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Figure 6.14
Processor and I/O time horizon. |
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Thus, the number of busy modules in an ensemble of m modules is |
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B(m,n) =m (1 - (1 - 1/m)n). |
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The well-known Poisson distribution is a limiting case of the binomial distribution. Suppose we let n grow very large as p approaches zero (m very large) for a finite T. Now n p is simply the expected number of arrivals at the designated server during T (n is the number of arrivals): |
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Substituting and taking limits, we get the Poisson distribution: |
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where P(k) is the probability that exactly k requests for service will be made during time T. |
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It may appear that the Binomial distribution is a more natural (accurate) description of memory behavior, as one rarely encounters memory systems with m ®¥ memory modules (i.e., p = 1/m ® 0). Indeed, this is true as we shall see. The Poisson distribution provides somewhat simpler expression and provides a more accurate description of I/O behavior, especially for disk access models. |
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6.4.2 Arrival Distribution |
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To better understand the variety and selection of an arrival distribution such as arises in memory and I/O modeling, consider an I/O storage device attached to a high-speed processor (Figure 6.14). The I/O service time is much longer than the time it takes to initiate a request. From a modeling point of view, we can treat the system as consisting of a single server with multiple requestors with (about) the same time basis. Let |
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and model the processor as n requestors each making a single request during Tserver. If n is large and the probability of a request in any particular processor cycle is small (i.e., p » 0), we use the Poisson arrival distribution. |
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