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(a) Derive a recursion scheme for computing the limiting distribution of the number of messages present and give an expression for the long-run average cost per time unit (b) Write a computer program for calculating the value of R which minimizes the average cost and solve for the numerical data = 08, = 1, 1 = 1, 2 = 15, h = 1, r0 = 0, r1 = 5 and r2 = 25 414 Customers asking for a certain product arrive according to a Poisson process with rate The demand sizes of the customers are independent random variables and have a common discrete probability distribution {pk , k = 1, 2, } Any demand that cannot be directly satis ed from stock on hand is back ordered The control rule is based on the inventory position, which is de ned as the stock on hand minus the amount back ordered plus the amount on order Each time the inventory position reaches the reorder level s or drops below it, the smallest multiple of the basic quantity Q is ordered to bring the inventory position level above s The lead time of any replenishment order is a xed constant L > 0 (a) Prove that the limiting distribution of the inventory position is a discrete uniform distribution (Hint : use relation (432) and verify that the one-step transition matrix of the embedded Markov chain is doubly stochastic) (b) Derive the limiting distribution of the stock on hand (c) What is the average replenishment frequency and what is the average stock on hand (d) What is the fraction of customers whose demands are (partially) back ordered What is the fraction of demand that is not satis ed directly from stock on hand 415 Consider the transient probabilities pij (t) in a continuous-time Markov chain with nite space I = {1, , n} Let the n n matrix Q be de ned as in the proof of Theorem 452 Assume that the matrix Q has n different eigenvalues 1 , , n Let ak be an eigenvector corresponding to the eigenvalue k for k = 1, , n and let S be the n n matrix whose kth column vector is ak For each initial state i, denote by pi (t) the vector whose j th element equals pij (t) Use results from Section 14 to verify the representation.

CAMEL: Intelligent Networks for the GSM, GPRS and UMTS Network 2006 John Wiley & Sons, Ltd Rogier Noldus

pi (t) =

cik e k t ak ,

t 0,

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Few things demonstrate confidence in your company and your future like advertising Walgreen had been a clever and motivated marketer since he set up his stand of pots and pans in his first store and watched them fly out the door; but during the Depression, when most firms were cutting back their advertising budget because the economy was bad, Walgreen stepped his up dramatically for the very same reason People were going to be tighter with their money, he figured, so Walgreens had to show them the tremendous deals their stores offered Walgreen was right In 1931, when Henry Ford was closing a plant that housed 75,000 jobs seven times the number of total Walgreens positions at the time while blaming the Depression on the laziness of workers, Walgreen was supporting his people with the $75,000 ad campaign to promote the chain s four-day sale that April When others were backing off, Walgreens was jumping in, increasing its advertising expenditures from $905,000 in 1929 to $1093 million in 1933 And it knew how to make a splash, too, at one time flooding 150 publications in 125 cities with

CAMEL phase 4 features and capabilities Feature Section 62 622 623 624 625 626 627 628 629 6210 6211 6212 6213 63 64 641 642 65 65 66 661 662 67 68 69

for constants ci1 , , cin , where the vector ci = (ci1 , , cin ) is given by ci = S 1 ei with ei denoting the ith unit vector (0, , 1, , 0) 416 An operating system has r + s identical units where r units must be operating and s units are in preoperation (warm standby) A unit in operation has a constant failure rate of , while a unit in preoperation has a constant failure rate of with < Failed units enter a repair facility that is able to repair at most c units simultaneously The repair of a failed unit has an exponential distribution with mean 1/ An operating unit that fails is replaced immediately by a unit from the warm standby if one is available The operating system goes down when less than r units are in operation Show how to calculate the probability distribution function of the time until the system goes down for the rst time when all of the r + s units are in good condition at time 0 417 An electronic system uses one operating unit but has built-in redundancy in the form of R standby units The standby units are not switched on (cold standby) The operating unit has an exponentially distributed lifetime with mean 1/ If the operating unit fails, it is immediately replaced by a standby unit if available Each failed unit enters repair immediately and is again available after an exponentially distributed repair time with mean 1/ It is assumed that the mean repair time is much smaller than the mean lifetime There are ample repair facilities The system is down when all R + 1 units are in repair Assuming that all R + 1 units are in perfect condition at time 0, let the random variable be the time until the rst system failure.

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