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with probability 1
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The constant gives the long-run average arrival rate of customers The limit exists when customers arrive according to a renewal process (or batches of customers arrive according to a renewal process with independent and identically distributed batch sizes) The existence of the above limits is suf cient to prove the basic relations Lq = Wq and L = W (231) (232)
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These basic relations are the most familiar form of the formula of Little The reader is referred to Stidham (1974) and Wolff (1989) for a rigorous proof of the formula of Little Here we will be content to demonstrate the plausibility of this result The
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RENEWAL-REWARD PROCESSES
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formula of Little is easiest understood (and reconstructed) when imagining that each customer pays money to the system manager according to some non-discrimination rule Then it is intuitively obvious that the long-run average reward per time unit earned by the system = (the long-run average arrival rate of paying customers) (233) (the long-run average amount received per paying customer) In regenerative queueing processes this relation can often be directly proved by using the renewal-reward theorem; see Exercise 226 Taking the money principle (233) as starting point, it is easy to reproduce various representations of Little s law To obtain (231), imagine that each customer pays $1 per time unit while waiting in queue Then the long-run average amount received per customer equals the long-run average time in queue per customer (= Wq ) On the other hand, the system manager receives $j for each time unit that there are j customers waiting in queue Hence the long-run average reward earned per time unit by the system manager equals the long-run average number of customers waiting in queue (= Lq ) The average arrival rate of paying customers is obviously given by Applying the relation (233) gives next the formula (231) The formula (232) can be seen by a very similar reasoning: imagine that each customer pays $1 per time unit while in the system Another interesting relation arises by imagining that each customer pays $1 per time unit while in service Denoting by E(S) the long-run average service time per customer, it follows that the long-run average number of customers in service = E(S) (234)
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If each customer requires only one server and each server can handle only one customer at a time, this relation leads to the long-run average number of busy servers = E(S) Finite-capacity queues Assume now there is a maximum on the number of customers allowed in the system In other words, there are only a nite number of waiting places and each arriving customer nding all waiting places occupied is turned away It is assumed that a rejected customer has no further in uence on the system Let the rejection probability Prej be de ned by Prej = the long-run fraction of customers who are turned away, assuming that this long-run fraction is well de ned The random variables L(t), Lq (t), Dn and Un are de ned as before, except that Dn and Un now refer to the queueing time and sojourn time of the nth accepted customer The constants Wq and W now represent the long-run average queueing time per accepted customer (235)
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POISSON ARRIVALS SEE TIME AVERAGES
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and the long-run average sojourn time per accepted customer The formulas (231), (232) and (234) need only slight modi cation: Lq = (1 Prej )Wq = (1 Prej )E(S) and L = (1 Prej )W, (236)
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the long-run average number of customers in service (237)
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Heuristically, these formulas follow by applying the money principle (233) and taking only the accepted customers as paying customers
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