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(Hint : use results from Section 26 to obtain the expected amount of time elapsed between two arrivals nding the channel free)
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The very readable monograph of Cox (1962) contributed much to the popularization of renewal theory A good account of renewal theory can also be found in the texts Ross (1996) and Wolff (1989) A basic paper on renewal theory and regenerative processes is that of Smith (1958), a paper which recognized the usefulness of renewal-reward processes in the analysis of applied probability problems The book of Ross (1970) was in uential in promoting the application of renewal-reward processes The renewal-reward model has many applications in inventory, queueing and reliability The illustrative queueing example from Section 26 is taken from the paper of Yadin and Naor (1963), which initiated the study of control rules for queueing systems Example 223 is adapted from the paper of Vered and Yechiali (1979) The rst rigorous proof of L = W was given by Little (1961) under rather strong conditions; see also Jewell (1967) Under very weak conditions a samplepath proof of L = W was given by Stidham (1974) The important result that Poisson arrivals see time averages was taken for granted by earlier practitioners A rigorous proof was given in the paper of Wolff (1982) The derivation of the Laplace transform of the waiting-time distribution in the M/G/1 queue is adapted from Cohen (1982) and the relation between this transform and the generating function of the queue size comes from Haji and Newell (1971)
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REFERENCES
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Artalejo, JR, Falin, GI and Lopez-Herrero, MJ (2002) A second order analysis of the waiting time in the M/G/1 retrial queue Asia-Paci c J Operat Res, 19, 131 148 Cohen, JW (1982) The Single Server Queue, 2nd edn North-Holland, Amsterdam Cox, DR (1955) The statistical analysis of congestion J R Statist Soc A, 118, 324 335 Cox, DR (1962) Renewal Theory Methuen, London Haji, R and Newell, GF (1971) A relation between stationary queue and waiting-time distribution J Appl Prob, 8, 617 620 Jewell, WS (1967) A simple proof of L = W Operat Res, 15, 1109 1116 Keilson, J (1979) Markov Chain Models Rarity and Exponentiality Springer-Verlag, Berlin Little, JDC (1961) A proof for the queueing formula L = W Operat Res, 9, 383 387 Miller, DR (1972) Existence of limits in regenerative processes Ann Math Statist, 43, 1275 1282 Ross, SM (1970) Applied Probability Models with Optimization Applications Holden-Day, San Francisco Ross, SM (1996) Stochastic Processes, 2nd edn John Wiley & Sons, Inc, New York Smith, WL (1958) Renewal theory and its rami cations J R Statist Soc B , 20, 243 302 Solovyez, AD (1971) Asymptotic behaviour of the time of rst occurrence of a rare event in a regenerating process Engineering Cybernetics, 9, 1038 1048
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Stidham, S Jr (1974) A last word on L = W Operat Res, 22, 417 421 Tak cs, L (1962) Introduction to the Theory of Queues Oxford University Press, New York a Vered, G and Yechiali, U (1979) Optimal structures and maintenance policies for PABX power systems Operat Res, 27, 37 47 Wolff, RW (1982) Poisson arrivals see time averages Operat Res, 30, 223 231 Wolff, RW (1989) Stochastic Modeling and the Theory of Queues Prentice Hall, Englewood Cliffs NJ Yadin, M and Naor, P (1963) Queueing systems with removable service station Operat Res Quart, 14, 393 405
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The notion of what is nowadays called a Markov chain was devised by the Russian mathematician AA Markov when, at the beginning of the twentieth century, he investigated the alternation of vowels and consonants in Pushkin s poem Onegin He developed a probability model in which the outcomes of successive trials are allowed to be dependent on each other such that each trial depends only on its immediate predecessor This model, being the simplest generalization of the probability model of independent trials, appeared to give an excellent description of the alternation of vowels and consonants and enabled Markov to calculate a very accurate estimate of the frequency at which consonants occur in Pushkin s poem The Markov model is no exception to the rule that simple models are often the most useful models for analysing practical problems The theory of Markov processes has applications to a wide variety of elds, including biology, computer science, engineering and operations research A Markov process allows us to model the uncertainty in many real-world systems that evolve dynamically in time The basic concepts of a Markov process are those of a state and of a state transition In speci c applications the modelling art is to nd an adequate state description such that the associated stochastic process indeed has the Markovian property that the knowledge of the present state is suf cient to predict the future stochastic behaviour of the process In this chapter we consider discrete-time Markov processes in which state transitions only occur at xed times Continuous-time Markov processes in which the state can change at any time are the subject of 4 The discrete-time Markov chain model is introduced in Section 31 In this section considerable attention is paid to the modelling aspects Most students nd the modelling more dif cult than the mathematics Section 32 deals with the n-step transition probabilities and absorption probabilities The main interest, however, is in the long-run behaviour of the Markov chain In Section 33 we discuss both the existence of an equilibrium distribution and the computation of this distribution
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A First Course in Stochastic Models HC Tijms c 2003 John Wiley & Sons, Ltd ISBNs: 0-471-49880-7 (HB); 0-471-49881-5 (PB)
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