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basement restuarant For that reason, Oveian wrote, he felt he deserved a gold watch from Walgreens Chuck asked Russell Plummer, who worked in Walgreens food service department, to fly out to New York to see if Oveian s story was on the level Plummer discovered that Oveian, a lifetime bachelor who rented a modest apartment near the Paramount, had naturally become something of a local legend with every Paramount Walgreens store employee, past and present, and had more seniority than all of them As far as he could, Plummer verified Oveian s claim no one there recalled him missing a single meal and recommended to Chuck Walgreen that he grant Oveian s request Walgreen agreed65 The store s fountain manager, Joseph Pape, and his supervisor, Fred Sparks (who remembered Oveian from the days Sparks worked there in the 1930s) presented the loyal customer with his gold watch in Times Square, as reported in the New Yorker s Talk of the Town section and Walter Winchell s nationally syndicated column When a Pepper Pod reporter asked Oveian if had ever grown weary of Walgreens menu after eating 31,799 meals there, he said, No, I had to stop because the store closed It should not come as any surprise that after Walgreens shut the doors on its Paramount Building shop, Oveian quickly switched his allegiance to the Walgreens at Times Square a few blocks away He and Chuck Walgreen continued their Christmas card correspondence every year until Oveian s death in the mid-1980s Walgreen had entertained the idea of taking the company public for some time The chain s incredible growth convinced him that they needed to go public to help fund the expansion underway, and the success of the first New York store convinced him they were ready On the morning of February 15, 1934, Walgreens could boast 483 stores (25 percent of which were outside the Midwest), 12,000 employees, and, for the first time, the letters WAG on the New York Stock Exchange.

E(reward earned during one cycle) E(length of one cycle)

with probability 1 Lemma 3510 states that E (number of visits to state j in one cycle) = j / r for any recurrent state j This relation is also valid for a transient state j , since a transient state is not accessible from a recurrent state and j = 0 for j transient Hence E(reward earned during one cycle) =

f (j )

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Since E(length of one cycle) = rr = 1/ r by (355), the assertion of the theorem is now proved when X0 = r Take next any initial state X0 = i As in the proof of Lemma 3510, let = (i0 , i1 , i2 , ) be any realization of the Markov chain with i0 = i and let ik denote the realized state at the kth state transition Since fir = 1, we have for almost all that there is a nite integer t = t ( ) such that it = r Hence 1 n

Detection points for the MO-SMS state model Description The subscriber has sent the short message or the SMS command to the MSC or SGSN There are no conditions prohibiting the SMS or command submission The CAMEL service is started A failure has occurred in the SMS or command submission The failure may have occurred internally in the MSC or SGSN or may have occurred externally, eg in the SMSC The short message or SMS command is successfully submitted to the SMSC

f (Xk ( )) =

t ( )

f (Xk ( ))

Letting n , it follows from part (b) of Assumption 332 that the rst term on the right-hand side of the equation tends to zero, while by the above proof the second term converges to j I f (j ) j This completes the proof Markov s proof and exponential convergence It is interesting to examine the original proof of Markov (1906) for the existence of a limiting distribution in a nite-state Markov chain The proof is not just of historical interest and the ideas it uses are still very much alive The proof also establishes the rate of convergence to the limiting distribution An aperiodic nitestate Markov chain with no two disjoint closed sets is assumed The Markov chain is said to be aperiodic when the period of the recurrent states is equal to 1; see Lemma 355 Theorem 3512 Let {Xn } be a nite-state Markov chain with no two disjoint closed sets Suppose that the Markov chain is aperiodic Then there exists a probability distribution { j , j I } and numbers > 0 and 0 < < 1 such that for all

Walgreens was on the board If you had bought a single share that day for $24625, it would have split nine times into 512 shares, each valued at $3625, by December 15, 2003 and would be worth a total of $18,560

132 i, j I ,

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