J. Zucker. The correspondence between cutelimination and normalization, Part I & II. Annals of Mathematical Logic, 7:1-112,113-155, 1974. North-Holland Publishing Company. 14  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is based on the subsequent three lemmas which derive preliminary results on a discrete time nite bu er queue, de ned as follows. Consider two i.i.d. sequences of positive random variables fX; X n g and fY; Y n g; n 2 N. Let W : 6. 1) Assuming that P[X n = Y n ] 1, in Chapter III.5 of [8] it was shown that W n has a unique stationary distribution, and that for any initial condition W 0 , W n converges to that stationary distribution. Let W be a random variable that is equal in distribution to W n in stationarity. Proof: Instead of proving the statement for W ....

....to analyze queueing process V that stochastically upper bounds W . Let fV n=0 be de ned by V n 1 = V B: 18 The above recursion is similar to (6. 1) and under the non triviality condition P[X n = Y n ] 1, V converges to a stationary distribution (see Chapter III.4 of [8]) again, let V be a random variable that is equal in distribution to V n in stationarity. Now, we show that W n V n for all n 0. Clearly, this is implied by W 0 = V 0 = 0 and the next inductive step (V B = V n 1 ; where the inequality follows from (x B y) x ....

J. W. Cohen. The Single Server Queue. North-Holland Publishing Company, 1982.

....namely proof simplifications. Consider for example the following proof: 1) FDGDF GDF 71 1 F This may be simplified to just the proof h. The idea that proofs may be simplified until no simplification can be made any longer, obtaining equivalent normal proofs, was pioneered by Gerhard Gentzen [48] to give the first finitist proof (in the sense of Hilbert) of the consistency of first order Peano arithmetic. If the logical system is presented in a proper way, as with Gentzen s sequent calculus, it is easy to see that false has no normal proof (no rule can lead to a proof of false) So false ....

M.E. Szabo, The collected papers of Gerhard Gentzen, North-Holland Pub- lishing Company, Amsterdam, 1969.

....process is then exactly the proof simplification process described above. 3.3. Natural Deduction and the Lambda Calculus The language of Hilbert systems and combinatory logic is not easy to work with, although this can be done [28] A more comfortable choice is given by Church s A calculus [2], the programming language associated with minimal logic in natural deduction format [47] Its terms are given by the grammar: M, N, P : x[Ax. M[MN where x ranges over variables, Ax. M is A abstraction ( the function that maps x to M , where M typically depends on x) For convenience, we write ....

Henk Barendregt, The lambda calculus, its syntax and semantics, Studies in Logic and the Foundations of Mathematics, vol. 103, North-Holland Publishing Company, Amsterdam, 1984.

....c #(0) F 1 From (12) and (11) we see that F is continuous at K and thus there is no probability mass at v = K. Remark 1: If there was no threshold discarding, i.e. K = then (13) is the standard Takacs Integro differential equation (see for e.g. [1], 6] This is a standard Volterra Integral Equation [15] whose solution is given as c #(0) F 1 ( 14) where the iterated kernels M i are given as g )M 1 (t, g)dt . M n 1 (t, g)dt . Substituting the expression for from ....

J. W. Cohen. The Single Server Queue. North-Holland Publishing Company, 1982.

....there are many ways that a proof using only cuts can be rearranged by permuting and distributing cuts across each other. Normal proofs will then be of specific forms, allowing one to reduce non determinism in proof search. This is very much in the spirit of Gentzen s Hauptsatz for sequent calculi [13]. While permuting and distributing cuts is easy, showing that this process terminates is slightly more challenging. Note that weak termination would be enough, but it is not much harder to show strong termination: all rewrites terminate. Second, to provide a simple argument by which ....

M. E. Szabo. The Collected Papers of Gerhard Gentzen. North-Holland Publishing Company, Amsterdam, 1969. 16

....that #n is stationary and ergodic and that the system is stable, i.e. E #n c. Under this assumption, it has been shown that Qn converges to a stationary and ergodic process [12] It has also been shown that Qn converges to a stationary process when the system is a GI GI m x type of queue [13], 14] and when the system is a G M m x type of queue [15] Since proving the convergence of Qn is not the focus of this paper, and moreover, practical measurements of PL (x) and P Q x are based on time averaging assuming ergodicity (see (1) and (2) we assume that both Qn and Qn ....

J. W. Cohen, The Single Server Queue, North-Holland Publishing Company, Amsterdam, 1969.

....the following proof: 1) F G F 1 F (MP ) G F 2 G (MP ) F This may be simpli ed to just the proof 1 . The idea that proofs may be simpli ed until no simpli cation can be made any longer, obtaining equivalent normal proofs, was pioneered by Gerhard Gentzen [45] to give the rst nitist proof (in the sense Homology, Homotopy and Applications, vol. 2001 6 of Hilbert) of the consistency of rst order Peano arithmetic. If the logical system is presented in a proper way, as with Gentzen s sequent calculus, it is easy to see that false has no normal ....

M. E. Szabo, The collected papers of Gerhard Gentzen, North-Holland Publishing Company, Amsterdam,

....described above. Homology, Homotopy and Applications, vol. 2001 7 3.3. Natural Deduction and the Lambda Calculus The language of Hilbert systems and combinatory logic is not easy to work with, although this can be done [25] A more comfortable choice is given by Church s calculus [2], the programming language associated with minimal logic in natural deduction format [44] Its terms are given by the grammar: M;N;P : xj x M jMN where x ranges over variables, x M is abstraction ( the function that maps x to M , where M typically depends on x) For convenience, we ....

....rewrites proofs to proofs of the same sequents. Lemma 2 (Subject Reduction) If the typing judgment M : F is derivable and M N then N : F is derivable. Proposition 3 (Con uence) If M N 1 and M N 2 , then there is P such that N 1 P and N 2 P . Proof. We closely follow [2]. The plan of the proof is as follows. We rst replace by another notion of reduction 1 , which will be strongly con uent (if M 1 M 1 and M 1 M 2 , then there is P such that M 1 1 P and M 2 1 P ) hence con uent. This will be proved by showing that a related calculus S4 ....

[Article contains additional citation context not shown here]

Henk Barendregt, The lambda calculus, its syntax and semantics, Studies in Logic and the Foundations of Mathematics, vol. 103, North-Holland Publishing Company, Amsterdam, 1984.

....times is given in Remark 2.2. The well known relation for the busy period of the M G 1 queue, x = #(s #(1 x) readily leads to (4.10) We immediately have that for Re(s) # 0, E e sCP is equal to the (unique) root of (4. 10) inside (or on) the unit circle (see for instance Cohen [4] p. 250) We emphasise that the non empty period is not equal to the busy period of the M G 1 queue with exceptional first service as described in Remark 2.2. In fact, in the context of that model, the non empty period is equal to the busy period initiated by a customer with a regular service. ....

J.W. Cohen. The Single Server Queue. North-Holland Publishing Company, Amsterdam, 2nd edition, 1982.

....by definition, Y (t) 4 is not influenced by X(t) i.e. if we define p i : PfY = ig : lim t 1 PfY (t) ig: p 0 = 1 N X i=1 i Y k=1 q ( k Gamma1 q ( k Gamma1 ; p i = p 0 i Y k=1 q ( k Gamma1 q ( k ; i = 1; N; 2. 1) see for instance Part I of [2]. By p we denote the vector of these steady state probabilities : p = p 0 ; p 1 ; p N ) We define the simultaneous equilibrium probabilities j;i : PfX = j; Y = ig : lim t 1 PfX(t) j; Y (t) ig; 2.2) and partition them into vectors j : j;0 ; j;1 ; j;N ) of length ....

J.W. Cohen. The Single Server Queue. North-Holland Publishing Company, Amsterdam, 2nd edition, 1982.

....a discrete time finite buffer queue, defined as follows. Consider two i.i.d. sequences of positive random variables fX; X n g and fY; Y n g; n 2 N. Let W B 0 = 0 and W B n 1 = Gamma min(W B n X n 1 ; B) Gamma Y n 1 Delta : 8. 1) Assuming that P[X n = Y n ] 1, in Chapter III.5 of [9] it was shown that W B n has a unique stationary distribution, and that for any initial condition W B 0 , W B n converges to that stationary distribution. Let W B be a random variable that is equal in distribution to W B n in stationarity. Then, we show that Lemma 8.1 If X r 2 IR and EX ....

....defined by V B 0 = 0 and the following recursion V B n 1 = min Gamma (V B n X n 1 Gamma Y n 1 ) B Delta : 8.2) The above recursion is similar to (8. 1) and under the non triviality condition P[X n = Y n ] 1, V B n converges to a stationary distribution (see Chapter III.4 of [9]) again, let V B be a random variable that is equal in distribution to V B n in stationarity. Now, we show that W B n V B n for all n 0. Clearly, this is implied by W B 0 = V B 0 = 0 and the next inductive step W B n 1 = Gamma min(W B n X n 1 ; B) Gamma Y n 1 Delta min ....

J. W. Cohen. The Single Server Queue. North-Holland Publishing Company, 1982.

....a discrete time finite buffer queue, defined as follows. Consider two i.i.d. sequences of positive random variables fA; A n g and fC; C n g; n 2 N. Let W B 0 = 0 and W B n 1 = Gamma min(W B n A n 1 ; B) Gamma C n 1 Delta : 8. 1) Assuming that P[A n = C n ] 1, in Chapter III.5 of [9] it was shown that W B n has a unique stationary distribution, and that for any initial condition W B 0 , W B n converges to that stationary distribution. Let W B a random variable that is equal in distribution to W B n in stationarity. Then, we show that Lemma 8.1 If A r 2 IR and EA EC, ....

....defined by V B 0 = 0 and the following recursion V B n 1 = min Gamma (V B n A n 1 Gamma C n 1 ) B Delta : 8.2) The above recursion is similar to (8. 1) and under the non triviality condition P[A n = C n ] 1, V B n converges to a stationary distribution (see Chapter III.5 of [9]) again, let V B be a random variable that is equal in distribution to V B n in stationarity. Now, we show that W B n V B n for all n 0. Clearly, this is implied by W B 0 = V B 0 = 0 and the next inductive step W B n 1 = Gamma min(W B n A n 1 ; B) Gamma C n 1 Delta min ....

J. W. Cohen. The Single Server Queue. North-Holland Publishing Company, 1982.

....known. We now show how the C unknown probabilities P(X =0) P(X = C 1) may be determined. The function r ] has exactly C zeros r 1 , r C with #1, i = 1, C. Consider the circle C 1 # around zero with radius 1 #, for small but positive #. According to Rouche s theorem ([4], p. 652) the function r ] has exactly as many zeros as r inside C 1 # if (# (1 #)r) on C 1 # . We apply Rouche s theorem with the circle C 1 # instead of with the unit circle, since r = 1 is a zero that lies on the unit circle. The right hand term equals (1 ....

J.W.Cohen.The Single Server Queue. North-Holland Publishing Company, Amsterdam, 2nd revised edition, 1982.

....differential equations (PDE) see [8, 32] Especially for such kind of equations the finite element method (FEM) is a basic and powerful tool to solve them in an accurate way. The mathematical literature in this field is quite large, and vicarious for the people who work on this topic we refer to [6]. A more engineering literature on FEM is given in [20] Another method for solving such kind of problems is the so called boundary element method (BEM) We will not go into detail on this topic and refer to [34] The major problem of FEM is the solution of the arising finite element equations. ....

....form 2.4.1 A weak formulation in 2D Let us consider (26) 27) 28) and a geometrical setting like in Figure 3 as basis for our weak formulation. Without going into detail for weak formulations we give just the result. For the theory on Sobolev spaces see [1] for weak formulations see [6, 25]. An appropriate function space for elliptic PDEs of second order is the H ( Further we set V : H ( and Yg : v V: v glo 0 (47) where g is the given Dirichlet boundary condition on 0fL In a straight forward manner the following bilinear and linear form is obtained after ....

[Article contains additional citation context not shown here]

P.G. Ciarlet The Finite Element Method for Elliptic Problems, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1987

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As The Basis For Speci#cation Validation," Theory and Practice of Software Technology, North-Holland Publishing Company, pp. 21-49, 1983.

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J. Zucker. The correspondence between cutelimination and normalization, Part I & II. Annals of Mathematical Logic, 7:1-112,113-155, 1974. North-Holland Publishing Company. 14

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J. W. Cohen. The Single Server Queue. North-Holland Publishing Company, 1st edition, 1969.

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Thomas Jech. The Axiom of Choice. North-Holland Publishing Company, 1973.

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E. W. Beth. The Foundations of Mathematics. North Holland Publishing Company, 1959.

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J. Miklowitz, The Theory of Elastic Waves and Waveguides, North-Holland Publishing Company, New York, 1978. 8

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J. W. Cohen, The single server queue, North-Holland publishing company, 1982.

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Cohen, J. W ., The Single Server Queue, North-Holland Publishing Company, 1982.

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J. W. Cohen, The single server queue, North-Holland publishing company, 1982.

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J.W. Cohen. The Single Server Queue. North-Holland Publishing Company, Amsterdam, 1982 (revised edition).

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J. W. Cohen. The Single Server Queue. North-Holland publishing company, 1982.

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