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Retrial queues are in general much more dif cult to analyse than queues without retrials The Laplace transform for the waiting-time distribution in the M/G/1 queue with retrials is very complex; see also Artalejo et al (2002) 234 Consider again the production system from Section 26 except that the system is now controlled in a different way when it becomes idle Each time the production facility becomes empty of orders, the facility is used during a period of xed length T for some other work in order to utilize the idle time After this vacation period the facility is reactivated for servicing the orders only when at least one order is present; otherwise the facility is used again for some other work during a vacation period of length T This utilization of idle time is continued until at least one order is present after the end of a vacation period This control policy is called the T -policy The cost structure is the same as in Section 26 Use renewal-reward theory to show that K(1 1 )(1 e T )/T + 1 h T + 1 h 2 2 /(1 1 ) 2 2 gives the long-run average cost per time unit under a T -policy 235 Suppose that, at a communication channel, messages of types 1 and 2 arrive according to independent Poisson processes with respective rates 1 and 2 Messages of type 1 nding the channel occupied upon arrival are lost, whereas messages of type 2 are temporarily stored in a buffer and wait until the channel becomes available The channel can transmit only one message at a time The transmission time of a message of type i has a general probability distribution with mean i and the transmission times are independent of each other It is assumed that 2 2 < 1 Use the renewal-reward theorem to prove that the long-run fraction of time the channel is busy equals ( 1 + 2 )/(1 + 1 ), where i = i i for i = 1, 2.

(Hint : use results from Section 26 to obtain the expected amount of time elapsed between two arrivals nding the channel free)

Wait_for_ Request (WFR)

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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)

To anyone who lived through the 1990s, a thumbnail sketch of the Roaring Twenties will sound very familiar, indeed: a decade of unprecedented peace and prosperity, fun and frivolity, which came crashing down at the decade s close The landing in the 1920s, however, was a lot rougher than that in the 1990s Fatigued by the drudgery of farm life and worn out by the horrors of the Great War, young Americans moved to cities like Chicago by the millions There they discovered hair bobs and hip flasks, jazz clubs and ball clubs The decade marked the Golden Era of Sports Fans flocked to see Babe Ruth, Bobby Jones, Bill Tilden, and Jack Dempsey A brand new circuit, the National Hockey League, started a franchise in Chicago called the Blackhawks, in honor of the tribe Abraham Lincoln s unit fought outside Dixon, Illinois, in 183225 The public s immense interest in college football justified the con-

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