6.4.1 Performance Models of Processor Memory Interactions
Review of Stochastic Models:
Arrival Process: Requests are made to a system. The interarrival times are random variables with arrival time probability distribution.
Server: Service is provided by the system; service times are random variables with service time probability distribution.
Certain distributions are very important in modeling complex systems. In the general case, the state of a complex system depends on the probability distribution of arrivals, the service distribution, and the previous state of the system (i.e., the usage history). For Markovian distributions, statistically the current (and future) state of the system does not depend on the previous state, but only on the request and service distributions. They are said to have a memoryless distribution.
One important such distribution is the binomial distribution. Suppose there are n items that enter a system in time interval T. Of these, k items request service each with probability p. For purposes of memory system analysis and interpretation, assume that n requests are made to m memory modules and each request occurs with probability p = 1/m each memory cycle T. We want to know the number k of these requests that are directed to a particular module (since requests > 1 will queue up waiting for service). The probability that exactly k out of n requests are made to the designated server is:
This is the binomial distribution.
We can derive the Strecker model discussed in the preceding section directly from the binomial distribution. We compute the probability that exactly 0 out of n requests are made to a designated module (i.e., the module is idle). This is referred to as the null binomial and
