolation model.Comment: Revised version after referee of SPA: 39 page
Monotonicity and error bounds for networks of Erlang loss queues
Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably\ud
(i) classical circuit switch telephone networks (loss networks) and\ud
(ii) present-day wireless mobile networks.\ud
\ud
Performance measures of interest such as loss probabilities or throughputs can be obtained from the steady state distribution. However, while this steady state distribution has a closed product form expression in the first case (loss networks), it has not in the second case due to blocked (and lost) handovers. Product form approximations are therefore suggested. These approximations are obtained by a combined modification of both the state space (by a hyper cubic expansion) and the transition rates (by extra redial rates). It will be shown that these product form approximations lead to\ud
\ud
- secure upper bounds for loss probabilities and\ud
- analytic error bounds for the accuracy of the approximation for various performance measures.\ud
\ud
The proofs of these results rely upon both monotonicity results and an analytic error bound method as based on Markov reward theory. This combination and its technicalities are of interest by themselves. The technical conditions are worked out and verified for two specific applications:\ud
\ud
- pure loss networks as under (i)\ud
- GSM-networks with fixed channel allocation as under (ii).\ud
\ud
The results are of practical interest for computational simplifications and, particularly, to guarantee blocking probabilities not to exceed a given threshold such as for network dimensioning.\u
Monotonicity and error bounds for networks of Erlang loss queues
Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably are (i) classical circuit switch telephone networks (loss networks) and (ii) present-day wireless mobile networks. Performance measures of interest such as loss probabilities or throughputs can be obtained from the steady state distribution. However, while this steady state distribution has a closed product form expression in the first case (loss networks), it does not have one in the second case due to blocked (and lost) handovers. Product form approximations are therefore suggested. These approximations are obtained by a combined modification of both the state space (by a hypercubic expansion) and the transition rates (by extra redial rates). It will be shown that these product form approximations lead to (1) upper bounds for loss probabilities and \ud
(2) analytic error bounds for the accuracy of the approximation for various performance measures.\ud
The proofs of these results rely upon both monotonicity results and an analytic error bound method as based on Markov reward theory. This combination and its technicalities are of interest by themselves. The technical conditions are worked out and verified for two specific applications:\ud
(1)âą pure loss networks as under (2)âą GSM networks with fixed channel allocation as under.\ud
The results are of practical interest for computational simplifications and, particularly, to guarantee that blocking probabilities do not exceed a given threshold such as for network dimensioning
A probabilistic threshold-based bandwidth sharing policy for wireless multirate loss networks
We propose a probabilistic bandwidth sharing policy, based on the threshold (TH) policy, for a single cell of fixed capacity in a homogeneous wireless cellular network. The cell accommodates random input-traffic originated from K service-classes. We distinguish call requests to new and handover, and therefore, the cell supports 2K types of arrivals. If the number of in-service calls (new or handover) of a service-class exceeds a threshold (different for new and handover calls of a service-class), a new or handover arriving call of the same service-class is not always blocked, as it happens in the TH policy, but it is accepted in the system with a predefined state-dependent probability. The cell is analyzed as a multirate loss system, via a reversible continuous-time Markov chain, which leads to a product form solution (PFS) for the steady state distribution. Thanks to the PFS, the calculation of performance measures is accurate, but complex. To reduce the computational complexity, we determine performance measures via a convolution algorithm
Efficient Simulation of Large-Scale Loss Networks
Recently Rajasekaran and Ross [1] presented an algorithm that takes an expected 0(1) time to generate a nonuniform discrete random variate. In this paper we discuss how this algorithm can be employed in the efficient simulation of large-scale telephone networks. In a simulation based upon a standard event-list approach, the generation of a new event in the systems take 0(log n) time. With this new algorithm, event generation becomes an 0(1) process, and simulation times for large networks can be reduced
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