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It will be seen in 9 that the structural form (254) is very useful to design approximations in more complex queueing models Another important feature shown by the Pollaczek Khintchine formula is that the average delay and average queue size increase in a non-linear way when the offered load increases A twice as large value for the offered load does not imply a twice as large value for the average delay! On the contrary, the average delay and the average queue size explode when the average arrival rate becomes very close to the average service rate Differentiation of Wq as a function of shows that the slope of increase of Wq as a function of is proportional to (1 ) 2 As an illustration a small increase in the average arrival rate when the load = 09 causes an increase in the average delay 25 times greater than it would cause when the load = 05 This non-intuitive nding demonstrates the danger of designing a stochastic system with too high a utilization level, since then a small increase in the offered load will in general cause a dramatic degradation in system performance We have not yet proved the Pollaczek Khintchine formula First we give a heuristic derivation and next we give a rigorous proof Heuristic derivation Tag a customer who arrives when the system has reached statistical equilibrium Denote its waiting time in queue by the random variable Dtag Heuristically, E(Dtag ) = Wq By the PASTA property, the expected number of customers in queue seen upon arrival by the tagged customer equals Lq Noting that is the long-run fraction of time the server is busy, it also follows that the tagged customer nds the server busy upon arrival with probability Using the result (218) for the excess variable, it is plausible that the expected remaining service time of the customer seen in service by a Poisson arrival equals 1 E(S 2 )/E(S) Putting the 2 pieces together, we nd the relation E(Dtag ) = Lq E(S) + E(S 2 ) 2E(S).

Table 518 PDP context subscription record Description The PDP type to be used for this PDP context The PDP address to be used for this PDP context The APN for which this PDP context subscription record applies This APN consists of the network identi er only The subscribed QoS is the maximum QoS that may be applied for this PDP context This element indicates whether the subscriber may establish a PDP context using an APN that belongs to the VPLMN domain or only a PDP context using an APN that belongs to the HPLMN This element contains characteristics for the PDP context related to charging When CAMEL is used for on-line charging for this PDPc, then this element will have the value prepaid service However, this element is not used by the SGSN to determine whether CAMEL shall be invoked 3GPP TS 32015 [115] speci es the permissible values for this element

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Substituting E(Dtag ) = Wq and Lq = Wq , the relation becomes Wq = E(S)Wq + E(S 2 ) 2E(S)

yielding the Pollaczek Khintchine formula for Wq Rigorous derivation A rigorous derivation of the Pollaczek Khintchine formula can be given by using the powerful generating-function approach De ne rst the random variables L(t) = the number of customers present at time t, Qn = the number of customers present just after the nth service completion epoch, Ln = the number of customers present just before the nth arrival epoch The processes {L(t)}, {Qn } and {Ln } are regenerative stochastic processes with nite expected cycle lengths Denote the corresponding limiting distributions by pj = lim P {L(t) = j }, qj = lim P {Qn = j }

and j = lim P {Ln = j }

for j = 0, 1, The existence of the limiting distributions can be deduced from Theorem 224 (the amount of time elapsed between two arrivals that nd the system empty has a probability density and the number of customers served during this time has an aperiodic distribution) We omit the details The limiting probabilities can also be interpreted as long-run averages For example, qj is the long-run fraction of customers leaving j other customers behind upon service completion and j is the long-run fraction of customers nding j other customers present upon arrival The following important identity holds: j = pj = qj, j = 0, 1, (256)

Information element PDP type PDP address Access point name QoS pro le subscribed VPLMN address allowed

Since the arrival process is a Poisson process, the equality j = pj is readily veri ed from Theorem 241 To verify the equality j = qj , de ne the random variable (j ) Ln as the number of customers over the rst n arrivals who see j other customers (j ) present upon arrival and de ne the random variable Qn as the number of service completion epochs over the rst n service completions at which j customers are left behind Customers arrive singly and are served singly Thus between any two arrivals that nd j other customers present there must be a service completion at which j customers are left behind and, conversely, between any two service completions at which j customers are left behind there must be an arrival that sees j other customers present By this up- and downcrossing argument, we have for

(j ) (j ) (j )

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