PAPER 39 STOCHASTIC NETWORKS

Transcription

1 MATHEMATICAL TRIPOS Pat III Tuesday, 2 June, :30 pm to 4:30 pm PAPER 39 STOCHASTIC NETWORKS Attempt no moe than FOUR questions. Thee ae FIVE questions in total. The questions cay equal weight. STATIONERY REQUIREMENTS Cove sheet Teasuy Tag Scipt pape SPECIAL REQUIREMENTS None You may not stat to ead the questions pinted on the subsequent pages until instucted to do so by the Invigilato.

2 1 2 Pove that in equilibium the depatue pocess fom an M/M/K queue is a Poisson pocess. Ailine passenges aive at passpot contol in accodance with a Poisson pocess of ate ν. The contol opeates as a two seve queue at which sevice times ae independent and exponentially distibuted with mean µ < 2/ν and ae independent of the aival pocess. Afte leaving passpot contol, a passenge must pass though a secuity check. This opeates as a single-seve queue at which sevice times ae all equal to a constant τ(< ν 1 ). Show that in equilibium the pobability both queues ae empty is 2 νµ 2+νµ (1 ντ). Ifit takes atimeσ towalk fomthefistqueuetothesecond, whatis theequilibium pobability that both queues ae empty and thee is no passenge walking between them? 2 Deive Elang s fomula, E(ν,C), fo the popotion of calls lost at a esouce of capacity C offeed a load of ν. State clealy any assumptions you make in you deivation. Define a loss netwok with a fixed outing, and descibe biefly how the Elang fixed point equations ( ) B j = E (1 B j ) 1 A j ν (1 B i ) A i,c j, j = 1,2,...J (1) i aise as a natual appoximation fo the link blocking pobabilities in the netwok, whee A is the link-oute incidence matix, assumed to be a 0 1 matix. Show that a solution to the equations (1) also locates the minimum of a stictly convex function ν e j y ja j + yj U(z,C j )dz j 0 ove the positive othant y 0, fo a function U to be detemined. Descibe a fom of epeated substitution that conveges to a solution of equations (1), and pove that it conveges.

3 3 3 Define a Wadop equilibium fo the flows x = (x, R) in a congested netwok with outes R and links j J. Show that if the delay D j (y j ) at link j is a continuously diffeentiable, stictly inceasing function of the thoughput, y j, of the link j then a Wadop equilibium exists and solves an optimization poblem of the fom minimize ove j J yj 0 x 0, y D j (u)du subject to Hx = f, Ax = y, whee f = (f s,s S) and f s is the (fixed) aggegate flow between souce-sink pai s. What ae the matices A and H? Ae the equilibium thoughputs, y j, unique? Ae the equilibium flows, x, unique? Justify you answes. Suppose now that, in addition to the delay D j (y j ), uses of link j incu a tafficdependent toll T j (y j ), and suppose each use selects a oute in an attempt to minimize the sum of its toll and its delays. Show that it is possible to choose the functions T j ( ) so that the equilibium flow patten minimizes the aveage delay in the netwok. [TURN OVER

4 4 4 The dynamical system ( ) d dt µ j(t) = κ j µ j (t) A j x (t) C j j J x (t) = w k µ k(t)a k R is poposed as a model fo esouce allocation in a netwok, whee R is a set of outes, J is a set of esouces, A is a 0 1 matix, w > 0 fo R, C j is the capacity of esouce j and κ j > 0 fo j J. Povide a bief intepetation of this model as a pocess that attempts to balance supply and demand. By consideing the function V(µ) = Rw log j µ j A j µ j C j j J o othewise, show that if the matix A has full ank then all tajectoies of the dynamical system convege towad a uniqueequilibium point. What happensif A is not of full ank?

5 5 5 Let J be a set of esouces, and R a set of outes, whee a oute R identifies a subset of J. Let C j be the capacity of esouce j, and suppose the numbe of flows in pogess on each oute is given by the vecto n = (n, R). Define a popotionally fai ate allocation. Conside a netwok with esouces J = {1, 2}, each of unit capacity, and outes R = {{1},{2},{1,2}}. Given n = (n, R), find the ate x of each flow on oute, fo each R, unde a popotionally fai ate allocation. Suppose now that flows descibe the tansfe of documents though a netwok, that new flows oiginate as independent Poisson pocesses of ates ν, R, and that document sizes ae independent and exponentially distibuted with paamete µ fo each oute R. Detemine the tansition intensities of the esulting Makov pocess n = (n, R). Show that the stationay distibution of the Makov pocess n = (n, R) takes the fom ( ) n{1} +n {2} +n {1,2} ( ) n ν π(n) = B, n {1,2} whee B is a nomalizing constant, povided the paametes (ν,µ, R) satisfy cetain conditions. Detemine these conditions, and also the constant B. Show that, unde the distibution π, n {1} and n {2} ae independent. Ae n {1}, n {2} and n {1,2} independent? R µ END OF PAPER