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Performance Analysis of Spillover-Partitioning Call Admission ControlAuthor ProofE 1 hP3E 1 nP3E 2 hP3Fig. 2 SPN model for the third partitionUC 1 hP3UC 1 nP3UC 2 hP3227 and it requires the use of a more sophisticated performance model for P 3 .Wedevelopa228 mathematic model based on Stochastic Petri Net (SPN) [25] as shown in Fig. 2 to model229 P 3 and derive the spillover rates from P 3 to P 4 . The reason we use SPN for performance230 assessment is that SPN provides a concise representation of the underlying Markov model231 which potentially may contain tens of thousands of states. A SPN model also allows general232 time distributions rather than just exponential distributions to be associated with event times233 if necessary. We use places (circles) to hold calls being admitted and serviced by the system,234 and transitions (vertical bars) to model event arrivals. A transition is enabled when its input235 place (if exists) is non-empty and its associated enabling predicate (if exists) is evaluated236 true. A transition rate is associated with a transition to specify how often the event associated237 with the transition occurs. Since P 3 can only accommodate class 1 handoff and new callsand class 2 handoff calls, in Fig. 2 we use places UC 1 hP3 , UC1 nP3 and UC2 hP3to hold class 1handoff calls, class 1 new calls, and class 2 handoff calls, respectively. We use M(UC 1 hP3 ),M(UC 1 nP3 ), and M(UC2 hP3 ) to represent the number of calls they hold. Transition E1 hP3 modelsarrivals of class 1 handoff calls at rate λ 1 hP 3; E 1 nP3models arrivals of class 1 new call arrivals242 at rate λ 1 nP 3; E 2 hP3 models class 2 handoff call arrivals at rate λ2 h ; S1 hP3models departures of243 class 1 handoff calls with a service rate of M(UC 1 hP3 ) multiplied by per-call service rate µ1 h ;S 1 nP3 models departures of class 1 new calls with a service rate of M(UC1 nP3) multiplied byper-call service rate µ 1 n ; S2 hP3models departures of class 2 handoff calls with a service rateof M(UC 2 hP3 ) multiplied by per-call service rate µ2 h .247 A new service request arrival is admitted in P 3 only if the partition has enough chan-248 nels to accommodate the new request. Therefore, we assign enabling predicates to guard249 E 1 hP3 , E1 nP3 and E2 hP3 , with C 3 being the constraint. Specifically, the enabling predicate of250 E 1 hP3 and E1 nP3 is [M(UC1 hP3 )+M(UC1 nP3 )]k1 + k 1 + M(UC 2 nP3 )k2 ≤ C 3 . The enabling pred-251 icate of E 2 hP3 is [M(UC1 nP3 ) + M(UC1 hP3 )]k1 + k 2 + M(UC 2 hP3 )k2 ≤ C 3 . Handoff calls and252 new calls in class 1 and handoff calls in class 2 rejected by P 3 will be spilled into P 4 with253 rates λ 1 hP 4,λ 1 nP 4and λ 2 hP 4, respectively, given by:254255256S 1 hP3S 1 nP3S 2 hP3λ 1 hP 4= λ 1 hP 3− rate(E 1 hP 3) (6)λ 1 nP 4= λ 1 nP 3− rate(E 1 nP 3) (7)λ 2 hP 4= λ 2 hP 3− rate(E 2 hP 3) (8)uncorrected proofHere, rate(E i 257 c ) is calculated by the expected value of a random variable X defined asX = λ i 258 c if Ei c is enabled; 0 otherwise. To obtain rate(Ei c ) the SPN model is first converted259 into a Markov model from which the steady state probability distribution is solved analytically.Then rate(E i c ) is computed as the probability-weighted average of X.260 123Journal: 11277 MS: WIRE823 CMS: 11277_2009_9673_Article TYPESET DISK LE CP Disp.:2009/2/19 Pages: 21 Layout: Small