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between gsmSCF and gprsSSF The state of the gprsSSF FSM indicates the type of control that the gsmSCF may assert on the state models that are under control by this gprsSSF This is re ected in Figures 545 and 546 In the example in Figure 545, one gprsSSF FSM is controlling multiple state models: one GPRS session state model and multiple PDP context state models In the example in Figure 546, each gprsSSF FSM is controlling exactly one PDP context state model Table 520 lists the permissible gprsSSF FSM states Figure 547 represents how the gprsSSF FSM may transit between the permissible states CAMEL Relationship The CAMEL dialogue between gsmSCF and gprsSSF is described by means of the type of CAMEL relationship it represents As long as the CAMEL dialogue between gsmSCF and gprsSSF is active, there exist one or more relationships For CS call control, a CAMEL dialogue has exactly one

This relation is indeed true since it states that the rate of new customers entering the system equals the rate of customers leaving the system This completes the proof

The partial balance equations (565) and (566) are characteristic for the productform solution These equations express that the rate out of a state due to a change at node j = the rate into that state due to a change at node j

(569)

for each j = 0, 1, , K, where node 0 corresponds to the outside world This property of node local balance is in general not satis ed in a stochastic network, but can indeed be veri ed for the Jackson network model The product-form solution (563) can be expressed as p(n) = p1 (n1 ) pK (nK ), (5610)

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Table 520 gprsSSF FSM states Description The gprsSSF is currently not invoked The gprsSSF is invoked and is waiting for the initial service event (eg PDP context establishment) The gprsSSF has established a relationship with the gsmSCF and is waiting for instructions from the gsmSCF The gprsSSF is active and is monitoring a GPRS session or PDP context There is a relationship with the gsmSCF

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where for any k the probability distribution {pk (n), n = 0, 1, } of the number of customers present at station k is the same as the equilibrium distribution of the number of customers present in an M/M/1 queue with arrival rate k and service rate k In other words, in steady state the number of customers at the different service stations are independent of each other and the number at station k behaves as if station k is an M/M/1 queue with arrival rate k and service rate k The result (5610) is remarkable in the sense that in the network model the composite arrival process at station k is in general not a Poisson process An easy counterexample is provided by a single-station network with feedback ; that is, a customer served at the station goes immediately back to the station with a positive probability Suppose that in this network the arrival rate from outside is very small and the service rate is very large Then, if the feedback probability is close to 1, two consecutive arrivals at the station are highly correlated and so the arrival process is not Poisson The Jackson network model can be generalized to allow each service station to have multiple servers with exponential service times If station j has cj servers, the ergodicity condition (562) is replaced by j /(cj j ) < 1 Then the node local balance equation (569) can again be veri ed and the equilibrium distribution {p(n)} of the numbers of customers present at the different stations has the product form (5610), where the probability distribution {pk (n)} of the number of customers present at station k is the same as the equilibrium distribution of the number of customers present in an M/M/c queue with arrival rate k , service rate k and c = ck servers Note that the multi-server M/M/c queue with service rate can be regarded as a single-server queue with state-dependent service rate (n) =min(n, c) when n customers are present Indeed it can be shown that the product-form solution also applies to the Jackson network model with statedependent service rates provided that the service rate at each station depends only on the number of customers present at that station More about the product-form solution and its rami cations can be found in the books of Boucherie (1992) and Van Dijk (1993) In these references the product-form solution is also linked to the concept of insensitivity Insensitivity of the stochastic network holds when the condition of node local balance is sharpened to job local balance, requiring that.

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