QueueingNetworks - CPE 619 Queueing Networks Aleksandar...

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CPE 619 Queueing Networks Aleksandar Milenković The LaCASA Laboratory Electrical and Computer Engineering Department The University of Alabama in Huntsville http://www.ece.uah.edu/~milenka http://www.ece.uah.edu/~lacasa
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2 Overview Queueing Network : model in which jobs departing from one queue arrive at another queue (or possibly the same queue) Open and Closed Queueing Networks Product Form Networks Queueing Network Models of Computer Systems
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3 Open Queueing Networks Open queueing network : external arrivals and departures Number of jobs in the system varies with time Throughput = arrival rate Goal: To characterize the distribution of number of jobs in the system
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4 Closed Queueing Networks Closed queueing network : No external arrivals or departures Total number of jobs in the system is constant “OUT” is connected back to “IN” Throughput = flow of jobs in the OUT-to-IN link Number of jobs is given, determine the throughput
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5 Mixed Queueing Networks Mixed queueing networks : Open for some workloads and closed for others Two classes of jobs. Class = types of jobs All jobs of a single class have the same service demands and transition probabilities. Within each class, the jobs are indistinguishable
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6 Series Networks k M/M/1 queues in series Each individual queue can be analyzed independently of other queues Arrival rate = λ. If μ i is the service rate for i th server:
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7 Series Networks (cont’d) Joint probability of queue lengths: product form network
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8 Product-Form Network Any queueing network in which: When f i (n i ) is some function of the number of jobs at the ith facility, G(N) is a normalizing constant and is a function of the total number of jobs in the system
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9 Example 32.1 Consider a closed system with two queues and N jobs circulating among the queues Both servers have an exponentially distributed service time. The mean service times are 2 and 3, respectively. The probability of having n 1 jobs in the first queue and n 2 =N-n 1 jobs in the second queue can be shown to be: In this case, the normalizing constant G(N) is 3 N+1 -2 N+1 . The state probabilities are products of functions of the number of jobs in the queues. Thus, this is a product form network .
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10 General Open Network of Queues Product form networks are easier to analyze Jackson (1963) showed that any arbitrary open network of m-server queues with exponentially distributed service times has a product form
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11 General Open Network of Queues (cont’d) If all queues are single-server queues, the queue length distribution is: Note: Queues are not independent M/M/1 queues with a Poisson arrival process In general, the internal flow in such networks is not Poisson. Particularly, if there is any feedback in the network, so that jobs can return to previously visited service centers, the internal flows are not Poisson
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12 Closed Product-Form Networks Gordon and Newell (1967) showed that any
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This note was uploaded on 12/13/2011 for the course CPE 619 taught by Professor Milenkovic during the Fall '09 term at University of Alabama - Huntsville.

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QueueingNetworks - CPE 619 Queueing Networks Aleksandar...

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