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11 Queuing Networks

# 11 Queuing Networks - CS-350 Fundamentals of Computing...

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Unformatted text preview: CS-350: Fundamentals of Computing Systems Page 1 of 11Lecture Notes © Azer Bestavros. All rights reserved. Reproduction or copying (electronic or otherwise) is expressly forbidden except for students enrolled in CS-350. Elementary Queuing Analysis Multiple Server Queuing Systems Multi-server Queues Assume that there are N servers in a system. We consider two scenarios. In the first scenario, (illustrated in Figure 1) upon arriving to the system, customers choose one of N queues, each of which is served by an individual server. This scenario is analogous to queues at the check-out counters of grocery stores. We call this system a multiple single-server queue and denote it with N*M/M/1. In the second scenario, (illustrated in Figure 2) upon arriving to the system, customers join a single queue and the customer at the front of the queue is served by the "next available server" out of the N servers in the system. This scenario is analogous to queues at airport airline check-in. We call this system a multiserver queue and we denote it with M/M/N. Multiple Single-Server Queues Figure 1 A system with N servers, each with its own independent queue Server Queue λ/N µServer Queue λ/N µServer Queue λ/N µServer Queue λ/N µλCS-350: Fundamentals of Computing Systems Page 2 of 11Lecture Notes © Azer Bestavros. All rights reserved. Reproduction or copying (electronic or otherwise) is expressly forbidden except for students enrolled in CS-350. The N queues in Figure 1 could be analyzed independently. Applying the M/M/1 results we obtained for the single server queue with an arrival rate of λ/N we can obtain the various metrics (such as w, q, Tw, and Tq). Multi-server Queues: M/M/N In this case, our analysis for the M/M/1 queue needs to be generalized. While we are not going to delve into the derivation of the stationary probabilities for Sj as we did for the M/M/1 queue, we will try to understand the reason why the performance of the multi-server system outperforms that of a multiple single-queue system. Figure 2 A system with N servers sharing a single queue (e.g., M/M/N system) The "difference" between the multi-server queue illustrated in Figure 2 and the multiple single-server queues illustrated in Figure 1 lies in the "sharing" of the queue. In the multiple single-server queues system, it is quite possible (and for that matter likely) that a server would be idle when another server's queue is not empty (i.e. there would be customers waiting in a queue when a server is in effect available). This situation does not exist for the multi-server queue because the only way a server would be idle is for the main queue to be empty, implying no requests are waiting anywhere in the system....
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11 Queuing Networks - CS-350 Fundamentals of Computing...

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