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The GPRS reference number is carried in the dialogue portion of the TCAP message ITU-T Q764 [138] speci es which TCAP primitives may contain a dialogue portion28 For example, the TC End messages in Figure 548 will not contain the GPRS Reference Number The closing of the TCAP dialogue when the gprsSSF FSM is in the monitoring state has the following impact on the operation de nitions for CAP v3 for GPRS (1) When the gprsSSF is in the monitoring state, both gprsSSF and gsmSCF should be able to start a new TCAP dialogue in order to send a CAP operation For example, the gprsSSF should be able to send a charging report during monitoring and the gsmSCF should be able to release a PDP context For this purpose, two ACs are speci ed: (a) One AC is used by the gprsSSF to start a TCAP dialogue with the gsmSCF This TCAP dialogue may be opened with the following CAP operations: IDP-GPRS, event report GPRS, ACR-GPRS and entity released GPRS (b) The other AC is used by the gsmSCF to start a TCAP dialogue with the gprsSSF This TCAP dialogue may be opened with the following CAP operations: ACH-GPRS, cancel GPRS, FCI-GPRS, release GPRS, request report GPRS event and SCI-GPRS (2) The CAP operation activity test GPRS will always open a new TCAP dialogue The GPRS reference number is used to verify the existence of a CAMEL relationship (3) In CS call control, there is a strict relation between CAMEL dialogue closure and TCAP dialogue closure When the CAMEL dialogue closes (eg due to pre-arranged), the TCAP dialogue is also closed and vice versa With CAMEL control of GPRS, there are separate rules for closing the TCAP dialogue and closing the CAMEL relationship: (a) the gprsSSF closes the TCAP dialogue when the gprsSSF FSM transits to monitoring state or idle and the gprsSSF does not expect a charging operation from the gsmSCF; (b) The gprsSSF and gsmSCF may close the CAMEL relationship with pre-arranged end Retrieving the gsmSCF GT from TCAP The TCAP segmentation places a requirement on the TCAP part of the SS7 stack in the SGSN, in order to support SCP dynamic load sharing Load sharing for CS call control is explained in 2 When the SGSN establishes a CAMEL relationship, it uses the gsmSCF address from GPRS-CSI to address the SCP for the required CAMEL service The gsmSCF address has the form of a GT; the GT will be translated in the HPLMN of the served CAMEL subscriber, into the SPC of one of the SCPs associated with this GT When the SCP responds to the gprsSSF, it includes its own GT in the response message; the TCAP layer in the SGSN will store this GT and use it in the remainder of the TCAP dialogue, to address this SCP However, during the lifetime of the CAMEL dialogue, the TCAP dialogue may be terminated This has the effect that the GT of this SCP is discarded29 Using the gsmSCF address from the GPRS-CSI, which was used to start the CAMEL service, may result in addressing the wrong SCP, due to dynamic load sharing For this reason, CAMEL speci es that the GT of the SCP, as received in the rst TC Continue message, is reported to the gprsSSF; the gprsSSF will store this GT When a TCAP dialogue is opened for an existing CAMEL dialogue, the gprsSSF will use the stored gsmSCF address CAMEL also allows the gsmSCF address from GPRS-CSI to be used for opening a TCAP dialogue for an existing CAMEL dialogue However, that method cannot be used in combination with dynamic load sharing That method should therefore not be used when inbound roaming subscribers are served with CAMEL capability, as the VPLMN operator may not know whether the served subscriber s HPLMN uses dynamic load sharing for the GPRS SCP.

(n) fij = P {Xn = j, Xk = j for 1 k n 1 | X0 = i},

2 1 + 2

1 1 + 2

113 The M/G/ Queue Suppose that customers arrive at a service facility according to a Poisson process with rate The service facility has an ample number of servers In other words, it is assumed that each customer gets immediately assigned a new server upon arrival The service times of the customers are independent random variables having a common probability distribution with nite mean The service times are independent of the arrival process This versatile model is very useful in applications An interesting question is: what is the limiting distribution of the number of busy servers The surprisingly simple answer to this question is that the limiting distribution is a Poisson distribution with mean :

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28 The TCAP dialogue portion is supported by Q764 1994 onwards The dialogue portion is also used to carry the CAP application context 29 The ITU-T TCAP recommendation prescribes that the GT of the responding entity, as received in the rst TC Continue message, should be stored in the TCAP layer and not be provided to the TC-User

( )k k!

.

(116)

for k = 0, 1, This limiting distribution does not require the shape of the service-time distribution, but uses the service-time distribution only through its mean This famous insensitivity result is extremely useful for applications The M/G/ model has applications in various elds A nice application is the (S 1, S) inventory system with back ordering In this model customers asking for a certain product arrive according to a Poisson process with rate Each customer asks for one unit of the product The initial on-hand inventory is S Each time a customer demand occurs, a replenishment order is placed for exactly one unit of the product A customer demand that occurs when the on-hand inventory is zero also triggers a replenishment order and the demand is back ordered until a unit becomes available to satisfy the demand The lead times of the replenishment orders are independent random variables each having the same probability distribution with mean Some re ections show that this (S 1, S) inventory system can be translated into the M/G/ queueing model: identify the outstanding replenishment orders with customers in service and identify the lead times of the replenishment orders with the service times Thus the limiting distribution of the number of outstanding replenishment orders is a Poisson distribution with mean In particular,

( )k k!

.

Returning to the M/G/ model, we rst give a heuristic argument for (116) and next a rigorous proof Heuristic derivation Suppose rst that the service times are deterministic and are equal to the constant D = Fix t with t > D If each service time is precisely equal to the constant

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