515 d,forj≥3, EU/]=m-∑P EIUR-JJ U内+-1 We have thus proven the followin LUNi Remark 1. Equating the two expressions for elU I given by Propositions I and 2 yields explicit expression for the hyperharmonic number, which is defined in [2] by the recursive formula given in Proposition 2 3. The general case: Poissonization In the general case, we suppose that each item is of type k with probability Pk, 2kI Pk=1 To analyze this case, let us start by assuming that, rather than stopping when system l is filled, items continue coming forever. Suppose also that successive items arrive at times distributed according to a Poisson process with rate 1. Under this scenario, the arrival processes of the distinct types are independent Poisson processes, with respective rates Pk, k= 1 Because 1-P(a' denotes the probability that there have been less than j type-k arrivals when system I becomes full, we obtain upon conditioning on the arrival time of the jth item of type k 1-P(A4)= e e Pi)dx (-1)! The expected number of unfilled slots in system is now obtained from ∑Ⅱ-P(A),j≥2 5) k=1 The following lemma will be used to obtain bounds on E[UnI Lemma 1. For positive values xi, Ti=(l -e-xi) is a Schur concave function of yThe coupon-collector's problem revisited and, for j > 3, m E[U7] =m-E Pi- k=l -i = m-E(1 -E[UJ-k I _ k=l E[U_1] m E[O We have thus proven the following. Proposition 2. We have m E[U2 ] = k k=l and, for j > 3, m E[Uk_ ] E[Uj?] =j E k=l Remark 1. Equating the two expressions for E[UJ'] given by Propositions 1 and 2 yields an explicit expression for the hyperharmonic number, which is defined in [2] by the recursive formula given in Proposition 2. 3. The general case: Poissonization In the general case, we suppose that each item is of type k with probability Pk, Lkml Pk = 1. To analyze this case, let us start by assuming that, rather than stopping when system 1 is filled, items continue coming forever. Suppose also that successive items arrive at times distributed according to a Poisson process with rate 1. Under this scenario, the arrival processes of the distinct types are independent Poisson processes, with respective rates Pk, k = 1, ..., m. Because 1 - P(Ak) denotes the probability that there have been less than j type-k arrivals when system 1 becomes full, we obtain upon conditioning on the arrival time of the jth item of type k that 1 P(Ak 0I pkPkX (pkx)'j- 1- P(Aj) pke-PX ( ) n - -e-Pix)dx, j > 2. (4) The expected number of unfilled slots in system j is now obtained from m E[UJ7] = Z[1 - P(A5)], j > 2. (5) k=1 The following lemma will be used to obtain bounds on E[UJ]. Lemma 1. For positive values xi, f1=l (1 - e-xi) is a Schur concave function of y = (Y1, , Yr), where yi = ln(xi). 515