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Armed with an explicit expression for W0 , we are able to give a formula for the long-run average number of back orders outstanding at the bases For each base j the failed items arriving at base j can be thought of as customers entering service in a queueing system with in nitely many servers Here the service time should be de ned as the repair time in case of repair at the base and otherwise as the time until receipt of a replacement from the depot Thus the average service time of a customer at base j is given by j = rj j + (1 rj )( j + W0 ), j = 1, , N

The situation at base j can only be modelled approximately as an M/G/ queue The reason is that the arrival process of failed items interferes with the replacement times at the depot so that there is some dependency between the service times at base j Assuming that this dependency is not substantial, we nevertheless use the M/G/ queue as an approximating model and approximate the limiting distribution of the number of items in service at base j by a Poisson distribution with

mean j j for j = 1, , N In particular, the long-run average number of back orders outstanding at base j

.

( j j )k , k!

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This expression and the expression for W0 enables us to calculate the total average number of outstanding back orders at the bases for a given assignment (S0 , S1 , , SN ) Next, by some search procedure, the optimal values of S0 , S1 , , SN can be calculated 114 The Poisson Process and the Uniform Distribution In any small time interval of the same length the occurrence of a Poisson arrival is equally likely In other words, Poisson arrivals occur completely randomly in time To make this statement more precise, we relate the Poisson process to the uniform distribution Lemma 114 For any t > 0 and n = 1, 2, ,

P {Sk x | N (t) = n} =

(117)

Herbert Hoover, he replied, Why not I had a better year than he did Few argued2 It is not surprising that the American people voted to oust Herbert Hoover in the 1932 election in favor of an unheralded and underestimated presidential candidate named Franklin Delano Roosevelt By the time he took office in 1933, FDR had his work cut out for him Five thousand banks had failed and this, in the days before the Federal Deposit Insurance Corporation (FDIC) protected investors from losing everything and unemployment had quadrupled to some 16 million people, roughly a third of the nation s workers In Chicago, from 1929 to 1933, employment had been cut in half, and payroll was one-quarter of its pre-Depression levels Foreclosures quintupled, from 3,148 in 1929 to 15,201 in 1933 Land values plummeted from $5 billion in 1928 to $2 billion in 1933 Chicago in the thirties was a place which few people of the twenties could recognize, and a city hard to imagine even now 3 In an Amos and Andy radio skit, a man walks up to a hotel desk and asks for a room The clerk replies, For sleeping or jumping 4 FDR s first job was to calm his people, restore their confidence, and remind them that they had nothing to fear but fear itself Then he went to work to get the country moving again

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