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Clearly, the open-queueing model is limited in its applicability to inter-leaved memory. Memory systems are not open queues. When too many requests are made on a memory system, the overall system simply slows down. The processor cannot and does not keep making demands at a peak rate on the memory system, since its buffers are finite. Yet the simple queueing model has its appeal; it is very easy to use and compute, and it provides an initial guess to the low-order interleave partition. Also, it is a conservative estimate of expected performance for relatively low values of occupancy ratio (r < .5). |
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Baskett [29] makes the following observation: Suppose we have an n, m system in overall stability, with n and m very large (i.e., n, m ®¥); the number of busy servers becomes a constant fraction of the total equal to r. The distribution of arrivals becomes Poisson in the limit [87]. Then the average queue size (including the item in service) is known to be simply |
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where ra is the achieved occupancy (as contrasted with r, the offered occupancy). |
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Recall from our discussion on open queues that |
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N = Qo + r. |
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That is, the number of items in the system is equal to the number waiting for service, plus the (average) number of items in service; all under the constraint that the arrival rate is unaffected by queue size (Qo) or resulting service time (Tw). |
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The closed-queue model assumes that the arrival rate is immediately affected by service contention. We may offer l as an arrival rate, but we achieve only lathis corresponds to occupancy r and ra. Now the difference, r - ra, is the number of items denied service. They are held (queued in Qc) for service at the next time interval, but the arrival rate immediately slows down so that only la items/sec now arrive. |
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If we know the queue size, then we can use the M/D/1 model to find the achieved occupancy (ra), which now gives us the overall effective bandwidth per module. Thus, a queueing model of an n processor, m memory module system can be reduced to an approximate M/D/1 model as shown in Figure 6.20. This solution to the bandwidth is called the asymptotic solution, and it exactly corresponds to the closed-form solution where n and m are large. |
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