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Every station in P2 monitors the channel and updates accordingly the value of its prefix sum In one additional time slot, station Sk reveals its identity The reader should have no difficulty confirming that when the protocol terminates, conditions (1) (3) above are satisfied To summarize, we have proved the following result Lemma 41 Assuming that the n (n m) stations of a single-hop, single-channel, radio network have unique IDs in the range [1, m], the prefix sums problem can be solved in 2m 2 time slots, with no station being awake for more than 2log m time slots We next extend the energy-efficient prefix sums protocol for single-channel RNs to the case where k (k 1) channels are available We begin by partitioning the stations into k equal-sized groups G1, G2, .



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, Gk such that Gi = {Sj | (i 1) m/k + 1 j i m/k and Sj exists} By assigning one channel to each such group, the instance of the prefix sums problem local to each group Gi can be solved using the protocol for the single-channel case discussed above By Lemma 41, this task can be completed in 2m/k 2 time slots, with no station being awake for more than 2 log m/k time slots At this point, we have the local sum sum(Gi) of each Gi and we need to compute the prefix sums of sum(G1), sum(G2), , sum(Gk) This can be done by modifying slightly the protocol described for the single-channel RN Recall that in the single-channel protocol, the prefix sums of P1 are computed recursively, and those for P2 are computed recursively.





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For example, a $900,000 loan at 5 percent to nance a stock position would cost the hedge fund $875 ($900,000 5% 7 days/360 days) The repo market works similarly except the value of the bond position includes accrued interest For example, if the hedge fund owned $1 million face value of bonds with a current price of $102 and accrued interest of a half point, the value of the position is $1,025,000 ($1,000,000 face 1025% of face) Also, haircuts are much lower on bonds than on equities The haircut for a US Treasury with a maturity of ve years or less may be a quarter of 1 percent or less The hedge fund may be able to set a repo balance as high as $1,022,43750 ($1,025,000 9975%) Also, nancing rates are somewhat lower for bonds, especially for Treasury and agency bonds.

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Since k channels are available, the prefix sums of P1 and P2 are computed simultaneously by allocating k/2 channels to each of them After that, the overall solution can be obtained in two more time slots Using this idea, the prefix sums problem can be solved in.

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Assuming the fund can borrow money at 45 percent, interest on this repo for a week would equal $89463 ($1,022,43750 45% 7/360) In practice, the hedge fund would nance a signi cantly lower portion of the total value of stocks or bonds in the portfolio than in the example Usually, the principal amount would also be set to a rounded amount..

After studying this chapter, the reader should be able to 1 Discuss the advantages and disadvantages of various traditional pricing methods used in the hospitality industry and understand the difference between long-range and tactical pricing. 2 Explain the concept of using net income after tax as a cost. 3 Calculate total annual revenue required for a restaurant operation to cover all forecasted costs including net income after tax and convert the annual revenue to an average check amount. 4 Use existing information to calculate an average check per meal period and explain the effect that sales mix of the various menu items will have on the average check. 5 Discuss the considerations to be kept in mind when pricing a menu item and calculate seat turnover figures. Also discuss integrated pricing for a restaurant. 6 Complete a menu engineering worksheet and discuss how to adjust the menu to respond to the results. 7 Calculate an average room rate to cover all forecasted costs, including net income after tax, and convert the average rate to an average single and average double rate. 8 Calculate room rates based on the square footage of a room. 9 Discuss room rate discounting and calculate occupancy percentage for a discount grid. Calculate a potential average room rate and discounted rates for various market segments. 10 Discuss some of the important considerations in pricing, such as the objectives of an organization, elasticity of demand, cost structure, and competition.

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