Lecure 7 : Loss networks

Lecure 7 : Loss networks • Motivation: multiple services, single cell / link -- HSCSD • Generalisedstochasticknapsack • Admissioncontrol • GSM network / PSTN network • Model, Equilibrium distribution • Blockingprobabilities • Reducedloadapproximation • Monte-Carlosummation • Summary and exercises

Plan for today • multiple services, single cell / link -- HSCSD • Generalisedstochasticknapsack • Admissioncontrol • GSM network / PSTN network • Model, Equilibrium distribution • Blockingprobabilities • Reducedloadapproximation • Monte-Carlosummation • 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: eachcallgetsnumber of channels GSM speech: 1 channel data: 1 channel (CS, data rate 9.6 kbps) GSM/HSCSD speech: 1 channel data: 1≤ b,...,B≤ 8channels (technicalrequirements, data rate 14.4 kbps) Callacceptediff minimum channel requirementb is met: loss system Up / downgrading:data callsmayuse more channels (up to B) whenother services are notusing these channels video: better picture quality, butsame video length data: fastertransmissionrate, thus smaller transmission time

31 University of Twente - Stochastic Operations Research Generalised stochastic knapsack: model number of resource units C number of object classes K class k arrival rate (n) class k mean holding time (exp) (n) class k size state (number of objects) state space object of class k accepted only if state of process at time t stationary Markov process transition rates

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 is that for all for some function . Moreover, when such a function exists, the equilibrium distribution for the generalised stochastic knapsack is given by

University of Twente - Stochastic Operations Research 29 Generalised stochastic knapsack: equilibrium distribution Proof: We have to verifydetailedbalance: Ifexiststhatsatisfies the last expression, thensatisfiesdetailedbalance. As the right hand side of thisexpression is independent of the index kit must bethat the condition of the theoreminvolving is satisfied. Conversely, assumethat the conditioninvolving is satisfied. Then is the equilibrium distribution.

28 University of Twente - Stochastic Operations Research Generalised stochastic knapsack: examples Stochastic knapsack Finite source input State space constraints

27 University of Twente - Stochastic Operations Research Admission control Admission class k whenever sufficient room 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

26 University of Twente - Stochastic Operations Research Stochastic knapsack under admission control Admission policy Transition rates Recurrent states Examples: complete sharing trunk reservation

25 University of Twente - Stochastic Operations Research Stochastic knapsack under trunk reservation trunk reservation admits class k object iff after admittance at least resource units remain available Not reversible: so that equilibrium distribution usually not available in closed form

24 University of Twente - Stochastic Operations Research Stochastic knapsack coordinate convex policies Coordinate convex set Coordinate convex policy: admit object iff state process remains in Theorem: Under the coordinate convex policy f the state processis reversible, and

23 University of Twente - Stochastic Operations Research Coordinate convex policies: examples Note: Not all policies are coordinate convex, e.g. trunk reservation Complete sharing: always admit if room available Complete partitioning: accept class k iff Threshold policies: accept class k iff

22 University of Twente - Stochastic Operations Research Coordinate convex policies: revenue optimization revenue in state n long run average revenue example: long run average utilization long run average throughput Intuition:optimal policy in special cases blocking obsolete complete sharing complete partitioning with where is the optimal solution of the knapsack problem If integer, where maximizes per unit revenue then

21 University of Twente - Stochastic Operations Research 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 show that optimal policy is in certain class often threshold policies then problem reduces to finding optimal thresholds in full generality: Markov decision theory

20 University of Twente - Stochastic Operations Research Loss networks Sofar: stochasticknapsack equilibrium distribution blockingprobabilities throughput admissioncontrol coordinate convex policy -- complex state space model formulti service single link multi service multiple links / networks

18 University of Twente - Stochastic Operations Research PSTN / ISDN C4 C1 C3 C5 C7 C2 C6 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

17 University of Twente - Stochastic Operations Research C4 C1 C3 C5 C7 C2 C6 Loss network : notation number of links J capacity (bandwidth units) link j number of object classes K class k arrival rate class k mean holding time (exp) bandwidth req. class k on link j Route Class kadmitted iff bandwidth units free in each link Otherwise call is blocked and cleared Admitted call occupies bandwidth units in each link for duration of its holding time

16 University of Twente - Stochastic Operations Research Loss network : notation Set of classes set of classes that uses link j state state space class k blocked

15 University of Twente - Stochastic Operations Research Loss network : notation state of process at time t stationaryMarkovprocess aperiodic,irreducible equilibrium state utilization of link j long run fraction of blockedcalls long run throughput unconstrainedcousin(∞capacity) Poissonr.v. withmean unconstrainedcousin of utilization PASTA Little

14 University of Twente - Stochastic Operations Research Equilibrium distribution Theorem: Product form equilibrium distribution Blocking probability of class k call The Markov chain is reversible, PASTA holds, and the equilibrium distribution (and the blocking probabilities) are insensitive. PROOF: special case of generalised stochastic knapsack! V, C vectors

13 University of Twente - Stochastic Operations Research Computing blocking probabilities Direct summation is possible, but complexity Recursion is possible (KR), but complexity Bounds for single service loss networks ( ) For link j in loss network: probability that call on link j is blocked load offered to link j facility bound Call of class k blocked if not accepted at all links product bound

12 University of Twente - Stochastic Operations Research Computing blocking probabilities: single service networksReduced load approximation Facility bound Part of offered load blocked on other links : probability at least one unit of bandwidth available in each link in reduced load approximation: blocking independent from link to link reduced load approximation existence, uniqueness fixed point repeated substitution accuracy

11 University of Twente - Stochastic Operations Research Reduced load approximation: existence and uniqueness Notation: Theorem There exists a unique solution to the fixed point equation Proof The mapping is continuous, so existence from Brouwer’s theorem is not a contraction!

10 University of Twente - Stochastic Operations Research Reduced load approximation: uniqueness Notation: inverse of Er for capacity : value of such that is strictly increasing function of B Therefore is strictly convex function of B for Proof of uniqueness: consider fixed point and apply : Define which is strictly convex. Thus, if is solution of Then is unique minimum of over writing out the partial derivatives yields (*)

University of Twente - Stochastic Operations Research 9 Reduced load approximation: repeated substitution Start Repeat Theorem Let Then Thus Proof is decreasing operator: Thus

University of Twente - Stochastic Operations Research 8 Reduced load approximation: accuracy Corollary so that Proof Indeed from reduced load approximation does not violate the upper bound

University of Twente - Stochastic Operations Research 7 Monte Carlo summation For performance measures: computeequilibrium distribution forblockingprobability of classkcall In general: evaluate difficultdue to size of the state space : useMonte Carlo summation

University of Twente - Stochastic Operations Research 6 Monte Carlo summation Example: evaluate integral Monte Carlo summation: draw points at random in box abcd # points under curve / # points is measure for surface //// = value integral Method Let draw n indep. Samples Then unbiased estimator of with 95% confidence interval Powerful method: as accurate as desired c d a b

University of Twente - Stochastic Operations Research 5 Monte Carlo summation For blockingprobabilities ratio of multidimensionalPoisson(ρ) distributedr.v. Let then Estimateenumerator and denominator via Monte Carlo summation: for confidence interval?

University of Twente - Stochastic Operations Research 4 Monte Carlo summation For blockingprobabilities Estimateenumerator and denominator via Monte Carlo summation: confidence interval?Harvey-Hillsmethod(acceptancerejectionmethod HH) unbiasedestimator!

University of Twente - Stochastic Operations Research 3 C4 GSM network architecture C1 C3 C5 C7 C2 C6 TELEPHONE MS (G)MSC PSTN/ISDN BSC BTS RADIO INTERFACE Summary loss network models and applications: • Loss networks - circuit switchedtelephone • wirelessnetworks • Markovchain • equilibrium distribution • blockingprobabilities • throughput • admissioncontrol • Evaluation of performance measures: • recursivealgorithms • reducedloadmethod • Monte Carlo summationGSM/HSCSD

2 University of Twente - Stochastic Operations Research Exercises: Consider a HSCSD system carrying speech and voice. Let C=24. For speech and video loadincreasingfrom 0 to ∞and minimum capacityrequirementfor speech of 1 channel and for video rangingfrom 1 to 4 channels, investigate the optimality of complete partitioning versus complete sharingfor long run average utilization and long run average througput. To this end, first show thatfor complete partitioning the long run average revenue is givenby sothat the optimalpartitioning is obtainedfrom 7

University of Twente - Stochastic Operations Research 1 Exercises: • 8 For the stochasticknapsack show (directlyfrom the equilibrium distribution) that Prove the elasticityresult:

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 batchedPoisson 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 derivationhand 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

Lecure 7 : Loss networks