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We consider level crossing for the difference of independent renewal processes. Second-order expansions for the distribution function of the crossing time of level n are found, as n - oo. As a by-product several other results on the difference process are found. The expected minimum of the difference process appears to play an important role in the analysis. This makes this problem essentially harder than the level crossing for the sum process which was studied earlier.
Quasi-Birth-and-Death (QBD) processes with infinite ``phase spaces'' can exhibit unusual and interesting behaviour. One of the simplest examples of such a process is the two-node tandem Jackson network, with the ``phase" giving the state of the first queue and the ``level" giving the state of the second queue.
Quasi-birth-and-death (QBD) processes with infinite “phase spaces” can exhibit unusual and interesting behavior. One of the simplest examples of such a process is the two-node tandem Jackson network, with the “phase” giving the state of the first queue and the “level” giving the state of the second queue. In this paper, we undertake an extensive analysis of the properties of this QBD. In particular, we investigate the spectral properties of Neuts’s R-matrix and show that the decay rate of the stationary distribution of the “level” process is not always equal to the convergence norm of R. In fact, we show that we can obtain any decay rate from a certain range by controlling only the transition structure at level zero, which is independent of R. We also consider the sequence of tandem queues that is constructed by restricting the wai...
In this paper we propose a fast adaptive importance sampling method for the efficient simulation of buffer overflow probabilities in queueing networks. The method comprises three stages. First we estimate the minimum cross-entropy tilting parameter for a small buffer level; next, we use this as a starting value for the estimation of the optimal tilting parameter for the actual (large) buffer level; finally, the tilting parameter just found is used to estimate the overflow probability of interest. We recognize three distinct properties of the method which together explain why the method works well; we conjecture that they hold for quite general queueing networks. Numerical results support this conjecture and demonstrate the high efficiency of the proposed algorithm.
In this paper, we propose a fast adaptive importance sampling method for the efficient simulation of buffer overflow probabilities in queueing networks. The method comprises three stages. First, we estimate the minimum cross-entropy tilting parameter for a small buffer level; next, we use this as a starting value for the estimation of the optimal tilting parameter for the actual (large) buffer level. Finally, the tilting parameter just found is used to estimate the overflow probability of interest. We study various properties of the method in more detail for the M/M/1 queue and conjecture that similar properties also hold for quite general queueing networks. Numerical results support this conjecture and demonstrate the high efficiency of the proposed algorithm.
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