Stochastic Models

Conti, Pier Luigi

doi: 10.1081/STM-120016441pmid: N/A

In this paper nonparametric statistical analysis of a discrete-time queueing system is considered. Estimation of performance measures of the system is studied. The attention is first focused on the estimation of the waiting time probability distribution, as well as of functionals of interest (mean waiting time, variance of the waiting time, etc.). The approach is based on the estimation of the corresponding generating function. Attention is paid to the estimation of the probability of a “long delay”, in view of its importance for applications. Results for possibly unstable models are also obtained. Finally, an application to ATM teletraffic data is provided.

Takada, Hiroyuki; Miyazawa, Masakiyo

doi: 10.1081/STM-120016443pmid: N/A

We consider a preemptive priority fluid queue with two buffers for continuous fluid and batch fluid inputs. Those two types of fluids are governed by a Markov chain with a finite state space, called a background process, and the continuous fluid is preemptively processed over the batch fluid. The stationary joint distribution of the two buffer contents and the background state is obtained in terms of matrix transforms. Numerical computation algorithms are presented for various moments of the two buffer contents.

Perry, D.; Stadje, W.; Zacks, S.

doi: 10.1081/STM-120016444pmid: N/A

We derive a new formula for the probability that a compound Poisson process with positive jumps hits a lower straight line before it crosses a parallel upper line. This yields a new approach to determine the distribution of the cycle maximum of the queue. Moreover, we express the hitting probabilities in terms of the corresponding ruin probabilities.

Zheng, Shen; Seila, Andrew F.; Sriram1, T. N.

doi: 10.1081/STM-120016445pmid: N/A

In this paper, we consider the problem of optimally allocating a sampling budget between the interarrival times and service times for the stationary queue and stationary queue with the goal of minimizing the mean squared error of one of four system performance measures. The natural estimator of the mean waiting time, obtained by substituting estimators for the arrival rate and service rate in the formula for the mean waiting time, has infinite mean squared error and cannot be used. Using an alternative estimator, we propose a two-stage sampling plan and show that this plan is asymptotically risk-efficient in the sense that the mean squared error of the resulting estimator converges to the minimum mean squared error as the sampling budget gets large. Asymptotic normality and other properties of the two-stage estimator are also proved. Simulation results for the system are given to support the theoretical results. **Supported by grant MDA904-00-1-0025 from NSA.

Breuer, Lothar

doi: 10.1081/STM-120016446pmid: N/A

The present paper contains an analysis of the MAP/G/1 queue with last come first served (LCFS) preemptive repeat service discipline and Lebesgue-dominated service time distribution. The transient distribution is given in terms of a recursive formula. The stationary distribution as well as the stability condition are obtained by means of Markov renewal theory via a QBD representation of the embedded Markov chain at departures and arrivals.

Lomonosov, Michael

doi: 10.1081/STM-120016447pmid: N/A

Reliability of a consecutive r-of-s-of-n: F system with identical elements is expressed through the n-th degree of a certain substochastic matrix depending only on r, s and the elements failure probability. As a consequence, a simple approximate formula for the reliability is derived.