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a heuristic explanation of why the answer for the average waiting time is the same as the average residual life in a renewal process 218 Consider a renewal process in which the interoccurrence times have a positive density on some interval For any time t let the age variable t denote the time elapsed since the last occurrence of an event Use the renewal-reward model to prove that limt E( t ) = 2 /2 1 , where k is the kth moment of the interoccurrence times (Hint : assume a cost at rate x when a time x has elapsed since the last occurrence of an event) 219 A common car service between cities in Israel is a sheroot A sheroot is a seven-seat cab that leaves from its stand as soon as it has collected seven passengers Suppose that potential passengers arrive at the stand according to a Poisson process with rate An arriving person who sees no cab at the stand goes elsewhere and is lost for the particular car service Empty cabs pass the stand according to a Poisson process with rate An empty cab stops only at the stand when there is no other cab (a) De ne a regenerative process and identify its regeneration epochs (b) Determine the long-run fraction of time there is no cab at the stand and determine the long-run fraction of customers who are lost Explain why these two fractions are equal to each other 220 Big Jim, a man of few words, runs a one-man business This business is called upon by loan sharks to collect overdue loans Big Jim takes his profession seriously and accepts only one assignment at a time The assignments are classi ed by Jim into n different categories j = 1, , n An assignment of type j takes him a random number of j days and gives a random pro t of j dollars for j = 1, , n Assignments of the types 1, , n arrive according to independent Poisson processes with respective rates 1 , , n Big Jim, once studying at a prestigious business school, is a muscleman with brains He has decided to accept those type j assignments for which E( j )/E( j ) is at least g dollars per day for a carefully chosen value of g (in Exercise 74 you are asked to use Markov decision theory to determine g ) Suppose that Big Jim only accepts type j assignments for j = 1, , n0 An assignment can only be accepted when Big Jim is not at work on another assignment Assignments that are refused are handled by a colleague of Big Jim (a) De ne a regenerative process and identify its regeneration epochs (b) Determine the long-run average pay-off per time unit for Big Jim (c) Determine the long-run fraction of time Big Jim is at work and the long-run fraction of the assignments of the types 1, , n0 that are not accepted Explain why these two fractions are equal to each other 221 Consider the (S 1, S) inventory model with back ordering from Section 113 What is the long-run fraction of customer demand that is back ordered What is the long-run average amount of time a unit is kept in stock 222 Consider a machine whose state deteriorates through time The state of the machine is inspected at xed times t = 0, 1, In each period between two successive inspections the machine incurs a random amount of damage The amounts of damage accumulate The amounts of damage incurred in the successive periods are independent random variables having a common exponential distribution with mean 1/ A compulsory repair of the machine is required when an inspection reveals a cumulative amount of damage larger than a critical level L A compulsory repair involves a xed cost of Rc > 0 A preventive repair at a lower cost of Rp > 0 is possible when an inspection reveals a cumulative amount of damage below or at the level L The following control limit rule is used A repair is done at each inspection that reveals a cumulative amount of damage larger than some repair limit z with 0 z < L It is assumed that each repair takes a negligible time and that after each repair the machine is as good as new (a) De ne a regenerative process and identify its regeneration epochs (b) What is the expected number of periods between two successive repairs What is the.

Like his newsletter, Walgreen believed that all work and no play made for dull boys, so he had no qualms about diverting some of his expanding fortune for fun The family bought a new home on Drexel Boulevard catering to all Walgreen s favorite pastimes, including rooms for reading, pingpong, billiards, and a bar In 1910, the Walgreens joined the Waupansa Club on Drexel Boulevard, a respected social club, so Charles could frequent a more refined setting to play his favorite game, billiards, than the neighborhood halls (However, he never stopped visiting the local pool halls altogether)

(a) It will be obvious that the process {N (t)} satis es the properties (A) and (B) To verify property (C) note that P {one arrival in (t, t +

one arrival of type i and no arrival of the other type in (t, t + t]

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IDP RRB, CUE IDP CUE ERB[Answer] ERB[Disconnect]

= [ 1 t + o( t)][1 2 t + o( t)] + [ 2 t + o( t)][1 1 t + o( t)] = ( 1 + 2 ) t + o( t) Property (D) follows by noting that P {no arrival in (t, t + t]} = [1 1 t + o( t)][1 2 t + o( t)] = 1 ( 1 + 2 ) t + o( t) as t 0 as t 0

This completes the proof that {N (t)} is a Poisson process with rate 1 + 2 To prove the other assertion in part (a), denote by the random variable Yi the interarrival time in the process {Ni (t)} Then P {Zk > t, Ik = 1} = P {Y2 > Y1 > t} = =

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