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These equations together with the normalizing equation S k = 1 determine k=0 uniquely the equilibrium probabilities j , j = 0, 1, , S If one of the equilibrium equations is omitted to obtain a square system of linear equations, the solution of the resulting system is still uniquely determined Example 313 (continued) The GI /M/1 queue In this example the Markov chain {Xn } describing the number of customers present just prior to arrival epochs has the in nite state space I = {0, 1, } In order to ensure that Assumption 331 is satis ed, we have to assume that the arrival rate of customers is less than the service rate Thus, denoting by the reciprocal of the mean interarrival time, it is assumed that < (3312)

We omit the proof that under this condition Assumption 331 is satis ed (with state 0 as regeneration state r) In the GI /M/1 queueing example the equilibrium probability j can be interpreted as the long-run fraction of customers who see j other customers present upon arrival for j = 0, 1, In particular, 1 0 is the long-run fraction of customers who have to wait in queue Using the speci cation of the pij given in Section 31, we obtain the equilibrium equations

.

j =

( t)k+1 j a(t) dt, (k + 1 j )!

j 1

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Unlike on-line charging for call control, there is no release if duration exceeded mechanism, whereby the gsmSCF can instruct the gsmSSF to release the call when the call period expires Therefore, when the gsmSCF has determined that the subscriber has reached the maximum permissible PDP context duration and the charging report for the last PDP context has been sent, the gsmSCF will send release GPRS to release the PDP context When a subscriber who has an active PDP context receives a CS call, the PDP context may be suspended (on hold)31 While the PDP context is on hold, the subscriber may answer the CS call The PDP context is kept active, but no data transfer takes place through this PDP context After the CS call is terminated, data transfer through the PDP may resume This suspended PDP context state is, however, not noti ed to the gprsSSF or gsmSCF Hence, duration-based charging continues during the suspended state 5352 Volume-based Charging for PDP Context The volume based charging for the PDP context relates to the total volume of data that is transferred through the PDP context: PDP context volume =< volume sent from UE to network > + < volume received by UE from network > This charging method may also be used in scenario 1 and scenario 2 control When the PDP context is established, the gsmSCF determines the price per volume quantity The price may depend on APN, subscriber location, quality of service, etc When the PDP context has become active, the PDP context is ready for data transfer When the volume de ned by the threshold is transferred

(3313)

The equilibrium equation for 0 is omitted since it is not needed An explicit solution for the j can be given This solution is j = (1 ) j ,

j = 0, 1,

(3314)

31 Dual transfer mode (DTM) GSM terminals allow for simultaneous GPRS usage and CS calls UMTS terminals are always multi-RAB , which means that they support simultaneous PS sessions and CS calls

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