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To introduce the formula of Little, we consider rst two illustrative examples In the rst example a hospital admits on average 25 new patients per day A patient stays on average 3 days in the hospital What is the average number of occupied beds Let = 25 denote the average number of new patients who are admitted per day, W = 3 the average number of days a patient stays in the hospital and L the average number of occupied beds Then L = W = 25 3 = 75 beds In the second example a specialist shop sells on average 100 bottles of a famous Mexican premium beer per week The shop has on average 250 bottles in inventory What is the average number of weeks that a bottle is kept in inventory Let = 100 denote the average demand per week, L = 250 the average number of bottles kept in stock and W the average number of weeks that a bottle is kept in stock Then the answer is W = L/ = 250/100 = 25 weeks These examples illustrate Little s formula L = W The formula of Little is a law of nature that applies to almost any type of queueing system It relates long-run averages such as the long-run average number of customers in a queue (system) and the long-run average amount of time spent per customer in the queue (system) A queueing system is described by the arrival process of customers, the service facility and the service discipline, to name the most important elements In formulating the law of Little, there is no need to specify those basic elements For didactical reasons, however, it is convenient to distinguish between queueing systems with in nite queue capacity and queueing systems with nite queue capacity In nite-capacity queues Consider a queueing system with in nite queue capacity, that is, every arriving customer is allowed to wait until service can be provided De ne the following random variables: Lq (t) = the number of customers in the queue at time t (excluding those in service), L(t) = the number of customers in the system at time t (including those in service),.

Dn = the amount of time spent by the nth customer in the queue (excluding service time), Un = the amount of time spent by the nth customer in the system (including service time) Let us assume that each of the stochastic processes {Lq (t)}, {L(t)}, {Dn } and {Un } is regenerative and has a cycle length with a nite expectation Then there are constants Lq , L, Wq and W such that the following limits exist and are equal to the respective constants with probability 1: 1 t t lim 1 t t lim

(the long-run average number in queue), (the long-run average number in system), (the long-run average delay in queue per customer),

(the long-run average sojourn time per customer)

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Now de ne the random variable A(t) = the number of customers arrived by time t, It is also assumed that, for some constant ,

small The discretized rejection probability (K) is routinely found by using the continuous-time Markov chain approach In the discretized model, let the random variable X(t) = the number of data units in the buffer at time t The process {X(t)} is a continuous-time Markov chain with the nite state space I = {0, 1 , K( )} Denote the equilibrium distribution of the discretized process by {pj ( )} The process has the property that for each state i the transition rate qij = 0 for j i 2 Hence pj ( ) can recursively be computed By equating the rate out of the set {i, , K( )} to the rate into this set,

.

( )pi ( ) =

j =0

CAP v2 for pre-paid [A - D call]

pj ( )

bk ( ) ,

i = 1, 2, , K( )

K( )

.

(K) =

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