1 t z t zs ds 0 as in the proof of littles formula we

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Unformatted text preview: ated by Theorem 3.6. Little’s formula. L = aW . Why is this true? Suppose each customer pays \$1 for each minute of time she is in the system. When ` customers are in the system, we are earning \$` per minute, so in the long run we earn an average of \$L per minute. On the other hand, if we imagine that customers pay for their entire waiting time when they arrive then we earn at rate a W per minute, i.e., the rate at which customers enter the system multiplied by the average amount they pay. Example 3.6. Waiting time in the queue. Consider the GI/G/1 queue and suppose that we are only interested in the customer’s average waiting time in the queue, WQ . If we know the average waiting time W in the system, this can be computed by simply subtracting out the amount of time the customer spends in service WQ = W E si (3.3) For instance, in the previous example, subtracting o↵ the 0.333 hours that his haircut takes we see that the customer’s average time waiting in the queue WQ = 0.246 hours or 14.76 minutes. Let LQ be the average queue length in equilibrium; i.e., we do not count the customer in service if there is one. If suppos...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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