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Consider a production facility at which production orders arrive according to a Poisson process with rate The production times 1 , 2 , of the orders are independent random variables having a common probability distribution function F with nite rst two moments Also, the production process is independent of the arrival process The facility can only work on one order at a time It is assumed that E( 1 ) < 1/ ; that is, the average production time per order is less than the mean interarrival time between two consecutive orders The facility operates only intermittently and is shut down when no orders are present any more A xed set-up cost of K > 0 is incurred each time the facility is reopened Also a holding cost h > 0 per time unit is incurred for each order waiting in queue The facility is only turned on when enough orders have accumulated The so-called N -policy reactivates the facility as soon as N orders are present For ease we assume that it takes a zero set-up time to restart production How do we choose the value of the control parameter N such that the long-run average cost per time unit is minimal To analyse this problem, we rst observe that for a given N -policy the stochastic process describing jointly the number of orders present and the status of the facility (on or off) regenerates itself each time the facility is turned on De ne a cycle as the time elapsed between two consecutive reactivations of the facility Clearly, each cycle consists of a busy period B with production and an idle period I with no production We deal separately with the idle and the busy periods Using the memoryless property of the Poisson process, the length of the idle period is the sum of N exponential random variables each having mean 1/ Hence E(length of the idle period I ) = Similarly, E(holding cost incurred during I ) = h N 1 1 + + N.

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To deal with the busy period, we de ne for n = 1, 2, the quantities tn = the expected time until the facility becomes empty given that at epoch 0 a production starts with n orders present, and hn = the expected holding costs incurred until the facility becomes empty given that at epoch 0 a production starts with n orders present These quantities are independent of the control rule considered In particular, the expected length of a busy period equals tN and the expected holding costs incurred

during a busy period equals hN By the renewal-reward theorem, the long-run average cost per time unit = (h/2 )N (N 1) + K + hN N/ + tN

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with probability 1 To nd the functions tn and hn , we need aj = the probability that j orders arrive during the production time of a single order Assume for ease that the production time has a probability density f (x) By conditioning on the production time and noting that the number of orders arriving in a xed time y is Poisson distributed with mean y, it follows that aj = It is readily veri ed that

PDP context state model (1)

0

( y)j f (y) dy, j!

j = 0, 1,

j aj = E( 1 )

PDP context state model (2)

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two years later on Cottage Grove and 35th Street, just four blocks down from its first two stores, with Starshak running the new site Both new stores quickly proved to be as successful as their predecessors, convincing Walgreen to give the green light to more expansion plans The incredible growth to come and the remarkable risks Walgreen and his team would take to create it would be breathtaking to anyone but a Chicagoan living at the dawn of the twentieth century The company s leaders had been born in Chicago or drawn to the specter of its unequaled rise as a world-class city Since they could remember, they had seen miracles occurring all around them on an almost daily basis: the reversal of the Chicago River to better serve the young city; the town rising rapidly from the ashes of the Great Fire to become far bigger and better than before; the Columbian Exposition s fabulous White City emerging from a fetid swamp in just 18 months to wow the world; skyscrapers popping up every month, then every week, drawing neck-craning tourists from around the globe; and the growth of the city itself, from a backwater river town to one of the world s great cities in mere decades, arguably the most dramatic development of any city, anywhere, at any time Walgreen and his colleagues not only saw all this happen or read about it they felt it in their bones, breathed its spirit, and believed in its lessons In Chicago in the early twentieth century, more than in any other place or era, anything was possible if you were willing to work for it In short, Walgreens would never have become the Walgreens we know today if Chicago hadn t been Chicago At this fortuitous nexus of time and place existed the perfect storm of preconditions for a sensational success, including ambition, innovation, and confidence, coupled with the Midwestern work ethic and bedrock values that made Walgreens early growth possible With the success of the first three stores, Walgreen and his management team s outlook shifted from cautious optimism to voracious enthusiasm, snapping up every promising new location they discovered The company exploded from five stores in 1915 to nine the next year and to nineteen by.

j =1

j =1

2 j 2 aj = 2 E( 1 ) + E( 1 )

(261)

PDP context state model (3)

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