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Taking the ratio of the expressions for the expected reward earned during one cycle and the expected length of one cycle and using relation (432), we get the result

of Theorem 422 for initial state r It remains to verify that the result also holds for any initial state X(0) = i with i = r This veri cation proceeds along the same lines as the proof of the corresponding result in Theorem 3511 By choosing an appropriate reward structure, Theorem 422 provides a rigorous proof of earlier interpretations we gave to the quantities pj and qjk pj Corollary 432 Suppose that the continuous-time Markov chain {X(t)} satis es Assumptions 412 and 421 Then (a) For each state k I , the long-run fraction of time the process is in state k equals pk with probability 1, independently of the initial state X(0) = i (b) For all j, k I with j = k, the long-run average number of transitions from state k to state j per unit time equals pk qkj with probability 1, independently of the initial state X(0) = i

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Many practical (queueing) problems can be modelled as a continuous-time Markov chain {X(t)} on a semi-in nite strip in the plane That is, the Markov process has the two-dimensional state space I = {(i, s) | i = 0, 1, ; s = 0, 1, , m} (441)

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for some nite positive integer m Assuming that the continuous-time Markov chain {X(t)} satis es Assumption 421, denote its equilibrium probabilities by p(i, s) for i = 0, 1, and s = 0, 1, , m These probabilities are determined by an in nite system of linear equations In many cases, however, this in nite system can be reduced to a nite system of linear equations of moderate size This can be done by using the geometric tail approach, discussed for discrete-time Markov chains in Section 342 Under rather general conditions the equilibrium probabilities p(i, s) exhibit a geometric tail behaviour as i , where the decay factor does not depend on s That is, for constants s > 0 and a constant with 0 < < 1, p(i, s) s i as i , (442)

Walgreen wanted to make this point with his top managers more forcefully, face to face So he summoned the directors of Walgreens 18 territories representing zone offices from New York in the East to Salt Lake City in the West, from Minnesota in the North to Houston in the South their top assistants, and Walgreens executives for the first company-wide meeting at Chicago s Stock Yard Inn (an irony Upton Sinclair would have appreciated) on April 6 and 7, 1931 The speakers most of them well-known Walgreens executives offered advice, support, and motivation to surmount the obstacles that the Great Depression presented, covering everything from merchandising to hiring to advertising nationwide In his lecture on the importance of window displays, vice president Harry Goldstine maintained, Precedent should not stand in the way of progress, giving the managers a free hand to try new approaches representing the outgrowth of progressive ideas Robert Knight weighed in on economizing Fine results have been attained, he said, by eliminating unnecessary things and unnecessary people, by giving the honey makers more scope and letting the drones loaf elsewhere It is not clear if he was advocating layoffs, but the Walgreens workforce grew substantially during the Depression Robert H Riemenschneider, the company s director of promotion and advertising, ended the two-day meeting with a crescendo by announcing that the company was less than two weeks away from launching a nationwide, five-day sale, running from April 16 to 20, and would generate publicity for the first-time event by spending over $75,000 for ads in 122 newspapers, blanketing every one of Walgreens 98 markets11 The managers were understandably thrilled to get such overwhelming backing and returned to their towns with great anticipation for the upcoming event The sale s success encouraged similarly aggressive efforts throughout the decade At the end of the next fiscal year, at the very nadir of the Depression, Charles Walgreen Sr received a personal letter from Arthur Andersen, father of the eponymous Big Five accounting firm, who felt compelled to.

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