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By means of a general formula for stochastic processes with imbedded marked point processes (PMP) some necessary and sufficient condition is given for the validity of a relationship, which is well-known in the case of exponentially distributed service times, between stationary time and customer state probabilities for loss systems G/GI/s/O (Theorem 3). A result of Miyazawa for the GI/GI/l/∞ queue is generalized to the case of non-recurrent interarrival times (Theorem 4)-. Furthermore, bounds are derived for the mean increment of the waiting time process at arrival epochs and for the mean actual waiting time in multi-server queues.

In generalization of special cases of the literature a class of stochastic processes (PMP) is defined with an imbedded stochastic marked point process of "basic points" which must not be renewal points. A theorem ("intensity conservation principle") has been proved concerning a relation between stationary distribution of PMP at arbitrary points in time and distributions and intensities connected with the basic points. This relationship simultaneously yields a general method for determination of stationary quantities at arbitrary points in time by means of the corresponding "imbedded" quantities. Some applications to concrete queueing systems have been demonstrated, where arrival or departure epochs of customers are used as basic points. Under weaker independence assumptions as till now done in the literature, new relations are given.

The weakness of stochastic independence conditions, assumed in queuing theory, is a great importance to take into consideration the properties of real queueing systems as well as possible. The present article gives a review about publications that assume weaker independence conditions than in GI/GI queues. Especially, the theory of stochastic marked point processes is a proper tool for description and, partially, investigation of such queueing systems. Conditions for the unique existence of stationary state probabilities and relations between these probabilities are discussed for several types of queuing systems. Continuity theorems and results for systems with special dependence structure are given. Next, some methods are mentioned, which have been applied to the study of queues with stochastic dependences.

In the present paper we consider the stationary distributions of the number of customers in G/G/s queuing systems at arbitrary points in time and at arrival or departure epochs. Using the theory of stationary marked point processes it can be shown that these distributions are comparable with respect to the stochastic order relation ≦. By this means we get, as a special case, the results of Marshall-Wolff [9] concerning first moments of the mentioned distributions. For GI/G/I queues we give and estimation for the difference between expected queue lengths at arbitray points in time and just before and arrival epoch.

BeiglböCK, W.D.Lineare Algebra. Eine anwendngsorientierte Einführung. Springer.. Verlag Berlin, Heidelberg, New York, Tokyo 1983. XXV, 296 S., D:NI39,50, ISBN 3-540-12477-2.Bourgin, R.D.:Geometrie Aspects of Convex Sets with the Radon Nikodým Property, Lecture Notes in Mathematics. Vol. 993. Springer Verlag Berlin, Heidelberg, New York, Tokyo 1983, XII, 474 S., DM 55,-, ISBN 3-540-09504-7.KÖVES, P.:Index Theory and Eeonomic Reality. Akademiai Kiadó Budapest 1983, 312 S., Ft. 290Nicolai, W.:Time-Oriented Analysis of Projects Using Stochastic Network Techniques.Mathematical Systems in Economics, Vol. 61. Verlag Anton Hain, Meisenheim GmbH, Königstein/Ts. 1981, 116 S., DM 38,-, ISBN 3-445-12145-1.Dontchev, A.L.:Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems Lectures Notes in Control and Information Sciences Vol 52. Springer Verlag Heidelberg, .New York 1983, IV, 158 S., DM 28,-, ISBN 3-540-12463-2.Zielinski, R.;Neumann, P.:Stoehastische Verfahren zur Suche mach dem minimum einer Funktion. Mathematical Research, Bd. 16. Akademie-Verlag Berlin 1983, 133 S., 18,- M.Braun, M: Differential Equations and Their Applications. An Introduction to Applied Mathematics.:3.Ed. Applied Mathematical Sciences ·Vo1. 15. Springer Verlag New York, Heidelberg, Berlin 1983,546 S., DM 66.-, ISBN 3-540-90806-4.Beckmann, M.J.: Tinbergen Lectures on Organization Theory. Texts and Monographs in Economics and Mat.hernat.ical Systems. Springer-Verlag Berlin, Heidelberg, New York, Tokyo 1983, XI + 176 S. DM:39,-, ISBN 3-540-12646-5.LIU, C.H.:General Deeoupling Theory of Multivariable Process Control Systems. Lecture Notes in Control and .Inforrnat.ion Sciences. VoL 53 Springer Verlag Berlin, Heidelborg New York, Tokyo 1983. XI, 474 p. DM 62,-, ISBN 3-540-12496-9.Brønsted, A..An Introduction to Convex Polytopes. Graduate Texts in Mathematics, Vol. 90.Springer Verlag New York, Heidelberg, Berlin 1983, 162 S. DM 69,-, ISBN 3-387-90722-X.Wäscher, G.:Innerbetriebliche Standortplanung bei einfacher und mehrfacher Zielsetzung.Verlag Gabler GmbH Wiesbaden 1982,389 S.Kallianpub, G. (ed.): Theorie and Appllications of Random Fields. Proceedings of the IFIP-WG 7/1 Working Conference held under the joint auspices of the Indian Statistical Institute Bangalore, India, January 1982. Lecture Notes in Control and Information Sciences Vol.. 49.Springer-Verlag Berlin, Heidelberg,. New York 1983, 290 33,50 DM, ISBN 3-540-12232-X.Yadrenko, M.I.:Spectral Theory of Random Fields. Translation Series in Mathematics and Engineering. Optirnization Software, Inc. Springer-Verlag Berlin, Heidelberg, New York 1983, 259 S., DM 68,-, ISBN 3-540-90823-4.

Ito, K.;T.Hida (eds); Stochastic Processes and Their Applications. Proceedings of the International Conference held in Nagoya, Jul 2-6, 1985. Lecture Notes in Mathematics,vol.1203, Springer-Verlag 1986, VI, 222 p., DM 35,-, ISBN 3-540-16773-0.Berg, L.; Lineare Gleichungssysteme mit Bandstruktur. VEB Deutscher Verlag der Wissenschaften Berlin 1986, 247 S., 16 Abb., 9 Tab., 19,-M. ISBN 3-326-00022-7Fröberg, C.E.: Numerical Mathematics. Theory and Computer Applications. The Benjamin/Cummings Publishing Company, Inc. 1985, 436 p. S 37,95, ISBN 0-8053-2530-1Hairer, E.; S.P.Norsett; G.Wanner: Solving Ordinary Differential Equations I - Nonstiff Problems.Springer-Verlag Berlin, Heidelberg 1987, 480 p., DM 124,-, ISBN 3-540-17145-2Grubb, G.:Functional Calculus of Pseudo-Differential Problems. Birkhäuscr Verlag Basel. Boston, Stuttgart 1986, 512 p., DM 122,-, ISBN 3-7643-3349-9Albers, J.; St. B.Rodi; A.E.Watkins: New Directions in Two-Year College Mathematics. Proceedings of the Sloan Foundation Conference on Two-Year College Mathematics, held on Jul 11-14, 1984, at Menlo College in Atherton, California. New York 1985, 492 P., DM 94,-, ISBN 3-540-96145-3Lau, H.T.: Combinatorial Heuristic Algorithms with FORTRAN. Lecture Notes in Economics and Mathematical Systems Vol.280, Springer-Verlag Berlin-Heidelberg, New York, London, Tokyo1986, 126 p., DM 29,-, ISBN 3-540-17161-4Kowaljow, M.M.(ed): Matroide in der diskreten Optimierung. 1987 Minsk, Universitetskoje, 222 S., russKarhunen, J.(ed): Recursive Estimation of Eigenvectors of Correlation Type Matriees for Signal Processing Applications. Dissertation 56 of the Helsinki University of Technology, Dep. of Technical Physics Espoo, Finland 1984, 57 ppWeiss, P.: Stochastisehe Modelle für Anwender, B.G. Teubner Verlag, Stuttgart 1987, 192 S., 47 Abb., 16 Tab., DM 36,-, ISBN 3-519-02621-XDemyanov, V.F.: A.M.Rubinov: Quasidifierential Calculus, Translations Servies in Mathematics and Engineering. Optimization Software. Inc. Publications Division, New york 1986, XI, 288 S., ISBN 0-911575-35-9Kohlas, J.: Zuverlässigkeit und Verfügbarkeit. B.G.Teubner, Stuttgart 1987, DM 38,-, 252 p., ISBN 3-519-02357-1Schofield, N.J.: Social Choice and Democracy. Springer-Verlag Berlin, Heidelberg, New York, Tokyo 1985, XVII., 321 pp.; DM 120,-, ISBN 3-540-15604-6Çinlar, E.; K.L.Chung; R.K.Getoor (eds): Seminar on Stochastic Processes 1985.and Statistics. Volume 12. Birkhäuser Boston, Basel, Stuttgart 1986, 336 p., sFr. 78.-. ISBN 3-7643-3331-6Henn,R.; D.Pallaschke:Methods of Operations Research.Vol. 56. Contributions to Automatical Economics and Optimization Thoeory. Verlag Anton Hain Meisenheim GmbH, Frankfurt/M. 1986, VII,175 P.ISBN 3-445-02412.XBraess, D.:Nonlinear Approximation Theory. Springer-Verlag Berlin, Heidelberg, New York 1986, 38 fig., XIV, 290 pp., DM 148,-, ISBN 3-540-13625-8Rubinstein, Y.;Monte Carlo Optimization, Simulation and Sensitivity of Queueing Neworks. John Wiley, New York, Chichester, Brisbane 1986, XI, 260 p., ISBN 0-471-81929-8

Spies, P.P.: Grundlagen stochastischer Modelle. Einfuhrung fur Studierende der Informatik. Carl Hanser Verlag, Munchen, Wein 1983, 268 S., DM 36