In this research note we present a procedure to characterize the set of all Coxian distributions of algebraic degree q that have Coxian representations of order p where p > q

Fackrell, Mark

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Characterizing Coxian Distributions of Algebraic Degree q andTriangular Order p (Research Note)Mark FackrellDepartment of Mathematics and StatisticsUniversity of MelbourneVictoria 3010AUSTRALIAemail: mfackrel@ms.unimelb.eu.auKeywords: phase-type distribution; Coxian distribution; rational Laplace-Stieltjestransform; algebraic degree; triangular orderIn this research note we present a procedure to characterize the set of all Coxian distri-butions of algebraic degree q with real poles that have Coxian representations of order pwhere p > q.Coxian distributions (Cox [3]) are a particular class of phase-type (PH ) distributions(Neuts [8]) whose representations have the formβ =(β1 β2 . . . βp)(1)S =−λ1 λ1 0 · · · 00 −λ2 λ2 · · · 00 0 −λ3. . . 0....... . .. . ....0 0 0 · · · −λp, (2)where, without loss of generality, 0 < λ1 ≤ λ2 ≤ . . . ≤ λp.Consider a general order p PH representation (α,T ). For s < −λ1, where λ1 is thedominant eigenvalue of T , the Laplace-Stieltjes transform of the PH distribution withrepresentation (α,T ) isφ(s) = −α(sI − T )−1Te + α0 (3)=aqsq−1 + . . .+ a2s+ a1sq + bqsq−1 + . . .+ b2s+ b1+ α0, (4)where the numerator and denominator are coprime polynomials. Here, the coefficients arereal, and the zeros of the denominator polynomial are a subset of the eigenvalues of T . Thepoint mass at zero is α0 = 1−αe. We define the algebraic degree of the PH distributionto be the degree of the denominator polynomial, q, see O’Cinneide [9].Typically, representations for PH distributions are not unique, and we say that anyrepresentation for a PH distribution of minimal order is a minimal representation. Theorder of the PH distribution itself, is defined to be the order of any of its minimal repre-sentations. The order of a PH distribution is greater than or equal to its algebraic degree,see O’Cinneide [9].O’Cinneide [11] showed that any PH distribution whose generator T has only realeigenvalues, has an equivalent Coxian representation of some order. This order is called1Dagstuhl Seminar Proceedings 07461 Numerical Methods for Structured Markov Chains http://drops.dagstuhl.de/opus/volltexte/2008/1391 the triangular order of the distribution. The order of such a PH distribution is less thanor equal to its triangular order. Indeed, He and Zhang [6] showed that any order 3 PHdistribution with only real eigenvalues has an equivalent Coxian representation of order nomore than 4.We can represent any PH distribution with Laplace-Stieltjes transform (4) with analternative, so-called, matrix-exponential representation:γ =(a1 a2 · · · aq)(5)R =0 1 0 · · · 00 0 1 · · · 0.......... . ....0 0 0 · · · 1−b1 −b2 −b3 · · · −bq(6)eq =(0 0 · · · 0 1)′, (7)see Asmussen and Bladt [1]. This representation is minimal, and its order (that is, q) isequal to the algebraic degree of the PH distribution. Clearly, the representation in (5)–(7)is not a PH representation. We have alternative expressions for the density function andLaplace-Stieltjes transform of the PH distribution:f(u) = γ exp(Ru)eq (8)φ(s) = γ (sI − T )−1 eq + α0. (9)From now on we will assume that α0 = 0 since it makes the analysis slightly simpler, butthe case when α0 > 0 can be dealt with similarly.O’Cinneide [9] showed that the rational Laplace-Stieltjes transform (4) (with α0 = 0)corresponds to a PH distribution if and only if1. the pole of φ(s) of maximal real part is real, negative, and unique,2. a1 = b1, and3. f(u) = γ exp(Ru)eq > 0, for all u > 0.The first two conditions are simple to check. Equivalently, the third condition can beexpressed as: φ(s) corresponds to a PH distribution if and only if the point (a2, . . . , aq) isin the setΩq =⋂u>0{x ∈ Rq−1 | a1ei exp(Tu)eq +q−1∑i=1xiei+1(u) exp(Tu)eq > 0}. (10)In Bean, Fackrell, and Taylor [2] or Fackrell [5] a complete characterization of the bound-ary of Ω3 when p = 3 was given. (In fact, because the paper was concerned with matrix-exponential distributions, the inequality in (10) was not strict, and hence there, the bound-ary was included in the set.) Figure 1 shows Ω3 when the eigenvalues of T are −1,−2,−3.Any point (x1, x2) in Ω3 corresponds to a such a PH (or Coxian as the eigenvalues are all2−2 0 2 4 6 8 10−1012345x1x 2Ω3Figure 1: The set Ω3 contains all points (x1, x2) that correspond to Coxian distributionsof algebraic degree 3 with poles −1,−2, −3.real) distribution. Here, because the curved boundary is not in the set it is indicated by adashed line.Let 0 < λ1 ≤ λ2 ≤ . . . ≤ λp. For any subset {i1, i2, . . . , ir} of {1, 2, . . . , p}, let Fi1,i2,...,irrepresent the point in Rp−1 that corresponds to the Coxian distribution with Laplace-Stieltjes transformφi1i2...ir(s) =λi1λi2 . . . λir(s+ λir)(s+ λi2) . . . (s+ λir). (11)Every Coxian distribution with (triangular) order p (or less) and generator with eigenvalues(diagonal entries) in {λ1, λ2, . . . , λp} is contained in the (p− 1)-dimensional hypertetrahe-dron (a subset of Ωp) with vertices F12...p, F23...p, . . . , Fp, see Dehon and Latouche [4]. LetTp denote this hypertetrahedron. Figure 2 shows the 2-dimensional tetrahedron (triangle)with vertices F123, F23, and F3 - the set T3 contains all points (x1, x2) that correspond toCoxian distributions of algebraic degree 3 with poles −1,−2,−3 and triangular order 3.We note here that some points in T3 correspond to Coxian distributions of algebraic degree1 or 2. For example, F3 and F23 have algebraic degrees 1 and 2, respectively. Also, the lineF23F3 contains points that correspond to distributions of algebraic degree 2, except for F2whose algebraic degree is 1.The question now is: how do we characterize all Coxian distributions of algebraic degreeq, with poles −λ1,−λ2, . . . ,−λq, that have triangular order p with Coxian representationwhose generator has eigenvalues (diagonal entries) −λ1,−λ2, . . . ,−λq,−λq−1, . . .− λp (or-dered from largest to smallest)? The idea is to embed Ωq in Rp−1, and intersect the(q − 1)-dimensional hyperplane generated by the q vertices of Tq with the hypertetrahe-dron Tp. The resultant set will contain all points (in Rp−1) that correspond to Coxian3−2 0 2 4 6 8 10−1012345x1x 2F123F3F23T3Ω3 − T3Figure 2: Plot of Ω3 and T3. T3 contains all points that correspond to Coxian distributionsof algebraic degree 3 with poles −1,−2,−3 and triangular order 3.distributions of algebraic degree q (with the designated poles), that have triangular orderp (whose Coxian generators have the designated eigenvalues). Projecting this intersectionback onto Ωq gives the required subset of Ωq, that is, the set of all points in Rq−1 thatcorrespond to Coxian distributions of algebraic degree q and triangular order p.The approach is best illustrated with a simple example. We find all points in Ω3 thatcorrespond to Coxian distributions with poles −1, −2, −3 that have a Coxian represen-tation with the four eigenvalues −1, −2, −3, and −λ ≤ −3. We start by considering thecase when λ = 4. Figure 3 shows the tetrahedron T4 (which has vertices F1234, F234, F34,and F4) intersected with Ω3 (embedded in R3). The intersection is the quadrilateral withvertices F123, F23, F3, and X. Figure 4 shows the intersection projected back onto Ω3 - thetriangle above T3 contains all points that correspond to Coxian distributions with poles−1,−2,−3 whose triangular order is 4 that has a Coxian representation with eigenvalues−1,−2,−3,−4. Here, we have labeled the fourth vertex of the quadrilateral X despite thetwo points in Figures 3 and 4 being different, because they represent the same distribution.If we allow λ to vary so that λ ≥ 3 and take the union of all sets formed in themanner described above, we get the set of all points corresponding to Coxian distributionsof algebraic degree 3 and triangular order 4. This set, denoted by S4, is shown in Figure5.If we repeat the process and embed Ω3 in R4, intersect it with the hypertetrahedronwith vertices F12345, F2345, F345, F45, and F5, and project the intersection back onto Ω3we get the set of all points that correspond to Coxian distributions of algebraic degree 34T3F1234 F123F234F23F3F4XT4F34Ω3 − T3Figure 3: The intersection of Ω3 (embedded in R3) and T4.−2 0 2 4 6 8 10−1012345x1x 2F123 F23F3XFigure 4: The projection of the intersection of Ω3 (embedded in R3) and T4 onto Ω3.The triangle above T3 contains all points (x1, x2) that correspond to Coxian distributionsof algebraic degree 3 with poles −1,−2,−3, that have triangular order 4 whose Coxianrepresentations have eigenvalues −1,−2,−3,−4.5−2 0 2 4 6 8 10−1012345x1x 2F123 F23F3S4Figure 5: The set of all points that correspond to Coxian distributions of algebraic degree3 with poles −1,−2,−3, that have triangular order 4.whose poles are −1,−2,−3 that have triangular order 5 whose Coxian representations haveeigenvalues −1,−2,−3,−4,−5. This set is the trapezium above T3 depicted in Figure 6.If we now consider Coxian representations with −1,−2,−3,−λ,−µ, where µ ≥ λ ≥ 3, andintersect all the resultant intersecting sets, we get the set S5 depicted in Figure 7. Sincethe boundary of S5 depends on the two parametrs λ and µ with λ ≤ µ it is difficult to plotit as a single curve. The “shaded” and the thin “unshaded” regions above the boundaryof S4 constitute S5.To further illustrate the process we find the set of points in Ω4 that correspond to Coxiandistributions of algebraic degree 4 with poles −1,−1,−1,−1 that have triangular order 5with eigenvalues −1,−1,−1,−1,−2. Figure 8 shows this set. Here, the set of points thatcorrespond to all Coxian distributions of algebraic degree 4 with poles −1,−1,−1,−1 thathave triangular order 5 with eigenvalues −1,−1,−1,−1,−2, consists of two tetrahedra ontop of T4. The vertices of the first tetrahedron are F11, F111, F1111, and A1, and the verticesof the second tetrahedron are F1, F11, F1111, and A2. Also, since A1 and A2 and the lineF1F1111 are in the same plane, the set is convex. When we allow λ ≥ 1 to vary, the pointsA1 and A2 trace out two curves which are labeled C1 and C2 in Figure 9. The set thatcontains all points that corresponds to all Coxian distributions with poles −1,−1,−1,−1that have triangular order 5 consists of two conical shapes, one with F1111 as its vertexand F1F11, F11F111, and the curve C1 forming its base, the other with F1 as its vertexand F11F111, F111F1111, and the curve C2 forming its base. We note here, that for λ ≥ 1,each set is convex (as previously remarked), but since the line segment F1F1111 is commonto each set, the union of all the (convex) sets is not convex. Thus, the set of all PHdistributions of algebraic degree 3, with poles −1,−1,−1,−1 that have triangular order 5is not convex. This phenomena was observed by He and Zhang [7].6−2 0 2 4 6 8 10−1012345x1x 2F123 F23F3Figure 6: The set of all points that correspond to Coxian distributions of algebraic degree 3with poles −1,−2,−3, that have triangular order 5 whose representations have eigenvalues−1,−2,−3,−4,−5.−2 0 2 4 6 8 10−1012345x1x 2S4S5Figure 7: The set of all points that correspond to Coxian distributions of algebraic degree3 with poles −1,−2,−3, that have triangular order 5.7F111F1111F1F11A1A2Figure 8: The set of all points that correspond to Coxian distributions of algebraicdegree 4 with poles −1,−1,−1,−1, that have triangular order 5 with eigenvalues−1,−1,−1,−1,−2.The procedure described in this research note enables us to describe the set of allCoxian distributions of algebraic degree q with real poles that have triangular order p withCoxian generators that have real eigenvalues. For some cases of low algebraic degree andlow triangular order we are able to characterize the whole set of Coxian distributions withalgebraic degree q and triangular order p. However, a general description eludes us becauseof the complexity of the algebra involved. However, given that the procedure only requiresoperations from linear algebra it seems plausible that such a description could be achievedentirely using the products and sums of matrices. This will be left to later work.References[1] Asmussen, S.; Bladt, M. Renewal theory and queueing algorithms for matrix-exponential distributions. In Matrix-analytic Methods in Stochastic Models,Chakravarthy, S. R.; Alfa, A. S., editors; Lecture Notes in Pure and Applied Mathe-matics; Marcel Dekkar: New York, 1997; Volume 183, 313–341.[2] Bean, N. G.; Fackrell, M.; Taylor, P. G. Characterization of matrix-exponential dis-tributions. In final preparation, 2007.[3] Cox, D. R. A use of complex probabilities in the theory of stochastic processes. Pro-ceedings of The Cambridge Philosophical Society 1955, 51, 313–319.8F1111F111F11F1C1C2Figure 9: The set of all points that correspond to Coxian distributions of algebraic degree4 with poles −1,−1,−1,−1, that have triangular order 5.[4] Dehon, M,; Latouche, G. A geometric interpretation of the relations between theexponential and the generalized Erlang distributions. Advances in Applied Probability1982, 14, 885–897.[5] Fackrell, M. Characterization of Matrix-exponential Distributions. PhD thesis, Schoolof Applied Mathematics, University of Adelaide, South Australia, 2003.Available at http://thesis.library.adelaide.edu.au/public/adt-SUA20051207.123257/[6] He, Q-M.; Zhang, H. Spectral polynomial algorithms for computing bi-diagonal repre-sentations for phase-type distributions and matrix exponential distributions. Stochas-tic Models 2006, 22, 289–317.[7] He, Q-M.; Zhang, H. PH -majorization and Coxianization of PH -generators. Researchnote.[8] Neuts, M. F. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Ap-proach. The John Hopkins University Press: Baltimore, 1981.[9] O’Cinneide, C. A. Characterization of phase-type distributions. Communications inStatistics – Stochastic Models 1990, 6, 1–57.9[10] O’Cinneide, C. A. Phase-type distributions and invariant polytopes. Advances in Ap-plied Probability 1991, 23, 515–535.[11] O’Cinneide, C. A. Triangular order of triangular phase-type distributions. Communi-cations in Statistics – Stochastic Models 1993, 9, 507–52910

Characterizing Coxian Distributions of Algebraic Degree q and Triangular Order p

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Characterizing Coxian Distributions of Algebraic Degree q and Triangular Order p

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