zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The matrix Laguerre transform. (English) Zbl 0553.44002
The Laguerre transform, introduced by {\it J. Keilson} and {\it W. R. Nunn} [ibid. 5, 313-359 (1979; Zbl 0449.65086)], {\it J. Keilson}, {\it W. R. Nunn} and the author [ibid. 8, 137-174 (1981; Zbl 0457.42011)], and further studied by the author [Development of the Laguerre transform method for numerical exploration of applied probability models, Ph. D. Diss., Grad. School Management, Univ. Rochester (1981)], provides an algorithmic basis for the computation of multiple convolutions in conjunction with other algebraic and summation operations. The methods enable one to evaluate numerically a variety of results in applied probability and statistics that have been available only formally behind the ”Laplacian curtain”. For certain more complicated models, the formulation must be extended. In this paper we establish the matrix Laguerre transform, appropriate for the study of semi-Markov processes and Markov renewal processes, as an extension of the scalar Laguerre transform. The new formalism enables one to calculate matrix convolutions and other algebraic operations in matrix form. As an application, a matrix renewal function is evaluated and its limit theorem is numerically exhibited. In a recent paper by the author and Kijima (1984) the bivariate Laguerre transform has also been developed for the study of bivariate processes and distributions.

MSC:
 44A30 Multiple transforms 44A15 Special transforms (Legendre, Hilbert, etc.) 44A35 Convolution (integral transforms) 60J25 Continuous-time Markov processes on general state spaces 60K05 Renewal theory
Full Text:
References:
 [1] Cinlar, E.: Markov renewal theory: A survey. Management sci. 21, No. 7, 726-752 (1975) · Zbl 0314.90095 [2] Dym, H.; Mckean, H. P.: Fourier series and integrals. (1972) · Zbl 0242.42001 [3] Gaver, D. P.: A waiting line with interrupted service, including priorities. J. roy. Statist. soc. Ser. B 24, 73-90 (1962) · Zbl 0108.31403 [4] Keilson, J.: Queues subject to service interruption. Ann. math. Statist. 33, No. No. 4 (1962) · Zbl 0104.36805 [5] Keilson, J.: On the matrix renewal function for Markov renewal processes. Ann. math. Statist. 40, 1901-1907 (1969) · Zbl 0193.45303 [6] Keilson, J.; Nunn, W. R.: Laguerre transformation as a tool for the numerical solution of integral equations of convolution type. Appl. math. And comp. 5, 313-359 (1979) · Zbl 0449.65086 [7] Keilson, J.; Nunn, W. R.; Sumita, U.: The bilateral Laguerre transform. Appl. math. Comput. 8, No. 2, 137-174 (1981) · Zbl 0457.42011 [8] Keilson, J.; Petrondas, D.; Sumita, U.; Wellner, J.: Significance points for some tests of uniformity on the sphere. J. statist. Comp. simul. 17, No. 3, 195-218 (1983) · Zbl 0533.62046 [9] Keilson, J.; Sumita, U.: Waiting time distribution response to traffic surges via the Laguerre transform. Proceedings of the conference on applied probability -- computer science: the interface (1981) [10] Keilson, J.; Sumita, U.: The depletion time for M / G /1 systems and a related limit theorem. Adv. appl. Prob. 15, 420-443 (1983) · Zbl 0528.60096 [11] Keilson, J.; Sumita, U.: A general Laguerre transform and a related distance between probability measures. Working paper series no. OM8309 (1983) · Zbl 0531.60017 [12] Keilson, J.; Wishart, D. M. G.: A central limit theorem for processes defined on a finite Markov chain. Proc. Cambridge philos. Soc. 60, 547-567 (1964) · Zbl 0126.33504 [13] Keilson, J.; Wishart, D. M. G.: Boundary problems for additive processes defined on a finite Markov chain. Proc. Cambridge philos. Soc. 61, 173-190 (1965) · Zbl 0138.40703 [14] Keilson, J.; Wishart, D. M. G.: Addenda to processes defined on a finite Markov chain. Proc. Cambridge philos. Soc. 63, 189-193 (1967) · Zbl 0147.16401 [15] Sumita, U.: Numerical evaluation of multiple convolutions, survival functions, and renewal functions for the one-sided normal distribution and the Rayleigh distribution via the Laguerre transformation. Working paper series no. 7912 (1979) [16] Sumita, U.: On sums of independent logistic and folded logistic variants. Working paper series no. 8001 (1980) [17] Sumita, U.: Development of the Laguerre transform method for numerical exploration of applied probability models. Ph.d. dissertation (1981) [18] Takács, L.: Introduction to the theory of queues. (1962) · Zbl 0106.33502