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812 Consider an unreliable production unit whose output is temporarily stored in a nite buffer with capacity K The buffer serves for the demand process as protection against random interruptions in the production process For the output there is a constant demand at rate When operating, the production unit produces at a constant rate P > if the buffer is not full and produces at the demand rate otherwise If demand occurs while the unit is down and the buffer is empty then it is lost The operating time of the unit is exponentially distributed with mean 1/ If a failure occurs, the unit enters repair for an exponentially distributed time with mean 1/ Determine the long-run fraction of demand lost and determine the average inventory level in the buffer (Hint : de ne the state of the system as (1, x) and (0, x) respectively when the inventory in the buffer is x and the unit is operating or down The process regenerates itself each time the system enters state (0,0) Use differential equations to get the desired performance measures) This problem is motivated by Wijngaard (1979)
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The key renewal theorem has a long history, and analytic proofs were given under rather restrictive conditions The reader is referred to the book of Feller (1971) for a transparent proof under the weak condition of direct Riemann integrability; see also Asmussen (1987) The results for the alternating renewal process in Section 83 are proved in greater generality in Tak cs (1957) The material from Example 831 is a based on the paper of Van der Heijden (1987) The renewal-theoretic method used in Section 84 to derive asymptotic estimates for ruin and waiting-time probabilities comes from Feller (1971) Another application of this powerful method to a storage problem for dams is given in De Kok et al (1984) A good reference on heavytailed distributions is Embrechts et al (1997)
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Asmussen, S (1987) Applied Probability and Queues John Wiley & Sons, Inc, New York Dekker, R, and Smeitink, E (1994) Preventive maintenance at opportunities of restricted duration Naval Res Logist , 41, 335 353 De Kok, AG, Tijms, HC and Van der Duyn Schouten, FA (1984) Approximations for the single-product production-inventory problem with compound Poisson demand and service level constraints Adv Appl Prob, 16, 378 401 Den Iseger, PW, Smith, MAJ and Dekker, R (1997) Computing compound Poisson distributions faster Insurance Mathematics and Economics, 20, 23 34 Embrechts, P, Kl ppelberg, C and Mikosch, T (1997) Modelling Extremal Events u Springer-Verlag, Berlin Feller, W (1971) An Introduction to Probability Theory and Its Applications, Vol II, 2nd edn John Wiley & Sons, Inc, New York Gaver, DP (1963) Time to failure and availability of paralleled systems with repair IEEE Trans Reliab, 12, 30 38 Lorden, G (1970) On excess over the boundary Ann Math Statist , 41, 520 527 Luss, H (1976) Maintenance policies when deterioration can be observed by inspections Operat Res, 24, 359 366 Tak cs, LJ (1957) On certain sojourn time problems in the theory of stochastic processes a Acta Mathematica Academiae Scientiarum Hungaricae, 8, 169 191 Van der Heijden, MC (1987) Interval availability distribution for a 1-out-of-2 reliability system Prob Engng Inform Sci, 2, 211 224 Wijngaard, J (1979) The effect of interstage buffer storage on the output of two unreliable production units in series with different production rates AIEE Trans, 11, 42 47 Xie, M (1989) On the solution of renewal-type integral equations Commun Statist, 18, 281 293
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Queueing models have their origin in the study of design problems of automatic telephone exchanges and were rst analysed by the queueing pioneer AK Erlang in the early 1900s In planning telephone systems to meet given performance criteria, questions were asked such as: How many lines are required in order to give a certain grade of service What is the probability that a delayed customer has to wait more than a certain time before getting a connection Similar questions arise in the design of many other systems: How many terminals are needed in a computer system so that 80% of the users get access to a terminal within 20 seconds What will be the effect on the average waiting time of customers when changing the size of a maintenance staff to service leased equipment How much storage space is needed in buffers at workstations in an assembly line in order to keep the probability of blocking below a speci ed acceptable level These design problems and many others concern, in fact, facilities serving a community of users, where both the times at which the users ask for service and the durations that the requests for service will occupy facilities are stochastic, so that inevitably congestion occurs and queues may build up In the rst stage of design the system engineer usually needs quick answers to a variety of questions like those posed above Queueing theory constitutes a basic tool for making rstapproximation estimates of queue sizes and probabilities of delays Such a simple tool should in general be preferred to simulation, especially when it is possible to have a large number of different con gurations in the design problem In this chapter we discuss a number of basic queueing models that have proved to be useful in analysing a wide variety of stochastic service systems The emphasis will be on algorithms and approximations rather than on mathematical aspects We feel that there is a need for such a treatment in view of the increased use of queueing models in modern technology Actually, the application of queueing theory in the performance analysis of computer and communication systems has
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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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