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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).
 Spring '10
 DURRETT
 The Land

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