Partial Products, Inc., has hired you as its vice president in charge
of marketing. Your immediate task is to determine the sale prices of
three newly announced multiplier modules. The top-of-the-line Cooker
is a pipelined multiplier. The Sizzler is a combinational
multiplier. The Grunter is a slower sequential multiplier. Their
performance figures are as follows (T is some constant time interval):
Throughput Latency
Cooker 1/T 5T
Sizzler 1/4T 4T
Grunter 1/32T 32T
Customers follow a single principle: Buy the cheapest combination of
hardware that meets my performance requirements. These requirements
may be specified as a maximum allowable latency time, a minimum
acceptable throughput, or some combination of these. Customers are
willing to try any paralleled or pipelined configuration of
multipliers in an attempt to achieve the requisite performance. You
may neglect the cost (both financial and as a decrease in performance)
of any routing, latching, or other hardware needed to construct a
configuration. Concentrate only on the inherent capabilities of the
arrangement of multipliers itself.
It has been decided that the Cooker will sell for $1000. The
following questions deal with determining the selling prices of
Sizzlers and Grunters.
How much can you charge for Sizzlers and still sell any? That is, is
there some price for Sizzlers above which any performance demands that
could be met by a Sizzler could also be met by some combination of
Cookers costing less? If there is no such maximum price, indicate a
performance requirement that could be met by a Sizzler but not by any
combination of Cookers. If there is a maximum selling price, give the
price and explain your reasoning.
If there is a performance requirement for the latency to be <= 4T, then
there is no combination of Cookers that will meet this performance
requirement. So it is in theory possible to sell some Sizzlers at any
price. Using multiple Cookers can further improve the overall
multiplier throughput, but their latency cannot be shortened.
How little can you charge for Sizzlers and still sell any Cookers? In
other words, is there a price for the Sizzler below which every
customer would prefer to buy Sizzlers rather than a Cooker? Give and
explain your answer, as above.
The minimum price for a Sizzler is $250.01 if we want to continue
to sell Cookers. If the price of a Sizzler is less than that, 4
Sizzlers could be used in parallel to achieve the same throughput
as a Cooker with a better latency in the bargain.
Is there a maximum price for the Grunter above which every customer
would prefer to buy Cookers instead? As before, give the price, if it
exists, and explain your reasoning in either case.
The maximum price for the Grunter is $999.99 since for applications
that can accept long latencies (>= 32T) it's worth buying a Grunter if
it saves any money at all.
Is there a minimum price for the Grunter below which every customer
would prefer to buy Grunters rather than a Cooker? Once again, give
the price, if it exists, and explain your reasoning in either case.
There is no minimum price for a Grunter that would cause every
customer to buy Grunters instead of Cookers. The latency of the
Grunter will always be 32T, so when performance requirements demand
latencies < 32T, Grunters won't do the job.
Suppose that, as a customer, you have an application in which 64 pairs
of numbers appear all at once, and their 64 products must be generated
in as short a time as practicable. You have $1000 to spend. At what
price would you consider using Sizzlers? At what price would you
consider using Grunters?
Sizzlers will be considered when they cost $250 or less. Grunters may
be considered when they cost $124.93 or less. To see this, consider
the case when Sizzlers cost $125.01. Buying seven Sizzlers would
yield a latency of 40T at a cost of $875.07. The customer cannot
afford another Sizzler, but adding a single Grunter for $124.93 will
reduce the latency to 36T. All optimal configurations are explored
below: