G. L. Choudhury, A. Mandelbaum, M. I. Reiman, and W. Whitt, "Fluid and Diffusion Limits for Queues in Slowly Changing Random Environments," Stochastic Models, vol. 13, no. 1, pp. 121-146, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that the queue is weakly stable Proof: Given in Appendix C. Remarks: 1) This result strongly supports the fluid models (see [29] that have much lower computational complexity; 2) related asymptotic results in the Markovian framework that justify the fluid approximation were obtained in [9] and [30]; 3) this result can be proved under the weaker assumption that the distribution of the renewal times is intermediately regularly varying [13] Due to the space constraints and the need for introducing new definitions, we avoid stating this result in its most general form. Intuitively, the result ....

G. L. Choudhury, A. Mandelbaum, M. I. Reiman, and W. Whitt, "Fluid and diffusion limits for queues in slowly changing environments," preprint, 1996.

....32 Rigorous mathematical treatment of a single server queue with nearly decomposible Markovian arrivals requires rescaling of the queue size random variable in the form fflQ t=ffl ; then, fflQ t=ffl to a finite random variable as ffl 0. Investigation under this type of scaling has been done in [105, 23]. In Chapter 5 of this dissertation we investigate weakly stable queue with i.i.d. arrivals that are modulated with long tailed (subexponential) renewal process. Without any rescaling, we prove that the queue length distribution is asymptotically equivalent to the corresponding fluid queue. ....

....arrivals that are modulated with long tailed (subexponential) renewal process. Without any rescaling, we prove that the queue length distribution is asymptotically equivalent to the corresponding fluid queue. Similar results with rescaling under the Markovian assumptions have been derived in [105, 23]. 5 10 15 20 25 30 7 6 5 4 3 2 1 EB approx. Buffer size x Log[10,Pr[Q=x] p12=10 ( 5) p12=10 ( 6) p12=10 ( 7) Figure 2.5: Queue length distribution with two dominant decay rates. Numerical Examples. At this point we provide numerical and statistical examples to illustrate ....

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G. L. Choudhury, A. Mandelbaum, M. I. Reiman, and W. Whitt. Fluid and diffusion limits for queues in slowly changing environments. preprint, 1996.

....tail for the lowpriority buffer occupancy, which is used in an admission control scheme for such a multiplexer. Since our model is not a fluid model, the results in [18] do not directly apply. It may be possible to connect our results to the results in [18] using appropriate fluid limits as in [12]. However, we do not pursue this avenue of research in this paper. Using the results in [18] Presti et al. 30] have obtained approximations for the more general case of GPS service discipline. The rest of this paper is organized as follows. In Section 2, we introduce MAPs and presents some ....

.... a simple model which provides us with a justification of the fact that one can indeed expect asymptotics of the form (27) This example proving the asymptotic form is not for MAP processes but it is for fluid models which are closely related to MAPs when the service times are small, as shown in [12]. However, it should be noted that non preemptive and preemptive priority are the same in fluid models, and thus, we lose the detailed structure of the MAP queueing system when considering the fluid model. Thus, this example is only suggests the asymptotics, but is not a conlcusive proof that the ....

G. L. Choudhury, A. Mandelbaum, M. Reiman, and W. Whitt. Fluid and diffusion limits for queues in slowly changing environments. Stochastic Models, 13(1), 1996.

....IP[Q n x] IP[Q f n x] 1: Proof: Given in Appendix C. Remarks: i) This result strongly supports the fluid models (see [29] that have much lower computational complexity; ii) related asymptotic results in the Markovian framework that justify the fluid approximation were obtained in [9] [30]. iii) This result can be proved under the weaker assumption that the distribution of the renewal times is intermediately regularly varying [13] Due to the space constraints, and the need for introducing new definitions, we avoid stating this result in its most general form. Intuitively the ....

G. L. Choudhury, A. Mandelbaum, M. I. Reiman, and W. Whitt, "Fluid and diffusion limits for queues in slowly changing environments, " preprint, 1996.

....tail for the lowpriority buffer occupancy, which is used in an admission control scheme for such a multiplexer. Since our model is not a fluid model, the results in [19] do not directly apply. It may be possible to connect our results to the results in [19] using appropriate fluid limits as in [12]. However, we do not pursue this avenue of research in this paper. Using the results in [19] Presti et al. 31] have obtained approximations for the more general case of GPS service discipline. The rest of this paper is organized as follows. In Section 2, we introduce MAPs and presents some known ....

.... a simple model which provides us with a justification of the fact that one can indeed expect asymptotics of the form (26) This example proving the asymptotic form is not for MAP processes but it is for fluid models which are closely related to MAPs when the service times are small, as shown in [12]. Motivated by this example and the results in [4] we then numerically study priority multiplexing with MAP arrival processes to demonstrate the form of the asymptotics. 4.1 Fluid Model Example As mentioned earlier, non exponential tails for waiting times of low priority waiting times was shown ....

G. L. Choudhury, A. Mandelbaum, M. Reiman, and W. Whitt. Fluid and diffusion limits for queues in slowly changing environments. Stochastic Models, 13(1), 1996.

....IP[Q n x] IP[Q f n x] 1: Proof: Given in Appendix C. Remarks: i) This result strongly supports the fluid models (see [29] that have much lower computational complexity; ii) related asymptotic results in the Markovian framework that justify the fluid approximation were obtained in [9] [30]. iii) This result can be proved under the weaker assumption that the distribution of the renewal times is intermediately regularly varying [13] Due to the space constraints, and the need for introducing new definitions, we avoid stating this result in its most general form. Intuitively the ....

G. L. Choudhury, A. Mandelbaum, M. I. Reiman, and W. Whitt, "Fluid and diffusion limits for queues in slowly changing environments, " preprint, 1996.

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G. L. Choudhury, A. Mandelbaum, M. I. Reiman, and W. Whitt, "Fluid and Diffusion Limits for Queues in Slowly Changing Random Environments," Stochastic Models, vol. 13, no. 1, pp. 121-146, 1997.

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