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1 Motivation: multiple services, single cell / link -- HSCSD 2 Generalised stochastic knapsack 3 Admission control 4 GSM network / PSTN network 5 Model, Equilibrium distribution 6 Blocking probabilities 7 Reduced load approximation 8 Monte-Carlo summation 9 Summary and exercises 1 multiple services, single cell / link -- HSCSD 2 Generalised stochastic knapsack 3 Admission control 4 GSM network / PSTN network 5 Model, Equilibrium distribution 6 Blocking probabilities 7 Reduced load approximation 8 Monte-Carlo summation 9 Summary and exercises University of Twente - Stochastic Operations Research 33 GSM /HSCSD: High Speed Circuit Switched Data University of Twente - Stochastic Operations Research 32 HSCSD characteristics (LB) Multiple types (speech, video, data) circuit switched: each call gets number of channels GSM speech: 1 channel data: 1 channel (CS, data rate 9.6 kbps) GSM/HSCSD speech: 1 channel data: 1≤ b,...,B ≤ 8 channels (technical requirements, data rate 14.4 kbps) Call accepted iff minimum channel requirement b is met: loss system Up / downgrading: data calls may use more channels (up to B) when other services are not using these channels video: better picture quality, but same video length data: faster transmission rate, thus smaller transmission time 1 multiple services, single cell / link -- HSCSD 2 Generalised stochastic knapsack 3 Admission control 4 GSM network / PSTN network 5 Model, Equilibrium distribution 6 Blocking probabilities 7 Reduced load approximation 8 Monte-Carlo summation 9 Summary and exercises University of Twente - Stochastic Operations Research 31 n' n e2 Generalised stochastic knapsack: model 2 ( n ) number of resource units C n' n e1 1 (n) number of object classes K n' n e1 n class k arrival rate k (n) 1 (n) class k mean holding time (exp) 1 / k (n) 2 ( n) class k size bk state (number of objects) n (n1 ,..., nK ) n' n e2 state space S {n N 0 : b n C} K object of class k accepted only if bk C b n state of process at time t X (t ) ( X 1 (t ),..., X K (t )) stationary Markov process { X (t ), t 0} k (n)1(bk C b n) n' n ek transition rates q(n, n' ) k ( n) n' n ek University of Twente - Stochastic Operations Research 30 Generalised stochastic knapsack: equilibrium distribution Theorem 1: For the generalised stochastic knapsack, a necessary and sufficient condition for reversibility of X (t ) ( X 1 (t ),..., X K (t )) is that k ( n ) (n ek ) for all n S \ Tk , k 1,..., K k (n ek ) ( n) for some function : S [0, ). Moreover, when such a function exists, the equilibrium distribution for the generalised stochastic knapsack is given by ( n) ( n) , nS ( n) nS University of Twente - Stochastic Operations Research 29 Generalised stochastic knapsack: equilibrium distribution Proof: We have to verify detailed balance: (n)q (n, n ek ) (n ek )q(n ek , n) (n)k (n) (n ek ) k (n ek ) k ( n ) (n ek ) k (n ek ) ( n) If exists that satisfies the last expression, then satisfies detailed balance. As the right hand side of this expression is independent of the index k it must be that the condition of the theorem involving : S [0, )is satisfied. Conversely, assume that the condition involving : S [0, ) is satisfied. Then ( n) ( n) , nS is the equilibrium distribution. ( n) nS University of Twente - Stochastic Operations Research 28 Generalised stochastic knapsack: examples Stochastic knapsack k (n) k 1(bk C b n) n' n ek K kn (n) k 1(b n C ) k (n) nk k n' n ek k 1 nk ! Finite source input k (n) ( M k nk )k 1(nk M k ) n' n ek K M (n) k knk k 1 nk k (n) nk k n' n ek State space constraints k (n) k 1(bk C b n; nk Ck ) n' n ek (n) K kn k 1(b n C; nk Ck ) k (n) nk k n' n ek k 1 nk ! 1 multiple services, single cell / link -- HSCSD 2 Generalised stochastic knapsack 3 Admission control 4 GSM network / PSTN network 5 Model, Equilibrium distribution 6 Blocking probabilities 7 Reduced load approximation 8 Monte-Carlo summation 9 Summary and exercises University of Twente - Stochastic Operations Research 27 Admission control Admission class k whenever sufficient room bk C b n Complete sharing Simple, but may be unfair (some classes monopolize the knapsack resources) may lead to poor long-run average revenue (admitted objects may not contribute to revenue) admission policies: restrict access even when sufficient room available calculate performance under policy determine optimal policy Coordinate convex policies In general: Markov decision theory University of Twente - Stochastic Operations Research 26 Stochastic knapsack under admission control Admission policy f ( f1 ,..., f K ) f k : S {0,1} 1 class k acceptedin state n f k (n) 0 class k rejected in state n k f k (n)1(bk C b n) n' n ek Transition rates q(n, n' ) nk k n' n ek Recurrent states S ( f ) S {n : b n C} 1 b n C bk Examples: complete sharing f k (n) 0 otherwise trunk reservation 1 b n C bk tk f k (n) 0 otherwise University of Twente - Stochastic Operations Research 25 Stochastic knapsack under trunk reservation trunk reservation admits class k object iff after admittance at least t k resource units remain available C4 K 2 b1 b2 1 t1 0 t2 2 1 b n C bk tk f k (n) 0 otherwise Not reversible: so that equilibrium distribution usually not available in closed form University of Twente - Stochastic Operations Research 24 Stochastic knapsack coordinate convex policies Coordinate convex set S : n , nk 0 n ek Coordinate convex policy: admit object iff state process remains in f k (n) 1 iff n ek Theorem: Under the coordinate convex policy f the state process X f (t ) ( X 1 (t ),..., X K (t )) is reversible, and 1 K kn k f (n) , n Gf k 1 nk ! University of Twente - Stochastic Operations Research 23 Coordinate convex policies: examples Note: Not all policies are coordinate convex, e.g. trunk reservation Complete sharing: always admit if room available S Complete partitioning: accept class k iff bk (nk 1) Ck C1 CK {0,..., } ... {0,..., } C2 b1 bK C1 ... CK C C1 S Threshold policies: accept class k iff bk (nk 1) Ck C2 b n bk C C {n : b n C , nk k , k 1,...,K } C1 S bk University of Twente - Stochastic Operations Research 22 Coordinate convex policies: revenue optimization K revenue in state n r (n) rk nk k 1 long run average revenue W ( f ) r (n) f ( n) n example: long run average utilization rk bk long run average throughput rk k Intuition:optimal policy in special cases k 0 k blocking obsolete complete sharing k k complete partitioning with C k bk sk * * * where ( s1 ,..., sK ) is the optimal solution of the knapsack problem K K max rk sk subject to bk sk C sk N 0 k 1 k 1 rk / bk for k k * * If C / bk * integer, where k maximizes per unit revenue rk / bk then s * k 0 otherwise University of Twente - Stochastic Operations Research 21 Coordinate convex policies: optimal policies number of coordinate convex policies is finite thus for each coordinate convex policy f compute W(f) and select f with highest W(f) infeasible as number of policies grows as O(C1...CK ) show that optimal policy is in certain class often threshold policies b n bk C C2 Ck {n : b n C , nk , k 1,...,K } bk C1 S then problem reduces to finding optimal thresholds in full generality: Markov decision theory 1 multiple services, single cell / link -- HSCSD 2 Generalised stochastic knapsack 3 Admission control 4 GSM network / PSTN network 5 Model, Equilibrium distribution 6 Blocking probabilities 7 Reduced load approximation 8 Monte-Carlo summation 9 Summary and exercises University of Twente - Stochastic Operations Research 20 Loss networks So far: stochastic knapsack equilibrium distribution blocking probabilities throughput admission control coordinate convex policy -- complex state space model for multi service single link multi service multiple links / networks University of Twente - Stochastic Operations Research 18 PSTN / ISDN C4 C1 C3 C5 C2 C6 C7 Link: connection between switches Route: number of links Capacity Ci of link i Call class: route, bandwidth requirement per link Stochastic knapsack: special case = model for single link But: is special case of generalised stochastic knapsack 1 multiple services, single cell / link -- HSCSD 2 Generalised stochastic knapsack 3 Admission control 4 GSM network / PSTN network 5 Model, Equilibrium distribution 6 Blocking probabilities 7 Reduced load approximation 8 Monte-Carlo summation 9 Summary and exercises University of Twente - Stochastic Operations Research 17 C4 C1 C3 Loss network : notation C5 C2 C6 C7 number of links J capacity (bandwidth units) link j Cj n' n e2 number of object classes K 2 class k arrival rate k n' n e1 1 class k mean holding time (exp) 1/ k n' n e1 n bandwidth req. class k on link j b jk `n11 Route Rk {1,..., J } n2 2 Class k admitted iff b jk bandwidth units free in each link j Rk n' n e2 Otherwise call is blocked and cleared Admitted call occupies b jk bandwidth units in each link j Rk for duration of its holding time University of Twente - Stochastic Operations Research 16 Loss network : notation Set of classes K {1,...,K} set of classes that uses link j K j {k K : j Rk } state n (n1 ,..., nK ) state space S {n N 0 : K kK j b jk nk C j , j 1,...,J } S {n N0 : An C} A (b jk ) K class k blocked Tk {n S : b j n b jk C j , some j} K j Tk {n N 0 : An C , A(n ek ) C} K / University of Twente - Stochastic Operations Research 15 Loss network : notation state of process at time t X (t ) ( X 1 (t ),..., X K (t )) stationary Markov process { X (t ), t 0} aperiodic,irreducible equilibrium state X ( X 1 ,..., X K ) utilization of link j U j : b jk X k kK j long run fraction of blocked calls Bk 1 Pr{U j C j b jk , j Rk } PASTA long run throughput TH k k (1 Bk ) k E[ X k ] Little unconstrained cousin (∞ capacity) X ( X 1 ,..., X K ) Poisson r.v. with mean k k / k unconstrained cousin of utilization U j : kK j b jk X k University of Twente - Stochastic Operations Research 14 Equilibrium distribution Theorem: Product form equilibrium distribution 1 K k k Pr{X n} k n n K Pr{X n} k , nS G , nS G k 1 nk ! Pr{U C} nS k 1 nk ! Blocking probability of class k call K n nS \Tk Pr{X n} nS \Tk 1 n ! Bk 1 1 V, C vectors nS Pr{X n} K n nS 1 n ! The Markov chain is reversible, PASTA holds, and the equilibrium distribution (and the blocking probabilities) are insensitive. PROOF: special case of generalised stochastic knapsack! 1 multiple services, single cell / link -- HSCSD 2 Generalised stochastic knapsack 3 Admission control 4 GSM network / PSTN network 5 Model, Equilibrium distribution 6 Blocking probabilities 7 Reduced load approximation 8 Monte-Carlo summation 9 Summary and exercises University of Twente - Stochastic Operations Research 13 Computing blocking probabilities Direct summation is possible, but complexity O( KC1...C J ) Recursion is possible (KR), but complexity O( KC1...C J ) Bounds for single service loss networks ( b jk {0,1} ) For link j in loss network: probability that call on link j is blocked jC / C j! j L j Er( j , C j ) Cj , j k b jk k load offered to link j kK j kK j k / k! facility bound k 0 Call of class k blocked if not accepted at all links Bk 1 (1 Er( j , C j )) jRk product bound 1 multiple services, single cell / link -- HSCSD 2 Generalised stochastic knapsack 3 Admission control 4 GSM network / PSTN network 5 Model, Equilibrium distribution 6 Blocking probabilities 7 Reduced load approximation 8 Monte-Carlo summation 9 Summary and exercises University of Twente - Stochastic Operations Research 12 Computing blocking probabilities: single service networks Reduced load approximation Facility bound L j Er( k , C j ) kK j Part of offered load k blocked on other links : L j Er( tk ( j ) k , C j ) kK j kK j tk ( j ) probability at least one unit of bandwidth available in each link in Rk \ { j} k t k ( j ) reduced load approximation: blocking independent from link to link tk ( j ) iRk \{ j } (1 Li ) reduced load approximation L j Er( k (1 Li ),C j ), j 1,...,J k K j iR k \{ j} existence, uniqueness fixed point repeated substitution Bk 1 (1 L j ), k 1,...,K j R k accuracy University of Twente - Stochastic Operations Research 11 Reduced load approximation: existence and uniqueness Notation: L : ( L1 ,..., LJ ) T j ( L) : Er ( k (1 Li ), C j ) kK j iRk \{ j } T ( L) : (T1 ( L),..., TJ ( L)) Theorem There exists a unique solution L* to the fixed point equation L T (L) Proof The mapping T : [0,1]J [0,1]J is continuous, so existence from Brouwer’s theorem T is not a contraction! University of Twente - Stochastic Operations Research 10 Reduced load approximation: uniqueness Notation: Er 1 j ( B) inverse of Er for capacity C j : value of such that B Er ( , C j ) is strictly increasing function of B B Therefore 0 Er 1 j ( z )dz is strictly convex function of B for B [0,1] Proof of uniqueness: consider fixed point L T (L) and apply Er 1 j : Er1 j (L j ) : k (1 Li ) (*) k K j iR k \{ j} Define L [0,1] J J Lj (L) : (1 L ) k i Er1 j (z)dz k K j iR k j1 0 which is strictly convex. ( L) Thus, if L* [0,1]J is solution of 0, i 1,..., J Li Then L* is unique minimum of over [0,1] J writing out the partial derivatives yields (*) University of Twente - Stochastic Operations Research 9 Reduced load approximation: repeated substitution L [0,1]J T j (L) : Er( k Start (1 Li ),C j ) k K j iR k \{ j} Repeat L0 : L Lm : T ( Lm1 ), m 1,2,... Theorem Let L (1,...,1) 0 Then (0,..., 0) L1 L2 n 1 L2 n 3 L* L2 n 2 L2 n L0 (1,...,1) Thus L L , 2n L2 n 1 L , L L* L Proof T is decreasing operator: T(L) T(L') if L' L componentwise Thus T 2n (L) T 2n (L') and T 2n 1(L) T 2n 1(L') if L' L componentwise University of Twente - Stochastic Operations Research 8 Reduced load approximation: accuracy T j (L) : Er( k (1 Li ),C j ) L j Er( k ,C j ) * Corollary k K j iR k \{ j} k K j Bk 1 (1 L j ) 1 Er( k ,C j ), * so that k 1,...,K j R k j R k kK j Tj (1...1) T(0...0) Er( k ,C j ) 2 Proof kK j (0,..., 0) L1 L2 n 1 L* L2 n L0 (1,...,1) L2 T (0,..., 0) Indeed Bk from reduced load approximation does not violate the upper bound 1 multiple services, single cell / link -- HSCSD 2 Generalised stochastic knapsack 3 Admission control 4 GSM network / PSTN network 5 Model, Equilibrium distribution 6 Blocking probabilities 7 Reduced load approximation 8 Monte-Carlo summation 9 Summary and exercises University of Twente - Stochastic Operations Research 7 Monte Carlo summation For performance measures: compute equilibrium distribution K k n n! k Pr{X n} k1 K k n , nS k n! k n S k1 k K l n l nl ! for blocking probability of class k call Bk n Tk l 1 K l n l nl ! n S l 1 K l n l In general: evaluate n U l 1 nl ! difficult due to size of the state space : use Monte Carlo summation University of Twente - Stochastic Operations Research 6 Monte Carlo summation Example: evaluate integral d c b f ( x)dx a a b Monte Carlo summation: draw points at random in box abcd # points under curve / # points is measure for surface //// = value integral Method d d Let X U (a, b) Y U (0, c) indep, and let Z 1(Y f ( X )) draw n indep. Samples Z1 ,..., Z n b 1 unbiased estimator of c(b a) Then Z Z1 ... Z n f ( x)dx n a with 95% confidence interval Z 1.96 S 2 (n) /n Powerful method: as accurate as desired University of Twente - Stochastic Operations Research 5 Monte Carlo summation K l n l For blocking probabilities n Tk l 1 nl ! e Bk K l n e l n S l 1 nl ! ratio of multidimensional Poisson( ρ) distributed r.v. d Pr{ X ( ) Tk } Let X Poisson( ) then Bk Pr{ X ( ) S} Estimate enumerator and denominator via Monte Carlo summation: draw Vi from Poisson( ), i 1,...,n iid Let g(Vi ) 1(Vi U) for U Tk and U S K l nl Eg(V ) Pr{X() U} e l confidence interval? n U l 1 nl ! University of Twente - Stochastic Operations Research 4 Monte Carlo summation K l n l For blocking probabilities n Tk l 1 nl ! e Bk K l n e l n S l 1 nl ! Estimate enumerator and denominator via Monte Carlo summation: confidence interval? Harvey-Hills method (acceptance rejection method HH) Pr{ X ( ) Tk } Bk Pr{ X ( ) Tk | X ( ) S} Pr{ X ( ) S} draw Vi from Poisson( ), i 1,...,n iid unbiased estimator! if sample S ignore if sample S count g (Vi ) 1(Vi Tk | Vi S ) if sample also Tk count as succes Eg(V ) Bk 1 multiple services, single cell / link -- HSCSD 2 Generalised stochastic knapsack 3 Admission control 4 GSM network / PSTN network 5 Model, Equilibrium distribution 6 Blocking probabilities 7 Reduced load approximation 8 Monte-Carlo summation 9 Summary and exercises University of Twente - Stochastic Operations Research 3 Summary loss network models and applications: GSM network architecture Loss networks - circuit switched telephone wireless networks TELEPHONE MS Markov chain (G)MSCPSTN/ISDN equilibrium distribution BSC BTS blocking probabilities throughput admission control RADIO Evaluation of performance measures: INTERFACE recursive algorithms reduced load method Monte Carlo summation GSM/HSCSD C4 C1 C3 C5 C2 C6 C7 University of Twente - Stochastic Operations Research 2 Exercises: 7 Consider a HSCSD system carrying speech and voice. Let C=24. For speech and video load increasing from 0 to ∞ and minimum capacity requirement for speech of 1 channel and for video ranging from 1 to 4 channels, investigate the optimality of complete partitioning versus complete sharing for long run average utilization and long run average througput. To this end, first show that for complete partitioning the long run average revenue is given by Ck bk nk 1 K nk k ( j ) K W (C ) rk rkWk (Ck ) nk 0 j 0 Ck k 1 k 1 bk nk 1 nk 0 j 0 k ( j) so that the optimal partitioning is obtained from K K max Wk (Ck ) subject to Ck C 0 Ck C , Ck N 0 k 1 k 1 University of Twente - Stochastic Operations Research 1 Exercises: 8 For the stochastic knapsack show (directly from the equilibrium distribution) that E[ X k ] k (1 Bk ) / k k 1,..., K Prove the elasticity result: B j Bk j , k 1,...,K k j University of Twente - Stochastic Operations Research 0 References: Ross, sections 5.1, 5.2, 5.5 (reduced load method) Kelly: lecture notes (on website of Kelly) Kelly F.P. Kelly Loss networks, Ann. Appl. Prob 1, 319-378, 1991 HH C. Harvey and C.R. Hills Determining grades of service in a network. Presented at the 9th International Teletraffic Conference, 1979. KR J.S. Kaufman and K.M. Rege Blocking in a shared resource environment with batched Poisson arrival processes. Performance Evaluation, 24, 249-263, 1996 Exercises: rules In all exercises: give proof or derivation of results (you are not allowed to state something like: proof as given in class, or proof as in book…) motivate the steps in derivation hand in exercises week before oral exam, that is based on these exercises All oral exams (30 mins) for this part on the same day, you may suggest the date… Hand in 5 exercises: 1 and 4 and select 2 or 3 and 5 or 6, and 7 or 8 Each group 2 persons: hand in a single set of answers G1: 1, 4, 2, 5, 7 G2: 1, 4, 2, 6, 8 G3: 1, 4, 3, 5, 8 Each group 3 persons: hand in a single set of answers G1: 1, 4, 2, 5, 7 G2: 1, 4, 3, 6, 8

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posted: | 10/28/2011 |

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