02. Operational Laws - CS4 Modelling and Simulation LN-2 2...

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CS4 Modelling and Simulation LN-2 2O p e r a t i o n a l L a w s 2.1 Introduction Operational laws are simple equations which may be used as an abstract representation or model of the average behaviour of almost any system. One of the advantages of the laws is that they are very general and make almost no assumptions about the behaviour of the random variables characterising the system 1 . Another advantage of the laws is their simplicity: this means that they can be applied quickly and easily by almost anyone. Based on a few simple observations of the system the performance analyst can, by applying these simple laws, derive more information. Using this information as input to further laws the analyst gradually builds up a more complete picture of the behaviour of the system. Note that although we will talk in this section about operational laws in the context of systems, the laws are equally applicable to the observations obtained from models and we will have occasion to use the laws in this way later in the course. The foundation of the operational laws are observable variables . These are values which we could derive from watching a system over a Fnite period of time. We assume that the system receives requests from its environment. Each request generates a job or customer within the system. When the job has been processed the system responds to the environ- ment with the completion of the corresponding request. System - Arrivals - Completions ±igure 1: An Abstract System If we observed such an abstract system we might measure the following quantities: T , the length of time we observe the system; A , the number of request arrivals we observe; C , the number of request completions we observe; B , the total amount of time during which the system is busy ( B T ); N , the average number of jobs in the system. ±rom these observed values we can derive the following four important quantities: λ = A/T , the arrival rate ; X = C/T , the throughput or completion rate , U = B/T , the utilisation ; S = B/C , the mean service time per completed job. 1 In contrast, Markovian analysis relies on very strong assumptions about the distribution function of the random variables which are used. 9
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CS4 Modelling and Simulation LN-2 We will assume that the system is job fow balanced . This means that the number of arrivals is equal to the number of completions during an observation period, i.e. A = C . Obviously this assumption is not true in all observation periods, but it is a testable assumption because an analyst can always test whether the assumption holds. Note that if the system is job flow balanced the arrival rate will be the same as the completion rate, that is, λ = X . 2.2 Little’s Law The best known and most commonly used operational law is Little’s law. It is named after the man who published the Frst formal proof of the law in 1961, although it had been widely used before that time. Little’s law is usually phrased in terms of the jobs in a system and relates the average number of jobs in the system N to the residence time W , the average time they spend in the system. Let
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02. Operational Laws - CS4 Modelling and Simulation LN-2 2...

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