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Except for a factor we have an integral over an Erlang (n + 1, ) density Thus Eij (t) = 1

.

n=0

( t)k k!

By interchanging the order of summation, we next get the desired result 453 First Passage Time Probabilities In this section it is assumed that the state space I of the continuous-time Markov chain {X(t)} is nite For a given set C of states with C = I , de ne C = the rst epoch at which the continuous-time Markov chain {X(t)} makes a transition into a state of the set C Also, de ne the rst passage time probability QiC (t) by QiC (t) = P { C > t | X(0) = i}, i C and t > 0 /

.

The computation of the rst passage time probabilities QiC (t) can be reduced to the computation of the transient state probabilities in a modi ed continuous-time Markov chain with an absorbing state The most convenient way to model an

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absorbing state is to take its leaving rate equal to zero In the modi ed continuoustime Markov chain the set C is replaced by a single absorbing state to be denoted by a The state space I and the leaving rates i in the modi ed continuous-time Markov chain are taken as I = (I \C) {a} and i = i , 0, i I \C, i = a

The in nitesimal transition rates qij are taken as i, j I \C with j = i, qij , i I \C, j = a, qij = k C qik , 0, i = a, j I \C Denoting by pij (t) the transient state probabilities in the modi ed continuous-time Markov chain, it is readily seen that QiC (t) = 1 pia (t),

i C and t 0 /

week, the resilient Chicagoans had already built 5,000 temporary structures and started 200 permanent ones, which inspired Chicago Tribune editor Joseph Medill to write, In the midst of a calamity without parallel in the world s history, the people of this once beautiful city have resolved that CHICAGO SHALL RISE AGAIN 13 And rise it did, perhaps like no other city in the world In contrast to Tokyo and Berlin, which lost much of their ancient charm during their similarly massive rebuilding projects, Chicago had little of lasting value to lose in the bargain Chicago happily began to replace its seamier sides in favor of stone buildings, planned streets, and an infrastructure built to last This ability to adapt to sudden changes (which gerontologists tell us is one of the most common traits of those who live to be 100 years old) plus the capacity to surmount daunting obstacles and take on great challenges with complete conviction have long been central to Chicago s identity and, not coincidentally, Walgreens , too As recent chief executive officer (CEO) Dan Jorndt wrote to his minions, Don t be afraid to bite off more than you can chew: You ll be amazed how big your mouth can get 14 It s no accident that Walgreens has drawn its trademark resilience, grit, and understated confidence from the city that gave it birth In addition to possessing more than a little moxie, Chicago had all the raw materials it needed to fuel a roaring renaissance after the Great Fire, including Lake Michigan s endless supply of fresh water to the east, Wisconsin s vast acres of lumber to the north, the Midwest s fertile fields to the west and south, and the all-important shipping channels running through it By the time a young architect named Louis Sullivan took the train from Philadelphia to Chicago in 1873, his new city was home to the largest livestock, lumber, and grain markets in the world, with the biggest rail system to distribute all of it around the country As Sullivan noticed, biggest was the most popular word in the Chicago lexicon I thought it all magnificent and wild, Sullivan said of Chicago at the end of the nineteenth century A crude extravaganza; an intoxicating rawness 15 No large city, Miller wrote, not even Peter the Great s St Petersburg,.

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