# MVA - CPE 619 Mean-Value Analysis Aleksandar Milenkovi The...

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CPE 619 Mean-Value Analysis 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 Analysis of Open Queueing Networks Mean-Value Analysis Approximate MVA Balanced Job Bounds
3 Analysis of Open Queueing Networks Used to represent transaction processing systems, such as airline reservation systems, or banking systems Transaction arrival rate is not dependent on the load on the computer system Arrivals are modeled as a Poisson process with a mean arrival rate λ Exact analysis of such systems Assumption : All devices in the system can be modeled as either fixed-capacity service centers (single server with exponentially distributed service time) or delay centers (infinite servers with exponentially distributed service time)

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4 Analysis of Open Queueing Networks For all fixed capacity service centers in an open queueing network, the response time is: R i = S i (1+Q i ) On arrival at the i th device, the job sees Q i jobs ahead (including the one in service) and expects to wait Q i S i seconds. Including the service to itself, the job should expect a total response time of S i (1+Q i ) Assumption : Service is memory-less (not operationally testable) Not an operational law Without the memory-less assumption, we would also need to know the time that the job currently in service has already consumed
5 Mean Performance Assuming job flow balance , the throughput of the system is equal to the arrival rate: X = λ The throughput of i th device, using the forced flow law is: X i = X V i The utilization of the i th device, using the utilization law is: U i = X i S i = X V i S i = D i The queue length of i th device, using Little's law is: Q i = X i R i = X i S i (1+Q i ) =U i (1+Q i ) or Q i = U i / (1-U i ) Notice that the above equation for Q i is identical to the equation for M/M/1 queues

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6 Mean Performance The device response times are: In delay centers, there are infinite servers and, therefore: Notice that the utilization of the delay center represents the mean number of jobs receiving service and does not need to be less than one
7 Example 34.1 File server consisting of a CPU and two disks, A and B With 6 clients systems:

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8 Example 34.1 (cont’d)
9 Example 34.1 (cont’d) Device utilizations using the utilization law are:

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10 Example 34.1 (cont’d) The device response times using Equation 34.2 are: Server response time:
Example 34.1 (cont’d) We can quantify the impact of the following changes Q: What if we increase the number of clients to 8? Request arrival rate will go up by a factor of 8/6

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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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MVA - CPE 619 Mean-Value Analysis Aleksandar Milenkovi The...

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