Essentials of Stochastic Processes by Richard Durrett

Author:Richard Durrett
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing

where N a (t) is the number of customers who arrive before time t and enter the system. Ignoring the problem of proving the existence of these limits, we can assert that these quantities are related by

Theorem 3.6 (Little’s Formula).

L= λ a W.

Why is this true? Suppose each customer pays $1 for each minute of time she is in the system. When ℓ customers are in the system, we are earning $ℓ 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 λ 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, W Q . 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

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