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The 1902 wedding of Charles Sr and Myrtle secured the Walgreens family life forever By all accounts, their marriage was a happy one, made stronger by the birth of Charles Jr (or Chuck, as he was soon known) in 1906 and Ruth, with bright red hair, in 1910 Despite working long, arduous hours, Charles Sr would come home and get on all fours to thrill his young children, barking like a dog and bucking like a bronco to make them laugh Walgreen s happy family life didn t answer the questions in his professional life, however Although he had just taken out a heavy loan to become a co-owner of Blood s store about the same time he met Myrtle and was only a year and change into paying off the loan when the two were married, he still appeared far from committed to a lifelong career in pharmacology

from the explicit expression for P (z) Note that each of the points zk = e 2 ik/n , k = 0, , n 1 satis es zn = 1 and thus lies on the unit circle |z| = 1 By the power series representation of P (z) and the choice of the integer n, we have ck 1 n

n 1 =0

This relation is of the same form as (D3) Thus the unknown probabilities p can be calculated by applying the inverse discrete FFT method to the known vector (c0 , , cn 1 ) Example D1 The M/D/1 queue Consider the M/D/1 queue with deterministic services In Section 25 it was shown that the generating function of the limiting distribution {pk } of the number of customers present is given by P (z) = (1 D)(1 z) , 1 ze D(1 z) |z| 1,

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The following enhancements are introduced in CAMEL phase 4 in 3GPP Rel-6: enhanced dialled service; handover noti cation criteria; enhancement to SCUDIF control; reporting of user-to-user information; enhancement to user interaction

where is the arrival rate of customers and D is the xed service time of a customer with D < 1 Hence the state probabilities {pk } can be calculated by applying the discrete FFT method In the speci c problem of the M/D/1 queue, the explicit expression for the generating function P (z) is of the form Q(z)/R(z) In such a situation one should verify whether or not R(z) has zeros on the unit circle |z| = 1 (each zero of R(z) on the unit circle must also be a zero of Q(z) since P (z) = pk zk is analytic for |z| 1) If a point zk = e 2 ik/n is a k=0 zero of R(z), the corresponding Fourier coef cient ck cannot be calculated directly from (D5) but can be found by applying L Hospital s rule to Q(z)/R(z) at the point z = zk (often z0 = 1 is a zero as is the case in the M/D/1 problem)

This appendix gives a brief outline of some results from Laplace transform theory that are useful in applied probability problems Suppose that f (x) is a continuous real-valued function in x 0 such that |f (x)| AeBx , x 0, for some constants A and B The Laplace transform of f (x) is de ned by the integral f (s) =

as a function of the complex variable s with Re(s) > B The integral always exists when Re(s) > B If f (x) is the probability density of a random variable X, the Laplace transform f (s) is de ned for all s with Re(s) > 0 and can be interpreted as (E1) f (s) = E(e sX ) Moreover, we then have E(Xk ) = ( 1)k lim d k f (s) , s 0 ds k k = 1, 2, (E2)

The results (a) to (c) below can easily be veri ed from the de nition of Laplace transform In the statements it is assumed that the various integrals exist (a) If the function f (x) = ag(x) + bh(x) is a linear combination of the functions g(x) and h(x) with Laplace transforms g (s) and h (s), then f (s) = ag (s) + bh (s) (b) If F (x) =

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