Karr, A.F. #1993#. Probability. New York: Springer-Verlag.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....g ) 48) It is easy to see that implies D g (q n ) lim dx g(x) x = 0 (49) see equation (24) in the proof of Lemma 6 in [20] i.e. q n (X) converges to X in mean square (here X has pdf g) This implies that P P g in the sense of weak convergence (see, e.g, Theorem 4. 2 of [31]) Furthermore, since by assumption there is a finite M such that f(x) g(x) M , D f (q n ) lim dx g(x) f(x) g(x) x M lim D g (q n ) 0 and hence by the same argument P P f (weak convergence) From [9] relative entropy is lower semicontinuous with respect to weak ....

A. N. Shiryaev, Probability. New York: Springer-Verlag, 2nd ed., 1996. 29

....below, s are considered as independent random variables. Define (3) Consider a block that is being processed by block based admission control at time It will be discarded if Therefore, the goal is to find conditions sufficient for (4) From a generalized version of the central limit theorem [19] (included in the Appendix as Theorem 3) we have (5) If the switch has a priori knowledge of the traffic characteristics of flows in the class, peak rate reservation may not be necessary. But it is unrealistic with today s networks. Note that BLP i is the probability of block losses through ....

....in may be for out of order service of a packet not in Similar to the proof for Theorem 1, we have (43) Moreover (44) 45) Combining (43) and (45) we have (46) From (41) 42) and (46) we have (47) 48) 49) Because of (38) can be substituted by in (49) Therefore, 37) holds. Theorem 3 ([19]) Assume that are independent random variables centered at expectations with distribution functions (d.f. s) and variance Let be their consecutive sums with variance Then (50) if and only if for all (51) as Theorem 4: For any random variable that can take value between 0 and , the following ....

M. Loeve, Probability Theory I, 4th ed. New York: Springer-Verlag, 1977.

....they are asymptotically globally convergent. Main Results: We now list the main results and organization of this paper: Section II formally presents the signal model and estimation objectives. In Section III, we use the DA algorithm, proposed by [21] together with the law of large numbers [20], to compute conditional mean state estimates of the finite state Markov chain and the state of jump Markov linear system. In Theorem 3.1, we prove the convergence of the DA applied to the state space model (1) and (2) In Section IV, a new SA algorithm is derived based on the DA algorithm. The ....

....the conditional mean estimates and for the signal model (1) 2) These estimates are theoretically obtained by integration with respect to the joint posterior distribution . If we were able to obtain (for large ) i.i.d. samples according to the distribution , then using the Law of Large Numbers [20], conditional mean estimates can be computed by averaging, thus solving the state estimation problem. Unfortunately, obtaining such i.i.d. samples from the posterior distribution is not straightforward. Thus, alternative sampling schemes must be investigated. A. Data Augmentation In this paper, ....

A. N. Shiryaev, Probability. New York: Springer-Verlag, 1996.

....that an incoming cell belongs to the connection can be expressed as . Let denote the interarrival time (in the unit of cell slots) between two consecutive cells in the connection, then , and . When is small, this discrete geometric distribution can be approximated as the exponential distribution [11]. Suppose the first cell of the packet arrives at time 0, the response time of the SPFM is cell time, and is the number of cells in the connection that arrive within time window (0, then is a random variable with Poisson distribution . No latency is experienced by the packet that consists of ....

....cell; without LCH, the cell that was delayed most is the first cell since it has to wait for the last cell to arrive. Therefore (3) where is the arrival time of the th cell, assuming the arrival time of the first cell is zero. The density function of is (4) where is the Erlang random variable [11]. Therefore (5) 6) B. The Numerical Results In this section, we calculate numerical results of the firewallswitch performance by plugging in the parameters measured from the Internet backbone traffic [4] The performance analysis of our firewall switch owes thanks to NLANR s Internet traffic ....

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R. Nelson, Probability, Stochastic Processes, and Queueing Theory. New York: Springer-Verlag, 1995.

....the norms given by #2#, we now have the result that g 2 B s p;q almost surely if and only if: 1 X j=0 2 js 0 q # q j jj# j jj q p = 1 X j=0 2 j#q j #q=2 jj# j jj q p # 1 almost surely. #21# It can be shown from the monotone convergence theorem and the three series theorem #see Karr, 1993, Theorems 4.10 and 7.5 respectively# that, if Zn are independent and identically distributed nonnegative random variables with strictly positive #nite mean, and an are nonnegative constants, then P anZn is convergent almost surely if and only if P an is convergent. It follows that #21# is ....

Karr, A.F. #1993#. Probability. New York: Springer-Verlag.

....can reparameterize the time axis such that an inhomogeneous PCP can be expressed as a homogeneous PCP on a nonuniform time axis. Let be a PCP with rate 1. Then to generate an inhomogeneous PCP with the rate , let (23) The random process is still Markovian, with independent increments. 4, p. 44] [15]. B. Poisson Device Models A two terminal Poisson device model (i.e. a shot noise model) consists simply of two independent forward and reverse current random processes (see Fig. 5) Each current is a Poisson counting process with a rate that is a function of the instantaneous applied voltage ....

P. E. Pfeiffer, Probability for Applications. New York: Springer-Verlag, 1990, p. 563.

....not possibly list here even all of the key references, and content ourselves with giving a small relation for the interested reader. Among the numerous books, it is worth mentioning the general books of Cox and Smith (1961) Tak acs (1962) Cooper (1981) Gross and Harris (1985) Medhi (1991) and Nelson (1995); Kleinrock (1975, 1976) and Newell (1982) are good sources of applications; Cooper (1990) is a nice survey with an extensive bibliography. Nevertheless, there is a plethora of survey books and papers in queueing, and Prabhu (1987) is a useful survey of such up to date. Most of this vast effort, ....

R Nelson. Probability, Stochastic Processes and Queueing Theory. New York: Springer Verlag, 1995.

....norms given by (2) we now have the result that g 2 B s p;q almost surely if and only if: 1 X j=0 2 js 0 q q j jj j jj q p = 1 X j=0 2 jffiq j flq=2 jj j jj q p 1 almost surely. 21) It can be shown from the monotone convergence theorem and the three series theorem (see Karr, 1993, Theorems 4.10 and 7.5 respectively) that, if Zn are independent and identically distributed nonnegative random variables with strictly positive finite mean, and an are nonnegative constants, then P anZn is convergent almost surely if and only if P an is convergent. It follows that (21) is ....

Karr, A.F. (1993). Probability. New York: Springer-Verlag.

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Karr, A.F. #1993#. Probability. New York: Springer-Verlag.

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R. Nelson, Probability, Stochastic Processes and Queueing Theory. New York: Springer-Verlag, 1995.

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Shiryaev, A. N. (1996): Probability. New York: Springer-Verlag.

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Loeve, M.: Probability Theory II, 4th Edition. New York: Springer-Verlag, 1977

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Loeve, M.: Probability Theory I, 4th Edition. New York: Springer-Verlag, 1977

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A. N. Shiryaev, Probability. New York: Springer-Verlag, 2nd ed., 1996. 29

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M. Love, Probability Theory, ,th edn. Vol. I. New York: Springer-Verlag (1977).

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Karr, A. F. (1993). Probability, Springer Texts in Statistics, New York: Springer-Verlag.

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