Analysis of Loss Networks with Routing

Transcription

1 Analysis of Loss Netwoks with Routing Nelson Antunes, Chistine Ficke, Philippe Robet, Danielle Tibi To cite this vesion: Nelson Antunes, Chistine Ficke, Philippe Robet, Danielle Tibi. Analysis of Loss Netwoks with Routing. 25. <inia-959v1> HAL Id: inia Submitted on 21 Dec 25 v1, last evised 26 Jan 26 v3 HAL is a multi-disciplinay open access achive fo the deposit and dissemination of scientific eseach documents, whethe they ae published o not. The documents may come fom teaching and eseach institutions in Fance o aboad, o fom public o pivate eseach centes. L achive ouvete pluidisciplinaie HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau echeche, publiés ou non, émanant des établissements d enseignement et de echeche fançais ou étanges, des laboatoies publics ou pivés.

2 ANALYSIS OF LOSS NETWORKS WITH ROUTING NELSON ANTUNES, CHRISTINE FRICKER, PHILIPPE ROBERT, AND DANIELLE TIBI Abstact. This pape analyzes stochastic netwoks consisting of finite capacity nodes with diffeent classes of equests which move accoding to some outing policy. The Makov pocesses descibing these netwoks do not have, in geneal, evesibility popeties so that the explicit expession of thei invaiant distibution is not known. A heavy taffic limit egime is consideed: the aival ates of calls as well as the capacities of the nodes ae popotional to a facto going to infinity. It is poved that, in the limit, the associated escaled Makov pocess conveges to a deteministic dynamical system with a unique equilibium point chaacteized by a non-standad fixed point equation. inia-959, vesion 1-21 Dec 25 Contents 1. Intoduction 1 2. The Stochastic Model 3 3. Convegence esults 5 4. Equilibium Points 8 5. Appendix 14 Refeences Intoduction In this pape a new class of stochastic netwoks is intoduced and analyzed. Thei dynamics combine the key chaacteistics of the two main classes of queueing netwoks: loss netwoks and Jackson type netwoks. 1 Each node of the netwok has finite capacity so that a equest enteing a satuated node is eected as in a loss netwok. 2 Requests visit a subset of nodes along some possibly andom oute as in Jackson o Kelly s netwoks. This class of netwoks is motivated by the mathematical epesentation of cellula wieless netwoks. Such a netwok is a goup of base stations coveing some geogaphical aea. The aea whee mobile uses communicate with a base station is efeed to as a cell. A base station is esponsible fo the bandwidth management concening mobiles in its cell. New calls ae initiated in cells and calls ae handed ove tansfeed to the coesponding neighboing cell when mobiles move though the netwok. A new o a handoff call is accepted if thee is available bandwidth in the cell, othewise, it is eected. Date: Decembe 22, 25. Key wods and phases. Stochastic Netwoks. Heavy Taffic Limits. Asymptotic Dynamical Systems. Fixed Point Equations. 1

3 2 N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI Figue 1. The motion of a mobile among the cells of the netwok Up to now these netwoks have been modeled at a macoscopic level as loss netwoks chaacteized by call aival ates, mean call lengths, handoff ates and capacity estictions on the numbe of calls in the case of exponential times. One of the main quantity of inteest is the blocking pobability of the netwok. Appoximations have been used to analyze these netwoks. See Antunes et al. [2], Boucheie and van Dik [5] and Sidi and Staobiski [11] and the efeences theein. Assuming Poisson aivals and exponential dwell times at each node, the time evolution of such a netwok with N nodes can be epesented as a Makov ump pocess Xt with values in some finite but lage set S. It tuns out that, contay to loss netwoks, the Makov pocess Xt is not in geneal evesible o quasi evesible. Consequently, contay to Jackson netwoks and the like o pue loss netwoks, these netwoks do not have a stationay distibution with a poduct fom. In this pape the time evolution of these netwoks is analyzed by consideing heavy taffic limits. The aival ates and capacities at nodes ae popotional to some facto N which gets lage. This scaling has been intoduced by Kelly [7] to study the invaiant distibution of loss netwoks. A study of the time evolution of loss netwoks unde this scaling has been achieved by Hunt and Kutz [6]. See Kelly [8] fo a suvey of these questions. A diffeent scaling is consideed in Antunes et al. [1]. The equilibium points. The time evolution of the netwok can be oughly descibed as follows. A stochastic pocess X N t associated with the state of the netwok fo the paamete N is intoduced: X N t is the vecto descibing the numbes of equests of diffeent classes at the nodes of the netwok. The equation of evolution of the netwok is d dt X Nt = F N XN t + M N t, t, whee M N t is a matingale which vanishes as N gets lage, F N is a quite complicated functional associated with the geneato of the coesponding Makov pocess conveging to some limit F. As N goes to infinity, it is poved that X N t conveges to some function xt, satisfying the deteministic equation d 1 dt xt = F xt, t. The equilibium points of the limiting pocess ae the solutions x of the equation Fx =. It is shown in this pape, and this is a difficult point, that thee is only one equilibium point in the heavy taffic egime.

4 ANALYSIS OF LOSS NETWORKS WITH ROUTING 3 Related Wok. Fo classical loss netwoks, the invaiant pobability has a poduct fom epesentation. Nevetheless, the heavy taffic scaling of the evolution of these netwoks tuns out to be quite inticate. Hunt and Kutz [6] has shown that at any x the vecto field Fx diving the limiting dynamical system is given in tems of some eflected andom walk in R d + with ump ates depending in x. At points x at which this andom walk is egodic, Fx is expessed in tems of the invaiant distibution and at x at which it is tansient, the exit paths to infinity detemine Fx. It is not known, in geneal, whethe thee always exists a unique limiting dynamical system. Hunt and Kutz [6], Bean et al. [3, 4] and Zachay [12] analyzed seveal examples with one o two nodes whee uniqueness is shown to hold. Results of the Pape. Using the teminology of cellula netwoks: uses aiving in the netwok coespond to new equests fo a connection in a cell. Diffeent classes of customes access the netwok; Classes diffe by thei aival ate, by thei dwell time at the nodes i.e. the amount of time that a mobile of on going call emains in a given cell and by thei call duation and also by thei outing though the netwok. Duing a call, a use moves fom one cell to anothe accoding to some Makovian mechanism depending on his class. When a use moves to anothe cell node, this cell has to be non-satuated to accommodate the use, othewise the use is eected the call is lost. If it is not eected duing the tavel though the netwok, the use call teminates afte the call duation time has elapsed. Fo the netwoks analyzed in this pape, the uniqueness of the limiting dynamical system is not difficult to establish. The main difficulty lies in the complexity of the system of equations defining the equilibium points of the dynamical system. Since thee does not seem to exist some easonably simple contacting scheme to solve these equations, the uniqueness of the equilibium points is theefoe a quite challenging poblem. As an example, in Section 4.2 the case of a vey simple netwok with two nodes and two deteministic outes is investigated, the explicit epesentation of the equilibium point is obtained, it is expessed with quite complicated polynomial expessions of the paametes. The pape is oganized as follows: Section 2 intoduces the Makovian desciption of these netwoks, Section 3 gives the convegence esults togethe with the desciption of the limiting dynamical system. Section 4 is devoted to the main esults of the pape, it is shown that, in the limit, thee exists a unique stable point fo the netwok. The ingedients used to obtain this uniqueness esult ae: a dual appoach fo the poblem of uniqueness: find the set of paametes such that a given point is an equilibium point of the dynamical system; a key inequality poved in Section 5; a convenient pobabilistic epesentation of a set of linea equations. The inequality poved in the appendix involves a quantity elated to elative entopy, but, cuiously, it does not seem to be a consequence of a standad convex inequality as it is usually the case in this kind of situation. 2. The Stochastic Model The netwok consists of a finite set I of nodes, node i I has capacity c i N, whee c i > and N N. This netwok eceives a finite numbe of classes of customes indexed by a finite set R; class R customes ente the netwok accoding to a Poisson pocess with ate λ N with λ >.

5 4 N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI Call duation. A class custome who is neve eected duing his tavel though the netwok spends an exponentially distibuted time with ate µ in the netwok call duation in the context of a cellula netwok. The case µ = is not excluded, it coesponds to the case of customes staying foeve in the netwok as long as they ae not eected, i.e. as long as they do not aive at a satuated node. Dwell time. The esidence time of a custome of class at any node i I is exponentially distibuted with paamete γ. Such a custome can leave the node befoe the end of his dwell time, due to the end of call, at ate µ. Routing. A class custome enteing the netwok aives at some andom node in I whose distibution is q, and then he moves fom one node to anothe o to the outside efeed to as node accoding to some tansition matix p on I I {}. By changing the paamete of the esidence time, it can be assumed without loss of geneality that the matix p is on the diagonal. Capacity Requiements. All customes equie one unit of capacity at each node. All andom vaiables used fo aivals, esidence times o call duations ae assumed to be independent. This class of netwoks includes the case of classes of customes with deteministic outing as in Kelly s netwoks and also classes of customes with Makovian outing as in Jackson netwoks. Figue 2 epesents a netwok with two classes of customes, class 1 obs follow a deteministic oute while class 2 customes can eithe each node 1 o node 3 fom node 4, the capacities of the nodes ae 5. Note that no assumption has been done on the tansition matices p,, so that some classes of obs may achieve infinite loops in the netwok. λ 1 N 1 2 λ 2 N 4 3 Figue 2. A Netwok with Two Classes of Customes Notations. Fo i I, R and t, Xi, N t denotes the numbe of class customes at node i at time t, X N t = Xi, N t, i I, R is the coesponding pocess. The enomalized pocess is defined as follows, X N i,t = 1 N XN i,t and X N t = X N i, t, i I, R. Denote by I I the set of nodes which can be visited by a class custome, i.e. i I when i is visited with positive pobability by the Makov chain with

6 ANALYSIS OF LOSS NETWORKS WITH ROUTING 5 tansition matix p and initial distibution q. It is assumed that I = R I. The state space of the Makov pocess X N t is { S N = x = x i, N I R : x i, c i N and x i, = if i I }. The Q-matix A N x, y of X N t Aival of a class custome at node i: A N x, x + e i, = λ Nq i½ {x+ei, S N }, Sevice completion, eection by a cell o a tansition to the outside: A N x, x e i, = x i, µ + γ p i, ½ {x+e, S N } + γ p i, Tansfe fom node i to node : A N x, x e i, + e, = γ x i, p i,, I whee e i, is the unit vecto at coodinate i,. The state space of the enomalized pocess is given by { X c = x = x i, R I R + : x i, c i and x i, = if i I }, the subscipt c = c i of X c stands fo the vecto of capacities. 3. Convegence esults The following poposition establishes the deteministic behavio of X N t as N goes to infinity. This esult is the consequence of the fact that the stochastic petubations of the oiginal system ae of the ode N and theefoe vanish because of the scaling in 1/N. To descibe the time evolution of the netwok, one intoduces the following Poisson pocesses: N ξ denotes a Poisson pocess with paamete ξ >, an uppe index N p ξ, p Nd, d N, is added when seveal such Poisson pocesses ae equied. Fo example, fo i I, R, N λq i is the extenal aival Poisson pocess of class customes at node i. In a simila way, fo k 1, N k γ p is the Poisson pocess i, associated with the tansfe of the kth class customes fom node i to I {}. Fo t and i, I R, denote by Yi N t = c i N XN i, t, Y i N t is the size of the fee space at node i. The pocess X N t can then be epesented

7 6 N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI as the solution of the following stochastic integal equation, 2 X N i, t = XN i, + + I {i} k 1 I {} i ½ {Y N i k 1 s >} N λnq ids ½ {k X N, s,yi N s >} N k γ p,i ds ½ {k X N i, s } N k γ p i, ds k 1 ½ {k X N i, s } N i,k µ ds, whee ft denotes the limit on the left of the function f at t. By compensating the Poisson pocesses, i.e. by eplacing the diffeential tem N ξ ds by the matingale incement N ξ ds ξ ds, one gets the identity 3 X N i,t = X N i, + M N i,t + λ Nq i + γ p, i I ½ {Y N i s >} ds X N, s ½ {Y N i s >} ds γ + µ X N i, s ds whee Mi, N t is the matingale obtained fom the compensated integals of the pevious expession. Denote the enomalized matingale M N i, t = MN i, t/n, one finally gets 4 X N i, t = XN i, + MN i, t + λ q i + γ p, i I ½ {Y N i s >} ds X N, s ½ {Y N i s >} ds γ + µ X N i, s ds. The evolution equations fo the enomalized pocess being witten, one can establish the main convegence esult. Theoem 1. If the initial state X N conveges to x X c as N goes to infinity, then X N t conveges in the Skoohod topology to the solution xt of the following diffeential equation: fo i, I R, 5 d dt x i,t = with x = x and λ q i + γ x, tp, i τ i xt γ + µ x i, t τ i x = whee a b = mina, b fo a, b R and ρ i x def. = { 1 if x i, < c i, ρ i x 1 othewise, γ + µ x i, [λ q i + γ x,p, i].

8 ANALYSIS OF LOSS NETWORKS WITH ROUTING 7 By convegence in the Skoohod topology, one means the convegence in distibution fo Skoohod topology on the space of taectoies. Poof. Recall that if N ξ1 and N ξ2 ae two independent Poisson pocesses and if M p t = N ξp, t] ξ p t, p = 1, 2 ae thei associated matingales, they ae othogonal in the sense that M 1 tm 2 t is a matingale, i.e. the backet pocess M 1, M 2 t is fo all t. See Roges and Williams [1]. The same popety holds fo stochastic integals of pevisible pocesses H 1 t and H 2 t, fo t, H 1 sdm 1 s, H 2 sdm 2 s t =. The inceasing pocess of the enomalized matingale defined above is N M i,, M 1 N i, t = M N N 2 i,, Mi, N t, the inceasing pocess in the ight hand side of the last equation can be evaluated by using the othogonality of independent Poisson pocesses mentioned above. By using the fact that, fo i, I R and t, Xi, N t c in, one gets that thee exists some constant K such that E [M N i, t ] 2 = E M N i,, M N i, t KNt, Doob s Inequality implies that the matingale M N i, t conveges a.s. to unifomly on compact sets. Hence the stochastic fluctuations epesented by the matingales vanish in the limit. Now, by using the esults of Kutz [9], similaly as in Hunt and Kutz [6] fo loss netwoks, one can pove that any weak limit X = X i, of the pocess X N satisfies the following equations: fo i, I R, 6 X i, t = X i, + λ q i + γ I p I, ix, s π Xs N i ds γ + µ X i, sds whee N I i = {m = m N {+ } I : m i 1} and fo x = x i X c, π x is some stationay pobability measue on N I = N {+ } I of the Makov ump pocess whose Q-matix B x, is defined as B x m, m e i = B x m, m + e i = λ q i, if m i 1, x i, µ + γ p i, + p i, ½ {m=}, I B x m, m e i + e = γ x, p, i, if m i 1, whee e i denotes the ith unit vecto of R I. Moeove, the pobability distibution π x has to satisfy the following condition I 7 π x m N : m i = + = 1 if x i, < c i. The Makov pocess m x t associated with the matix B x, descibes the evolution of Y N t/n = Yi N t/n, i.e. the time-escaled pocess of the numbes of

9 8 N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI fee units of capacity at diffeent nodes, duing a time inteval [t, t + Ndt[ when the enomalized pocess X N is aound x on the nomal time scale. Compaed to X N t, the pocess Y N t indeed evolves on a apid time scale, so that quantities +dt t I ½ {Y N i s >} ds π x N i dt i.e. can be eplaced, in the limit, by the aveage values of indicato functions unde some limiting egime π x of Y N when X N t x. Hunt and Kutz [6] gives a detailed teatment of these inteesting questions. See also Bean et al. [3, 4] and Zachay [12] fo the analysis of seveal examples. In ou case the maginals of m x t ae also Makov, due to the fact that each custome occupies only one node at a time so that the acceptance at node i only depends on the numbe of fee units at node i. Fo i I, the pocess m x i t of the numbe of fee units at node i when the enomalized pocess is aound x, is a classical bith and death pocess on N whose ates ae given by qm, m + 1 = N qm, m 1 = N γ + µ x i,, λ q i + γ x, p, i The point + is an absobing point. Unde the condition 8 γ + µ x i, < λ q i + γ x, p, i the geometic distibution with paamete if m 1. / γ + µ x i, λ q i + γ x, p, i = ρ i x and the pobability δ + ae the two exteme invaiant measues of this pocess. If x i, = c i and if Condition 8 holds, then the quantity π x N I i is necessaily some convex combination of 1 and ρ i x. Fo such an i I, by summing up Equations 6 ove it is easy to check that the quantity π x N I i cannot be moe than ρ i x, othewise the finite capacity condition x i, c i would be violated. One gets that π x N I i = ρ i x. The othe cases follow fom Condition 7 o the tansience of the pocess m x i t. Since the diffeential 5 clealy has a unique solution, the theoem is poved. 4. Equilibium Points Theoem 1 shows that equilibium points x X c of the limiting dynamical system, that is those x that satisfy x i, t = fo any i, I R and t when x i, = x, ae the solutions of the following set of equations 9 γ + µ x i, = λ q i + γ x, p, i τ i x, i, I R, whee τ i x is defined in Theoem 1. Note that τ i x, 1] and that the following dichotomy holds: eithe τ i x = 1 o x i, = c i.

10 ANALYSIS OF LOSS NETWORKS WITH ROUTING Chaacteizations and existence of Equilibium Points. Convesely, assume that some point x X c satisfies the elation 1 γ + µ x i, = λ q i + γ x, p, i t i, i, I R, fo some t = t i, 1] I and that fo any i I eithe t i = 1 o x i, = c i. Fo a fixed i I, if λ q i + γ x,p, i = fo all R, then x i, = fo all and Relations 1 hold with τ i x eplaced by t i. Othewise, by adding up these elations ove R, one gets the identity t i = γ + µ x i, λ q i + γ x,p, i. If t i = 1 then ρ i x = 1, and so, by definition of τ ix, τ i x = 1 = t i. If t i < 1, due to the above assumption, necessaily x i, = c i so τ i x = γ + µ x i, λ q i + γ x,p, i 1 = t i. Equations 9 ae thus satisfied fo x. The following chaacteization of equilibium points of the system has thus been obtained. Poposition 1 Chaacteization of Equilibium Points. The equilibium points of the limiting dynamical system ae the elements x X c such that thee exists some t, 1] I satisfying 1 Fo any i, I R 11 x i, = α q i + β x, p, i t i 2 Fo any i I, eithe t i = 1 o x i, = c i, whee α = λ /γ + µ and β = γ /γ + µ fo R. To pove existence of a fixed point, a second chaacteization of equilibium points will be useful. Poposition 2 Existence of Equilibium Points. The equilibium points of the dynamical system 5 of Theoem 1 ae the fixed points in X c of the function Φ c defined by, fo x X c, α 12 Φ c x = Θ ci q i + β x, p, i, R, i I, whee, fo z > and u [, + R, z Θ z u = u 1 u. The function Φ c has at least one fixed point. Poof. Note that the function Θ c maps [, + R into the subset {u [, + R : u c} and that Φ c x indeed belongs to X c : its i, th coodinate is wheneve i / I. The chaacteization of equilibium points follows fom Poposition 1 and by noting that, fo u [, + R, z > and v [, + R such that v z, thee

11 1 N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI is an equivalence between the identity Θ z u = v and the fact that thee exists some t, 1] such that v = tu and that eithe t = 1 o v = z. The existence of a fixed point is then a consequence of Bouwe s fixed point theoem since X c is a convex compact subset of R I R and Φ c is a continuous function fom X c into itself The Example of Deteministic Routes. Requests of class use a deteministic oute of length L N {+ } consisting of a sequence I = i p, p < L with values in I such that q i = 1, p i p, i p+1 = 1, fo p < L 1 and p i L 1, = 1 if L < +. Note that, since I is finite, the case L = + necessaily coesponds to a oute which is peiodic afte some point. Equilibium points as descibed in Poposition 1 can be witten moe explicitly in tems of t solving Equations 11 x i, = α q i + β x, p, i t i. as follows: 1 Fo a non peiodic deteministic oute, L < +, these equations educe to a ecusion, fo p < L, p x ip, = α β p t ik Fo a peiodic oute consisting in nodes i, i 1,... i k 1 and then the infinite loop i k, i k+1,...,i k+l 1, i k, i k+1,... these equations have a solution if and only if β l t k... t k+l 1 < 1 and in this case x ih, = α β h t im, h k 1, m h k= x ih, = α β h t i t i1... t ih 1 β l t i k... t ik+l 1, h k. The above calculations show that an equilibium point x i, has a polynomial expession in t = t whose degee is elated to the ank of i along the oute in the case of a non-peiodic oute; and x i, is given by a powe seies in t when the oute is peiodic. Moeove, these quantities have to satisfy the following constaints: fo i I, then eithe t i = 1 o x i, = c i. The exact expession of fixed points in the case of deteministic outes is theefoe vey likely to be non tactable. As it will be seen, even the uniqueness is not a simple poblem. The complexity of exact expessions is illustated by a simple example of a netwok with two nodes: I = {1, 2} and two deteministic non peiodic outes: fist class entes at node 1, goes to node 2 then exits, second class does the opposite. Take µ 1 = µ 2 = so that β 1 = β 2 = 1, then it is easy to show that: 1 An equilibium point associated to t 1, t 2 with t 1 = t 2 = 1 exists if and only if α 1 + α 2 c 1 and α 1 + α 2 c 2, in this case x 1,1 = x 2,1 = α 1 and x 1,2 = x 2,2 = α 2. 2 An equilibium point exists with t 1 = 1, t 2 < 1 if and only if α 1 + α 2 α 1 + α 2 c 2 c 1 and α 1 + α 2 > c 2,

12 ANALYSIS OF LOSS NETWORKS WITH ROUTING 11 unde these conditions it is then unique x 1,1 = α 1, x 2,1 = α 1 c 2 α 1 + α 2, x 1,2 = x 2,2 = α 2 c 2 α 1 + α 2, 3 By symmety analogous esults hold with t 1 = 1 and t 2 < 1. 4 An equilibium point exists with t 1 < 1 and t 2 < 1 if and only if α 1 + α 2 α 1 + α 2 c 2 > c 1 and α 2 + α 1 α 1 + α 2 c 1 > c 2, in this case the solution is unique: with x 1,1 = α 1 t 1, x 2,1 = α 1 t 1 t 2 x 1,2 = α 2 t 1 t 2, x 2,2 = α 2 t 2, t 1 = α 1c 1 α 2 c 2 α 1 α 2 + α 1 c 1 α 2 c 2 α 1 α c 1 α 2 α 2 1 2α 2, 1 and t 2 has a simila expession with the subscipts 1 and 2 exchanged. It is not difficult to check that these fou cases ae disoint and cove all the situations. Theefoe the uniqueness of the equilibium point holds in this case. It does not seem possible to cay out a simila appoach fo a moe complicated system of deteministic outes. Even poving uniqueness in such a context is challenging Uniqueness of Equilibium Points. In view of Poposition 2, to pove the uniqueness of equilibium points, a contaction popety of Φ c would be enough. But it can be shown that Φ c is geneally not a contaction fo classical noms. Fo example, in the simple netwok consideed above with β 1 = β 2 = 1, the equation Φ c x = y is y 1,1, y 1,2 = Θ c1 α 1, x 2,2 and y 2,1, y 2,2 = Θ c2 x 1,1, α 2. When c 1 > α 1 and c 2 > α 2, one can choose x and x in X c such that { α 1 + x 2,2 c 1, α 1 + x 2,2 c 1, x 1,1 + α 2 c 2, x 1,1 + α 2 c 2, x 1,2 = x 1,2, x 2,1 = x 2,1 then, in this case Φ c x Φ c x p = x x p fo p [1, + ], whee x p is the L p -nom x p p = i, x i, p fo p < + and x = max{ x i, : i, I R}. Unde the condition max{β : R}<1, in the case of deteministic non-peiodic outes, the function x α q i + β x, p, i, i, I R is a contaction fo any L p -nom but the same popety does not necessaily hold fo Φ c since it can be shown that the function Θ c, c >, is not a contaction fo any L p -nom on [, + R, except when R = 1 o when R = 2 and p = +.

13 12 N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI A Dual Appoach. To pove uniqueness in the geneal case, the point of view is changed: instead of looking fo x X c which ae equilibium points of the limiting dynamics associated to a given vecto c = c i, i I, + I of capacities, an element x is given and one looks fo the set of vectos c such that x is a equilibium point of the limiting dynamics. The uniqueness of the equilibium point fo a given c is then equivalent to the popety that those sets associated to two diffeent values of x do not intesect. Define def. X = { x [, + I R } : x i, = if i / I, it is of couse enough to conside the solutions x in X that satisfy Equations 11 fo some t, 1] I. The fist step of this analysis is to show that fo any t, 1] I, a solution x to the system of equations 11 is at most unique. Poposition 3 Pobabilistic Repesentation. If t, 1] I is such that Equations 11 have a solution in X, it is unique and can be expessed as + k 14 x t i, = α E, i, I R, β k k= p= t Z p ½ {Z k =i} whee Z n is a possibly killed Makov chain with tansition matix p, and initial distibution q. Note that the above expession fo x i, genealizes the fomula obtained fo peiodic deteministic Makovian outes since, using the same notations as in the example of peiodic deteministic outes, Equation 14 gives, fo h k, x ih = α β h t i t i1... t ih + = β p t ik... t ik+l 1 = α β h t i t i1... t ih 1 β p t ik...t ik+l 1, which is Fomula 13, and x ih = α β h t i t ih fo h < k. Poof. The system of equations 11 splits into R systems, one fo each R, with unknown vaiables x i,, i I. So, ust conside one of these R systems and emove the index fo simplicity: J is defined as the ange in I of the Makov chain Z k with initial distibution q and tansition matix p,. The system then wites: x i = αqi + β x p, i t i, i J. These equations have a solution since the system of equations 11 is assumed to have one. Set, fo i J, y i = x i /αt i emembe that both α and t i ae positive, then the vecto y = y i solves the equations 15 y i = qi + y P, i, i J. with P, i = βt p, i. The matix P = Pi, is sub-makovian, Z n denotes the Makov chain with initial distibution qi and tansition matix P. Set, fo i J, clealy y i qi = P Z = i, the above equation gives by induction that, fo n 1, y i E ½ { Z=i} + ½ { Z + + ½ 1=i} { Z n=i},

14 ANALYSIS OF LOSS NETWORKS WITH ROUTING 13 by letting n goes to infinity, one gets that + def. y i u i = E k= ½ { Zk =i} Fo any i J, the above inequality implies that + k= P k i, i < +,, i J. one concludes that the state i is tansient fo the Makov chain Z n. It is easy to check that u i is also a solution of Equations 15, consequently the non-negative vecto v i = y i u i satisfies the equation v i = v P, i, i J, which is the invaiant measue equation fo this Makov chain. Since all the states ae tansient, necessaily v i = fo all i J. The uniqueness is poved. It is easy to check that the epesentation of x i in tems of the Makov chain Z n is indeed given by the epesentation of u i in tems of the Makov chain Z n. The poposition is poved. Definition. The set T is the subset of t, 1] I such that the system of Equations 11 has a solution, denoted by x t = x t i, since it is unique by the above poposition. Fo t T and i I, define σ i t = x t i, = + k α E β k t Z ½ p {Z k =i}, whee Z n is, as befoe, a Makov chain with tansition matix p, and initial distibution q. Lemma 1 Stong monotonicity. If t = t i and t = t i ae elements of T with the following popety, fo any i I, then t = t. k= p= t i < t i σ i t σ i t and t i < t i σ i t σ i t, Poof. The assumption on t and t gives that the elation 16 log t i /t i σ i t σ i t i I holds. The definition of σ i gives the following epesentation fo the diffeence σ i t σ i t, σ i t σ i t = [ k ] k α E β k t t Z h Z ½ h {Z k =i}. k= Note that, as in the poof of Poposition 3, the infinite sums within the expectation ae integable, theeby allowing these algebaic opeations. By plugging this h= h=

15 14 N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI expession into Relation 16 and fist exchanging summations on i I and R and then on i I and k N emembe that I and R ae finite, one gets, [ / α E β t ] k log k k t Z k Z t t k Z h Z ½ h {Z k }, k= h= and, by extending the definitions of t and t to the coodinate so that t = t = 1, [ α / ] k k E log β t β β Z t k Z β t β k Z t h Z. h k= Poposition 4 of the Appendix applied to the expession inside the expectation, implies that, with pobability 1, this integand should be. Consequently, the same poposition gives that, fo any R, the identity t Z = t holds almost k Z k suely fo any k N. Hence, t i = t i fo any i I and any R by definition of I. One concludes that t = t since I = I. The lemma is poved. The main esult concening the equilibium points of the limiting dynamical system 5 can now be established. Theoem 2 Uniqueness of Equilibium. Thee is a unique equilibium point of the dynamical system x i, t, i, I R defined by Equations 5. Poof. Fo t T, define C t as the set of vectos c = c i, + [ I such that x t is a fixed point fo the dynamical system associated to capacities c i. Fo t T and c, + I, Poposition 1 shows that if c C t then, fo any i I, σ i t c i and when t i < 1 then σ i t = c i. Fo t, t T, assume that thee exists some c C t C t. If i I, the elation t i < t i implies that t i < 1 and theefoe that σ i t σ i t = c i. Fom Lemma 1 one concludes that necessaily t = t. The uniqueness of equilibium points eadily follows fom this esult: if z and z ae equilibium points of the dynamical system 5 associated to some vecto of capacities c, + I, then thee exist t and t T such that z = x t and z = x t. Since c C t C t, one gets that t = t and theefoe z = z. The theoem is poved. h= 5. Appendix This section is devoted to the poof of a key technical esult fo the poof of the uniqueness of equilibium points. It involves an expession which beas some similaity with a elative entopy. Poposition 4. Let u = u i i N and u = u i i N be two sequences of elements of, 1]. If the seies + log u i/u i u i= i u i h= h= conveges, its sum is non-negative and is if and only if u = u. Poof. It is fist poved by induction on n N that fo any u, u, 1] n, n 17 f n u, u def. = log u i/u i u. i= i u i

16 ANALYSIS OF LOSS NETWORKS WITH ROUTING 15 This is obviously tue fo n =. Now assume this inequality holds with fo any intege k < n. Let u and u be some fixed elements of, 1] n. If thee exists some k such that 1 k n and u u k 1 k 1 then f n u, u can be decomposed as follows, k k u k k u, 18 f n u, u = f k 1 u,...,u k 1, u,...,u k 1 + f n k u, u k+1,...,u n, u, u k+1,..., u n log k 1 u / k 1 u k u k u. Fom the induction hypothesis and the assumption on k, all tems of the ight hand side of this identity ae nonnegative, so f n u, u. Othewise fo any k n the quantity k u k u has a constant sign and is not positive say. Thee ae two cases: 1 if u k u k fo all k such that k n, all tems in the sum defining f n u, u ae non-negative, hence f n u, u. 2 if not, let k n be the fist index such that u k > u k. Since u < u then k 1 and we can wite f n u, u = f n 1 [u,..., u k 2, u k 1 u k, u k+1,...,u n, ] u,..., u k 2, u k 1u k, u k+1,..., u n 1 + log u k 1 /u k 1 u k u 1 u k k 1 k 1 The fist tem is non-negative fom the induction hypothesis, the second one also since u k 1 u k 1, u k u k and k 1 u k 1 u, theefoe f n u, u also in this case. The poof by induction is completed. Inequality 17 is thus tue fo any n N, it implies that fo any u, u, 1] N f u, u def. = + i= log u i /u i i u i u, wheneve the seies conveges. The fist pat of the poposition is poved. Assume now that f u, u = fo some u, u, 1] N such that the seies conveges. Using the same kind of decomposition as in Equation 18, f u, u u

17 16 N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI can be expessed as, fo some fixed k 1, f u, u = f k 1 u,..., u k 1, u,..., u k 1 + f k u, u k+1,..., log k 1 u k / u, u k+1,... k 1 u k u k u =. The second tem of the ight hand side is clealy well defined since f u, u is. The fist and second tems being non-negative, one gets that / log u u. k 1 u k 1 k u k Consequently, the diffeence u u 1 u k u u 1 u k has a constant sign fo any k N. It can be assumed that these expessions ae non-negative. 1 If u i u i holds fo any i, then each tem of the infinite sum defining f u, u is non-negative and theefoe null since f u, u =. It clealy implies that u i = u i fo all i N. 2 Othewise, since u u, define n 1 as the smallest intege such that u n > u n. Since u u 1 u n u u 1 u n, thee exists some index i < n satisfying u i < u i, define k as the lagest one. In paticula, fo k < i < n, one has u i = u i. Theefoe, f u, u =f u,...,u k 1, n =k + logu k /u k 1 u, u n+1,..., u n,...,u k 1, k< n u k u 1 =k k< n u, u n+1,... u u =. The fist tem is non-negative and it is easily checked by using the definitions of k and n that the second one is positive. This equality is theefoe absud. This second case is not possible. The poposition is poved. Refeences [1] Nelson Antunes, Chistine Ficke, Philippe Robet, and Danielle Tibi, On stochastic netwoks with multiple stable points, Novembe 25, In pepaation. [2] Nelson Antunes, António Pacheco, and Rui Rocha, A Makov enewal based model fo wieless netwoks, Queueing Systems. Theoy and Applications 4 22, no. 3, [3] N. G. Bean, R. J. Gibbens, and S. Zachay, Asymptotic analysis of single esouce loss systems in heavy taffic, with applications to integated netwoks, Advances in Applied Pobability , no. 1, [4], Dynamic and equilibium behavio of contolled loss netwoks, Annals of Applied Pobability , no. 4, [5] Richad J. Boucheie and Nico M. van Dik, On a queueing netwok model fo cellula mobile telecommunications netwoks, Opeations Reseach 48 2, no. 1, [6] P.J. Hunt and T.G Kutz, Lage loss netwoks, Stochastic Pocesses and thei Applications , [7] F.P. Kelly, Blocking pobabilities in lage cicuit-switched netwoks, Advances in Applied Pobability , k

18 ANALYSIS OF LOSS NETWORKS WITH ROUTING 17 [8], Loss netwoks, Annals of Applied Pobability , no. 3, [9] T.G. Kutz, Aveaging fo matingale poblems and stochastic appoximation, Applied Stochastic Analysis, US-Fench Wokshop, Lectue notes in Contol and Infomation sciences, vol. 177, Spinge Velag, 1992, pp [1] L. C. G. Roges and David Williams, Diffusions, Makov pocesses, and matingales. Vol. 2: Itô calculus, John Wiley & Sons Inc., New Yok, [11] M. Sidi and D. Staobinski, New call blocking vesus handoff blocking in cellula netwoks, Wieless Netwoks , no. 1, [12] Stan Zachay, Dynamics of lage uncontolled loss netwoks, Jounal of Applied Pobability 37 2, no. 3, addess: Nelson.Antunes@inia.f N. Antunes Univesidade do Algave, Faculdade de Ciências e Tecnologia, Campus de Gambelas, Fao Potugal C. Ficke, Ph. Robet INRIA, domaine de Voluceau, B.P. 15, Le Chesnay Cedex, Fance addess, C. Ficke: Chistine.Ficke@inia.f addess, Ph. Robet: Philippe.Robet@inia.f URL, Ph. Robet: D. Tibi Univesité Pais 7, UMR 7599, 2 Place Jussieu, Pais Cedex 5, Fance addess: Danielle.Tibi@math.ussieu.f