M. Shaked and J.G. Shanthikumar, Stochastic Orders and Their Applications, Academic Press, San Diego (CA), 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for any x F # # (x y) P # (dy) P (# x # = y) P # (dy) P(# x) F # (x) 4) See e.g. 7] for the definition of regularity. 4 Hence, F # # (x y) F # (x) x R, P # a.s. y R, together with (4) implies F # (x) F # (x) which is equivalent with # # (see 1.A.1. in [6]) ii) By the independence of the r.v. s. we have F # # (x y) F # (x) for x P # a.s. y R. iii) Fix y R and take a r.v. # with cdf F # (x) F # # (x y) x R. The SFSD property implies # #. Secondly, recall that # # if and only if Eg(#) for all non decreasing function g : ....

....of the r.v. s. we have F # # (x y) F # (x) for x P # a.s. y R. iii) Fix y R and take a r.v. # with cdf F # (x) F # # (x y) x R. The SFSD property implies # #. Secondly, recall that # # if and only if Eg(#) for all non decreasing function g : R R (see 1.A.1. in [6]) # Next, we collect some basic facts on the SSSD property. Theorem 2 (Some features of SSSD) i) For any r.v. s # and #, # # SSSD # implies # # SSD #. # SSSD # is equivalent with # # SSD #. iii) For any r.v. s # and #, # # SSSD # holds if and only if we have E (g(#) #) a.s. for all ....

Moshe Shaked, J. George Shanthikumar: "Stochastic Orders and Their Applications ", Academic Press, London, 1994

....first analyze a heavy loaded network where all nodes have packets waiting all the time. The heavy load assumption decouples an interacting queues problem into a series of queues problem whose stability is established using Loynes s theory [44] We then provide a stochastic ordering relation (e.g. [49]) between the 14 normal network and the heavy loaded network showing that the stability and the achievability in the heavy loaded network implies the stability and the achievability in the normal network. While these ideas are well known in analysis of stochastic networks, their application to ....

....relations. One approach is provided by the definition above which does not restrict X and Y to be defined in the same probability space. However, there is another, sometimes more convenient, way of looking at stochastic order. If X st Y , then this equivalent approach (given as Theorem 4.B. 1 in [49]) constructs new random vectors X and Y in some probability space such that Y = Y and Y with probability 1. In other words, we can view the stochastic order as the usual order in an appropriate probability space. In the proof of Lemma 3, it is the second approach we will be using. ....

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M. Shaked and J. G. Shanthikumar, Stochastic Orders and Their Applications. Academic Press, 1994.

....has an embedded matrix inversion which may require excessive computation. Stochastic comparison is a technique by which both transient and steady state performance measures of a MC may be bounded. There are several applications of this technique in different areas of applied probability [16] and in practical problems of engineering [13] 14] 9] The stochastic comparison of MCs is discussed in detail in [10] 19] 11] The comparison of two MCs may be established by the comparison of their transient probability vectors at each time instant. Obviously, if steady states exist, ....

.... most well known is the strong stochastic ordering (i.e. st ) Intuitively speaking, two random variables X and Y which take values on a totally ordered space being comparable in the strong stochastic sense (i.e. X st Y ) means that it is less probable for X to take larger values than Y (see [16], 19] First we give the definition of st ordering used in this paper. For further information on the stochastic comparison method, we refer the reader to [19] Definition 1 Let X and Y be random variables taking values on a totally ordered space. E[f(X) E[f(Y ) for all nondecreasing ....

M. Shaked, J.G. Shantikumar, Stochastic Orders and Their Applications, Academic Press, California, 1994.

....(resp. decreasing) is to be understood to mean non decreasing (resp. non increasing) III. STOCHASTIC ORDERINGS In this section, we summarize basic definitions concerning the stochastic orderings of random vectors. Additional information can be found in the monographs by Shaked and Shanthikumar [35], and by Stoyan [36] Definition 2: Let # be a class of Borel measurable functions IR. We say that the two IR valued rvs X and Y satisfy the relation X ## E [#(X) E [#(Y) 12) for all functions # in #, whenever the expectations exist. This generic definition has been specialized ....

....not depend on H , it is appropriate to simply refer to it as A t ,t= 1, without further reference to H in the notation. B. Comparisons for FGN traffic models For each , 1, A 0 , A t ] is normally distributed. Since a Gaussian rv is stochastically increasing in the mean [35], the SSI property will follow if we can show that the conditional mean E 0 = 0 , A t = t is an increasing function in ( 0 , t ) Although the covariance function of the underlying sequence is explicitly given, we were unable to obtain usable closed form expressions for these ....

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M. Shaked and J.G. Shanthikumar, Stochastic Orders and Their Applications, Academic Press, New York (NY), 1994.

...., so that it is indeed the mean of the pdf with density f # (t) # C # e #t f(t) where C # is chosen so that the total mass is 1. To establish the monotonicity of f # (#) f (#) we use stochastic order concepts; these are discussed in the Appendix of Ross [19] and Shaked and Shanthikumar [20]. Note that the ratio of the pdf s satisfies f #2 (t) f #1 (t) C #2 e (#2 #1 )t C #1 , t # 0, which is decreasing in t for # 2 # 1 , which implies that f #2 is smaller than f #1 in the likelihood ratio ordering, which in turn implies that f # 2 is less than f # 1 in stochastic ....

M. Shaked and J. G. Shanthikumar (1994) Stochastic Orders and Their Applications, Academic Press, New York.

....i be the random variables associated with the distributions G i and G i , respectively. Then, the above condition means that L i is more variable than L i in the sense of the convex ordering, denoted L i # cx L i ; i.e. Ef(L i ) # Ef( L i ) for all convex function f( see, e.g. [17]) and hence in particular, Var[L i ] E[L i # i ] 2 # E[ L i # i ] 2 = Var[ L i ] Similarly, we can address the issue of variability associated with the batch size distribution. Suppose the new system has a batch size of ( Q K i ) for component i of type K product. ....

....variates, Z n i and Z n i , the above implies that for any vector u = u 1 , um ) with u i # 0, 1 , we have P[Z n i # u i , #i] # P[ Z n i # u i , #i] That is, Z n i # uo Z n i . Since this ordering is preserved under i.i.d. summation (e.g. Theorem 4.B. 10 in [17]) we have Y n # uo Y n . Letting n ##, we obtain the desired X # uo X. Next, consider batch arrivals. To be specific, write P[Q K j = r j , j = 1, m = p K r 1 . r m . Each vector (r 1 , r m ) corresponds to a fixed bill of materials. We can decompose the arrival process ....

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Shaked, M. and Shanthikumar, J.G., Stochastic Orders and Their Applications, Academic Press, New York, 1994.

....the constraint IE(Z (1) b) 1, and the proof of optimality (see, e.g. Sundt [18, Ch. 9] slightly adapted) proceeds by showing that r SL Z (1) cx r Z (1) 5 for any other r, where cx is the convex ordering: U cx V means that Ef(U) Ef(V ) for any convex function f , see [17], and in intuitive terms the content is roughly that V is more variable than U . When looking for a general convex minorant in the framework of Theorem 1.1, which would maximize the adjustment coecient, Proposition 2.1 shows that (since convex ordering implies ordering of variances) the only ....

M. Shaked and J.G. Shanthikumar. Stochastic Orders and Their Applications. Academic Press, 1994.

....has a simple form, with one transition point per subset of sources. 1. 1 Previous work The concepts of increasing failure rate and decreasing failure rate are utilized primarily in the literature of systems reliability, where they are used to describe the lifetime of system components [12, 15, 16, 19] The lifetime distribution may a#ect the proper strategy for scheduling maintenance. In this sense, the idea that these distributions can yield algorithmic implications has been known for some time [16] The problem we examine here fits naturally into the scheme of Markov decision processes ....

....reliability, where they are used to describe the lifetime of system components [12, 15, 16, 19] The lifetime distribution may a#ect the proper strategy for scheduling maintenance. In this sense, the idea that these distributions can yield algorithmic implications has been known for some time [16] The problem we examine here fits naturally into the scheme of Markov decision processes [4, 13] there is an underlying Markov reward process, and one wants to maximize the reward. Indeed, it specifically fits into the framework of optimal stopping theory, where one wishes to find the optimal ....

M. Shaked and J. G. Shanthikumar. Stochastic Orders and Their Applications. Academic Press, San Diego, 1994.

....greedy strategy and any other xed strategy. This one to one correspondence matches results so that the maximum load for each possible result pair is smaller using Azar et al. s greedy placement strategy. This is an example of a simple stochastic comparison; for more on this area, see [Sto83, SS94] Theorem 6 Suppose that n balls are sequentially placed into n bins. Each ball is placed in the least full bin at the time of the placement, among d bins, d 2, chosen independently and uniformly at random. Then after all the balls are placed the number of balls in the fullest bin is at least ....

M. Shaked and J. Shantikumar. Stochastic Orders and Their Applications. Academic Press, Inc., 1994.

....variability order. Roughly, two distributions, F 1 and F 2 , are ordered in variability if, given a xed location, mass accumulates more rapidly as we move away from this location, for say F 1 relative to F 2 , in either direction. In fact, there is no unique de nition for variability ordering. Shaked and Shanthikumar (1994) note at least three possibilities: convex order, dispersive order and peakedness order. We adopt a natural and exible de nition based upon a sign changes condition. Under this de nition we develop semiparametric Dirichlet process mixture models for variability ordering, extending the ideas in ....

....(2) yielding F ( G; 2) R F ( 1) 2) G(d (1) a semiparametric speci cation. 4 3 A review of variability orders Three de nitions relating to the general concept of variability ordering, mentioned in the introduction, can be found in the literature. We refer to Shaked and Shanthikumar (1994, chapter 2) for a comprehensive review. Hereafter, let X and Y be two random variables with distribution functions F 1 and F 2 , density functions, assuming they exist, f 1 and f 2 , and generalized inverses F 1 1 and F 1 2 , respectively. Here, F 1 i (u) inffx : F i (x) u g, i ....

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Shaked, M., and Shanthikumar, J.G. (1994), Stochastic Orders and Their Applications, Boston: Academic Press, Inc.

....greedy strategy and any other fixed strategy. This one to one correspondence matches results so that the maximum load for each possible result pair is smaller using Azar et al. s greedy placement strategy. This is an example of a simple stochastic comparison; for more on this area, see [Sto83, SS94] Theorem 6 Suppose that n balls are sequentially placed into n bins. Each ball is placed in the least full bin at the time of the placement, among d bins, d 2, chosen independently and uniformly at random. Then after all the balls are placed the number of balls in the fullest bin is at least ....

M. Shaked and J. Shantikumar. Stochastic Orders and Their Applications. Academic Press, Inc., 1994.

....since for bin than contributes to X t , there are only m possible items that could enter and reduce its residual capacity to something less than m. For this range of m, however, the bound of k=2 Gamma m is better. We now use stochastic domination. Following standard definitions (see, e.g. [13]) we say that X stochastically dominates Y and write X Y if Pr(X u) Pr(Y u) for all real values u. Intuitively, X is more likely to take on larger values than Y . It is simple to show (say via induction) that X t Z t , where Z 0 = 0 Z t 1 = 8 : Z t 1 w.p. k(2ff Gamma1) 2kff ; Z ....

M. Shaked and J. G. Shanthikumar. Stochastic Orders and Their Applications. Academic Press, Inc., San Deigo, 1994.

....(See Theorem 2.4 in Chapter II of Liggett (1985) Such a measure is called a coupling measure. Our question then is to determine when n n 1 for each n 1. A lot of work has been done in which stochastic monotonicity is proved or used in various contexts. A recent book on the subject is Shaked and Shanthikumar (1994). The particular problem we are concerned with has apparently come up only a few times. In perhaps the first of these, Efron (1965) found a sufficient condition for n n 1 see the remarks following the statement of Theorem 1.9 below. Joag Dev and Proschan (1983) showed that in this situation, ....

....understanding negative dependence. A common way of checking stochastic monotonicity on partially ordered sets is to apply Holley s Theorem, which appears as Theorem 2.9 of Chapter II of Liggett (1985) or one of its extensions. See Preston (1974) Karlin and Rinott (1980) and Theorem 4.E. 5 of Shaked and Shanthikumar (1994), for example. It states that a sufficient condition for is (x) y) x y) x y) x; y 2 S; where xy and xy denote the coordinatewise minimum and maximum of x and y respectively. Note that this condition cannot be applied in the present context, since if x 2 Sn and y 2 Sn 1 , then x y and x ....

M. Shaked and J. G. Shanthikumar, Stochastic Orders and their Applications, Academic Press, 1994.

....general, since for bin than contributes to X t , there are only m possible items that could enter and reduce its residual capacity to something less than m. For this range of m, however, the bound of k 2 m is better. We now use stochastic domination. Following standard definitions (see, e.g. [12]) we say that X stochastically dominates Y and write X # Y if Pr(X u) # Pr(Y u) for all real values u. Intuitively, X is more likely to take on larger values than Y . It is simple to show (say via induction) that X t # Z t , where Z 0 = 0 Z t 1 = # # # Z t 1 w.p. k(2# 1) 2k# , ....

M. Shaked and J. G. Shanthikumar. Stochastic Orders and Their Applications. Academic Press, Inc., San Deigo, 1994.

....is characterized by the monotonicity of the ratio F (x) G(x) over the support set SG of the distribution G, where F = 1 Gamma F and G = 1 Gamma G. We say that F ( G if and only if F (x) G(x) is nonincreasing on SG . Among the many extant notions of ordering between distributions (see Shaked and Shanthikumar (1994) for a comprehensive discussion) uniform stochastic ordering has a particular relevance in the fields of reliability and survival analysis. It is easily shown that, in the absolutely continuous case where F and G have densities f and g, F ( G is equivalent to the ordering of the respective ....

.... notions, and various implications of a USO assumption, have been explored in papers by Yanagimoto and Sibuya (1972) Whitt (1980) Keilson and Sumita (1982) Bagai and Kochar (1986) and Boland, El Newehi and Proschan (1994) An excellent exposition of these and related results may be found in Shaked and Shanthikumar (1994). Statistical papers include Caperaa s (1988) treatment of a nonparametric testing problem, Dykstra, Kochar and Robertson s (1991) derivation of the likelihood ratio test for equality of distributions against a USO alternative, Rojo and Samaniego s (1991, 1993) studies of consistent estimation of ....

Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and their Applications.

....is decreasing on its interval of support. These monotonicity properties of the hazard rate and reversed hazard rate functions play a role in each of the sections below. For further discussion of these functions, see Anderson et al. 1993) Barlow and Proschan (1975) Block et al. 1998) and Shaked and Shantikumar (1994). 3. Order Statistics Let X 1 , X m be independent and identically distributed observations with pdf f # (x) which has the exponential form in (2) Then the pdf of the k th order statistic, Y = X (k) is (see David, 1970) g # (y) k # # # m k # # #F # (y) k 1 F # (y) m k f # (y) ....

Shaked, M. and Shantikumar, J. G. (1994). Stochastic Orders and their Applications.

.... sample, let Fn1 stand for the distribution of (X (n) X (1) and let al..so Gn1 be the distribution of (Y (n) Y (1) It is then straightforward to prove that Fn1 #LTD Gn1 # Fn1 # SI Gn1 ##t#t # F(t) G(t) # F (t # ) G(t # ) #F rh G where rh denotes the reversed hazard order (cf. Shaked and Shanthikumar 1994). Likewise, if F 1n and G 1n denote the distribution functions of (X (1) X (n) and (Y (1) Y (n) respectively, it can then be checked easily that F 1n #RTI G 1n # F 1n # SI G 1n ##t#t # 1 F(t) 1 G(t) # 1 F(t # ) 1 G(t # ) #G# hr F where # hr denotes the classic hazard rate order. ....

Shaked, M., and Shanthikumar, J. (1994). Stochastic Orders and Their Applications.

....and concave service rate, which includes as a particular case multi server queueing systems. In the literature many results have been obtained on the comparison of queues. The reader is referred to the books of Ross [14] Stoyan [16] Baccelli and Br#maud [3] and Shaked and Shanthikumar [15]. The results most closely related to our model are those on the comparison of queueing systems with Doubly Stochastic Poisson (DSP) processes, see Ross [13] Rolski [11, 12] and Svoronos and Green [17] More recently, Chang and Pinedo [6] obtained monotonicity results for the blocking ....

M. Shaked, J. G. Shanthikumar. Stochastic Orders and Their Applications. Academic Press, Boston, 1994.

....and concave service rate, which includes as a particular case multi server queueing systems. In the literature many results have been obtained on the comparison of queues. The reader is referred to the books of Ross [14] Stoyan [16] Baccelli and Br emaud [3] and Shaked and Shanthikumar [15]. The results most closely related to our model are those on the comparison of queueing systems with Doubly Stochastic Poisson (DSP) processes, see Ross [13] Rolski [11, 12] and Svoronos and Green [17] More recently, Chang and Pinedo [6] obtained monotonicity results for the blocking ....

M. Shaked, J. G. Shanthikumar. Stochastic Orders and Their Applications. Academic Press, Boston, 1994.

....if P = M . We see that a PCFB is the one that produces a set of subband variances that majorizes sets of subband variances obtained by all other filter banks. There are many other applications of majorization. The interested reader is referred to [9] for a beautiful treatment of the subject and to [11] for more recent applications. 4. FIR ORTHONORMAL FILTER BANKS The result developed in previous sections asserts that for a given input, whenever one filter bank produces a set of subband variances that majorizes sets of subband variances of all other filter banks, it is optimal for subband ....

M. Shaked and J. G. Shanthikumar. Stochastic orders and their applications. Academic Press, 1994.

....the author visited Technische Universitat Braunschweig, in part supported by KBN Grant Postal address: Mathematical Institute, Wroc law University, pl. Grunwaldzki 2 4, 50 384 Wroc law, Poland and Savits (1993) Beichelt (1993) Sim and Endrenyi (1993) and the chapter by Block and Savits in the Shaked and Shanthikumar (1994) book. It seems important to study various monotonicity properties for replacement policies to derive useful bounds for cost functions. There exisits a tendency in the recent literature to compare replacement policies by means of whole stochastic processes (e.g. over periods of time) not only for ....

Shaked, M. and Shanthikumar, J. G. (1994), Stochastic Orders and Their Applications, Academic Press, New York.

....that the chance of balking relates appropriately to the chance of reneging, as in the construction in Section 3, in particular, assuming (3.2) We now show that a strong comparison is possible. In particular, we establish likelihood ratio (MLR) ordering. See Chapter 1 of Shaked and Shanthikumar [24] for background on stochastic orderings. Consider two random variables X 1 and X 2 with values in the state space f0; 1; sg, 1 s 1, that have probability mass functions (pmf s) that are positive for all states. We say that X 1 is less than or equal to X 2 in the likelihood ratio (LR) ....

M. Shaked and J. G. Shanthikumar, Stochastic Orders and Their Applications, Academic Press, New York, 1994.

....systems under the same conditions we use for the standard routing problem. 1. 2 Previous Work Stochastic comparison techniques have been used previously primarily on single queues, for example in [15, 21] These techniques have also been applied and generalized to more complicated processes [1, 13, 14, 19, 22]. Indeed, although our work was derived independently, it strongly resembles the work of Niu [13] who examined tandem queues using a similar approach. A good modern treatment of the subject is given by Shaked and Shanthikumar [19] In the computer science literature, comparison results have ....

....and generalized to more complicated processes [1, 13, 14, 19, 22] Indeed, although our work was derived independently, it strongly resembles the work of Niu [13] who examined tandem queues using a similar approach. A good modern treatment of the subject is given by Shaked and Shanthikumar [19]. In the computer science literature, comparison results have primarily been based on the work of Stamoulis and Tsitsiklis [20] Using coupling and stochastic comparison they showed that if the underlying network is layered and Markovian, and servers use a First Come First Served (FCFS) policy, ....

M. Shaked and J. Shantikumar. Stochastic Orders and Their Applications. Academic Press, Inc., 1994.

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M. Shaked and J.G. Shanthikumar, Stochastic Orders and Their Applications, Academic Press, San Diego (CA), 1994.

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M. Shaked and J. G. Shanthikumar, Stochastic Orders and Their Applications. Academic Press, 1994.

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Shaked M. & Shanthikumar J.G. (1994). Stochastic orders and their applications, Academic Press, pp.545.

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Shaked M. and ShanthikumarJ.G 1994. Stochastic orders and their applications. New York: cademic Press.

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Shaked, M. and Shanthikumar, J.G. (1994). Stochastic orders and their applications. Probability and Mathematical Statistics, Academic Press. 11

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Shaked M. & Shanthikumar J.G 1994 Stochastic orders and their applications Academic press p. 545.

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Shaked, M.; Shanthikumar, J.G. (1994). "Stochastic orders and their applications", Academic Press, pp. 545.

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Shaked, M.; Shanthikumar, J.G. (1994). "Stochastic orders and their applications", Academic Press, pp. 545.

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Shaked,M.; Shanthikumar,J.G. (1994). "Stochastic orders and their applications", Academic Press,pp. 545.

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Shaked M, Shanthikumar JG. Stochastic Orders and Their Applications. San Diego, CA: Academic Press, 1994.

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Shaked, M. and Shanthikumar, J.G., Stochastic Orders and Their Applications, Academic Press, New York, 1994.

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M. Shaked and J. G. Shanthikumar. Stochastic Orders and Their Applications. Academic Press, Inc., San Diego, 1994.

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Shaked, M. and Shanthikumar, J.G. (1994). Stochastic Orders and their Applications. Academic Press, San Diego, CA.

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Shaked M. and Shanthikumar J. G. (1994), Stochastic orders and their applications, Academic Press, London.

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Shaked, M. and J.G. Shanthikumar (1994). Stochastic Orders and their Applications. Academic Press, London.

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M. Shaked and J. G. Shanthikumar. Stochastic Orders and Their Applications. Academic Press, Inc., San Deigo, 1994.

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Shaked, M. and J.G. Shanthikumar, \Stochastic Orders and their Applications ". Academic Press, 1994

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M. Shaked and J. G. Shanthikumar. Stochastic Orders and Their Applications. Acadmic Press, 1994.

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M. Shaked and J. G. Shantikumar, Stochastic orders and their applications, Academic Press 1994 3

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SHAKED, M. and G., SHANTHIKUMAR (1993): Stochastic orders and their applications, New York: Academic Press Inc. Harcourt Brace & Company.

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M. Shaked and J. G. Shanthikumar. Stochastic Orders and Their Applications. Acadmic Press, 1994.

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M. Shaked and J. G. Shanthikumar. Stochastic Orders and Their Applications. Acadmic Press, 1994.

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Shaked, M. and Shanthikumar, G., Stochastic Orders and Their Applications, Academic Press, California, 1994.

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Shaked, M. and Shanthikumar, G., Stochastic Orders and Their Applications, Academic Press, California (1994).

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