Allocation Schemes of Ressources with Downgrading
Christine Fricker, Fabrice Guillemin, Philippe Robert, Guilherme Thompson
AALLOCATION SCHEMES OF RESOURCES WITHDOWNGRADING
CHRISTINE FRICKER, FABRICE GUILLEMIN, PHILIPPE ROBERT,AND GUILHERME THOMPSON † Abstract.
We consider a server with large capacity delivering video files en-coded in various resolutions. We assume that the system is under saturationin the sense that the total demand exceeds the server capacity C . In such case,requests may be rejected. For the policies considered in this paper, insteadof rejecting a video request, it is downgraded. When the occupancy of theserver is above some value C 1. Introduction 12. Model description 43. Scaling Results 54. Invariant Distribution 145. Applications 19References 231. Introduction Video streaming applications have become over the past few years the dominantapplications in the Internet and generate the prevalent part of traffic in today’s IPnetworks; see for instance Guillemin et al. [10] for an illustration of the applica-tion breakdown in a commercial IP backbone network. Video files are currentlydownloaded by customers from large data centers, like Google’s data centers forYouTube files. In the future, it is very likely that video files will be delivered bysmaller data centers located closer to end users, for instance cache servers dissem-inated in a national network. It is worth noting that as shown in Guillemin etal. [11], caching is a very efficient solution for YouTube traffic. While this solution Date : Version of November 7, 2018.2010 Mathematics Subject Classification. Key words and phrases. Resource Allocation; Scaling Methods; Loss Systems. † G. Thompson’s research was supported by Brazilian Government/CAPES grant BEX 13748-13-0. a r X i v : . [ c s . N I] F e b C. FRICKER, F. GUILLEMIN, PH. ROBERT, AND G. THOMPSON can improve performances by reducing delays, the limited capacity of those serversin terms of bandwidth and computing can cause overload.One possibility to reduce overload is to use bit rate adaptation. Video files canindeed be encoded at various bit rates (e.g, small and high definition video). If anode cannot serve a file at a high bit rate, then the video can be transmitted ata smaller rate. It is remarkable that video bit rate adaptation has become verypopular in the past few years with the specification of MPEG-DASH standardwhere it is possible to downgrade the quality of a given transmission, see Schwarzet al. [17], Sieber et al. [18], A˜norga et al. [1], Vadlakonda et al. [21] and Frickeret al. [7]. Adaptive streaming is also frequently used in mobile networks wherebandwidth is highly varying. In this paper, we investigate the effect of bit rateadaptation in a node under saturation. Downgrading Policy. We assume that customers request video files encodedat various rates, say, A j for j =1 , . . . , J , with 1= A
To study this allocation scheme, a scaling approach is used.It is assumed that the server capacity is very large, namely scaled up by a factor N . The bit rate adaptation threshold and the request arrival rates are scaled upaccordingly, i.e.(1) (cid:40) λ j (cid:55)→ λ j N, ≤ j ≤ J,C (cid:55)→ c N and C (cid:55)→ cN. Performances of the algorithm. Our main result shows that, for the downgradingpolicy and if c is chosen conveniently, then(1) the equilibrium probability of rejecting a job converges to 0 as N goes toinfinity;(2) the equilibrium probability of accepting a job without downgrading it con-verges to π − def. = c µ − J (cid:88) j =1 λ j (cid:44) µ J (cid:88) j =1 λ j µ j A j − J (cid:88) j =1 λ j , as N goes to infinity. See Theorem 2 and Corollary 1. LLOCATION SCHEMES OF RESOURCES WITH DOWNGRADING 3 The above formula gives an explicit expression of the success rate of this allocationmechanism. The quantity 1 − π − , the probability of downgrading requests, can beseen as the “price” of the algorithm to avoid rejecting jobs.The scaling (1) has been introduced by Kelly to study loss networks. SeeKelly [14]. The transient behavior of these networks under this scaling has beenanalyzed by Hunt and Kurtz [12]. This last reference provides essentially a frame-work to establish convenient convergence theorems involving stochastic averagingprinciples. This line of research has been developed in the 1990’s to study uncon-trolled loss networks where a request is rejected as soon as its demand cannot beaccepted.When the demand can be adapted to the state of the network, for controlled lossnetworks, several (scarce) examples have been also analyzed during that period oftime. One can mention Bean et al. [4, 5], Zachary and Ziedins [22] and Zachary [23]for example. Our model can be seen as a “controlled” loss networks instead of apure loss network. Controlled loss networks may have mechanisms such as trunkreservation or may allocate requests according to some complicated schemes de-pending on the state of the network. In our case, the capacity requirements ofrequests are modified when the network is in a “congested” state.Contrary to classical uncontrolled loss networks, as it will be seen, the Markovprocess associated to the evolution of the vector of the number of jobs for eachclass is not reversible. Additionally, the invariant distribution of this process doesnot seem to have a closed form expression. Kelly’s approach [13] is based on anoptimization problem, it cannot be used in our case to get an asymptotic expressionof some characteristics at equilibrium. For this reason, the equilibrium behavior ofthese policies is investigated in a two step process:(1) Transient Analysis. We investigate the asymptotic behavior of some charac-teristics of the process on a finite time interval when the scaling parameter N goes to infinity.(2) Equilibrium. The stability properties of the limiting process are analyzed,we prove that the equilibrium of the system for a fixed N converges to theequilibrium of the limiting process.For our model, the transient analysis involves the explicit representation of theinvariant distribution of a specific class of Markov processes. It is obtained withcomplex analysis arguments. As it will be seen, this representation plays an impor-tant role in the analysis of the asymptotic behavior at equilibrium.It should be noted that related models have recently been introduced to investi-gate resource allocation in a cloud computing environment where the nodes receiverequests of several types of resources. We believe that this domain will receive arenewed attention in the coming years. See Stolyar [19, 20] and Fricker et al. [8]for example. In some way one could say that the loss networks are back and thisis also a motivation of this paper to shed some light on the methods that can beused to study these systems. Outline of the paper. We consider a system in overload. Because of bit rateadaptation, requests may be downgraded but not systematically rejected as in apure loss system. As it will be seen, the stability properties of this algorithm arelinked to the behavior of a Markov process associated to the occupation of the link.Under exponential assumptions for inter-arrival and service times, this process turns C. FRICKER, F. GUILLEMIN, PH. ROBERT, AND G. THOMPSON out to be, after rescaling by a large parameter N , a bilateral random walk insteadof a reflected random walk as in the case of loss networks. Using complex analysismethods, an explicit expression of the invariant distribution of this random walkis obtained. With this result, the asymptotic expression of the probability that,at equilibrium, a job is transmitted at its requested rate (and therefore does notexperience a bit rate adaptation) is derived.This paper is organized as follows: In Section 2, we present the model used tostudy the network under some saturation condition. Convergence results when thescaling factor N tends to infinity are proved in Section 3. The invariant distributionof a limiting process associated to the occupation of the link is computed in Section 4by means of complex analysis techniques. Applications are discussed in Section 5. Acknowledgments The authors are very grateful to an anonymous referee for pointing out a gap inthe proof of Theorem 2 in the first version of this work.2. Model description One considers a service system where J classes of requests arrive at a server withbandwidth/capacity C . Requests of class j , 1 ≤ j ≤ J , arrive according to a Poissonprocess N λ j with rate λ j . A class j request has a bandwidth requirement of A j units for a duration of time which is exponentially distributed with parameter µ j .For the systems investigated in this paper, there is no buffering, requests have tobe processed at their arrival otherwise they are rejected. Without any flexibility onthe resource allocation, this is a classical loss network with one link. See Kelly [14]for example.This paper investigates allocation schemes which consist of reducing the band-width allocation of arriving requests to a minimal value when the link has a highlevel of congestion. In other words the service is downgraded for new requests ar-riving during a saturation phase. If the system is correctly designed, it will reducesignificantly the fraction of rejected transmissions and, hopefully, few jobs will infact experience downgrading.2.1. Downgrading policy D ( C ) . We introduce C 1, it isaccepted but with a minimal allocation, as a class 1 job. Finally it is rejected if thelink is fully occupied, i.e. (cid:104) A, (cid:96) (cid:105) = C . It is assumed that µ ≤ µ j , for 1 ≤ j ≤ J , i.e.class 1 jobs are served with the smallest service rate.Mathematically, the stochastic model is close to a loss network with the restric-tion that a job may change its requirements depending on the state of the network.This is a controlled loss network, see Zachary and Ziedins [23]. It does not seemthat, like in uncontrolled loss networks, the associated Markov process giving theevolution of the vector (cid:96) has reversibility properties, or that its invariant distri-bution has a product form expression. Related schemes with product form are LLOCATION SCHEMES OF RESOURCES WITH DOWNGRADING 5 trunk reservation policies for which requests of a subset of classes are systemati-cally rejected when the level of congestion of the link is above some threshold. SeeBean et al. [4] and Zachary and Ziedins [22] for example. Concerning controlledloss networks, mathematical results are more scarce. One can mention networkswhere jobs requiring congested links are redirected to less loaded links. Severalmathematical approximations have been proposed to study these models. See thesurveys Kelly [14] and Zachary and Ziedins [23]. In our model, in the language ofloss networks, the control is on the change of capacity requirements instead of achange of link.2.2. Scaling Regime. The invariant distribution being, in general, not known, ascaling approach is used. The network is investigated under Kelly’s regime, i.e.under heavy traffic regime with a scaling factor N . It has been introduced inKelly [13] to study the equilibrium of uncontrolled networks. The arrival rates arescaled by N : λ j is replaced by λ j N as well as the capacity C by C N and thethreshold C by C N which are such that(2) C N = cN + o ( N ) and C N = c N + o ( N ) , for 0 For ≤ j ≤ J and t ≥ , L Nj ( t ) denotes the number of class j jobs attime t in this system and L N ( t )=( L Nj ( t ) , ≤ j ≤ J ) . It will be assumed that the system is overloaded when the jobs have their initialbandwidth requirements( R ) (cid:104) A, ρ (cid:105) > c and Λ µ < c, with Λ= λ + · · · + λ J and ρ j = λ j /µ j , 1 ≤ j ≤ J . The first condition gives that, withoutany change on the bandwidth requirement of jobs, the system will reject jobs. Thesecond condition implies that the network could accommodate all jobs withoutlosses (with high probability) if all of them would require the reduced bit rate A =1 and service rate µ .It should be noted that, from the point of view of the design of algorithms, theconstant c has to be defined. If one takes c ∈ (Λ /µ , c ) then, (cid:104) ρ, A (cid:105) > c , ( R ) Λ µ < c . ( R )hold.If (cid:104) A, ρ (cid:105) In this section, we prove convergence results when the scaling parameter N goesto infinity. These results are obtained by studying the asymptotic behavior of the C. FRICKER, F. GUILLEMIN, PH. ROBERT, AND G. THOMPSON occupation of the link around C N ,(3) m N ( t ) = (cid:10) A, L N ( t ) (cid:11) − C N . In the context of loss networks, the analogue of such quantity is the number ofempty places. The following proposition shows that, for the downgrading policy,the boundary C N does not play a role after some time if Condition ( R ) holds. Proposition 1. Under Condition ( R ) and if the initial state is such that lim N → + ∞ (cid:32) L Nj (0) N (cid:33) = (cid:96) (0) = ( (cid:96) j, ) ∈ S def. = { x ∈ R J + : (cid:104) A, x (cid:105) < c } , then, for ε> , there exists t ε ≥ such that, for T >t ε , lim N → + ∞ P (cid:18) sup t ε ≤ t ≤ T (cid:10) A, L N ( t ) (cid:11) < ( c + ε ) N (cid:19) = 1 . Proof. Define (cid:16)(cid:101) L Nj ( t ) (cid:17) def. = (cid:0) D N ( t )+ X N ( t ) , D N ( t ) , . . . , D NJ ( t ) (cid:1) , where ( X N ( t )) is the process of the number of jobs of an independent M/M/ ∞ queue with X N (0)=0, service rate µ and arrival rate Λ= λ + · · · + λ J and, for1 ≤ j ≤ J , D Nj ( t ) = L Nj (0) (cid:88) k =1 { E µj,k >t } , where ( E µ j ,k ) is a sequence of i.i.d. exponentially distributed random variableswith rate µ j .The quantity D Nj ( t ) is the number of initial class j jobs still present attime t . Using Theorem 6.13 of Robert [15], one gets the convergence in distributionlim N → + ∞ (cid:18) X N ( t ) N (cid:19) = Λ µ (cid:0) − e − µ t (cid:1) , and, consequently,(4) lim N → + ∞ (cid:18) N (cid:68) A, (cid:101) L N ( t ) (cid:69)(cid:19) = Λ µ (cid:0) − e − µ t (cid:1) + J (cid:88) j =1 A j (cid:96) j, e − µ j t . Since µ ≤ µ j for 1 ≤ j ≤ J ,Λ µ (cid:0) − e − µ t (cid:1) + J (cid:88) j =1 A j (cid:96) j, e − µ j t ≤ Λ µ (cid:0) − e − µ t (cid:1) + e − µ t (cid:104) A, (cid:96) (0) (cid:105) ≤ max( c , (cid:104) A, (cid:96) (0) (cid:105) ) , by Condition ( R ). Note that the asymptotic occupancy, when N is large, remainsbelow the initial occupancy.If 0 <εN 0, or not, m N ( t ) < 0. In pure loss networks, when N is large, upto a change of time scale, the analogue of this process, the process of the numberof empty places converges to a reflected random walk in N . In our case, as it willbe seen, the corresponding process is in fact a random walk on Z . Definition 2. For (cid:96) =( (cid:96) j ) ∈S , let ( m (cid:96) ( t )) be the Markov process on Z whose Q -matrix Q (cid:96) is defined by, for x ∈ Z and ≤ j ≤ J , (6) Q (cid:96) ( x, x − A j ) = µ j (cid:96) j ,Q (cid:96) ( x, x + A j ) = λ j , if x < ,Q (cid:96) ( x, x + 1) = Λ , if x ≥ , with Λ def. = λ + λ + · · · + λ J . The following proposition summarizes the stability properties of the Markovprocess ( m (cid:96) ( t )). Proposition 2. If (cid:96) =( (cid:96) j ) ∈S , then the Markov process ( m (cid:96) ( t )) is ergodic if (cid:96) ∈ ∆ with (7) ∆ = (cid:26) x ∈S : (cid:104) A, x (cid:105) = c , J (cid:88) j =1 ( λ j − µ j x j ) A j > and Λ < J (cid:88) j =1 µ j x j A j (cid:27) π (cid:96) denotes the corresponding invariant distribution.Proof. The Markov process ( m (cid:96) ( t )) on Z behaves like a random walk on each of thetwo half-lines N and Z ∗− . Definition (7) implies that if (cid:96) ∈ ∆ , then the drift of therandom walk is positive when in Z ∗− and negative when in N . This property impliesthe ergodicity of the Markov process by using the Lyapounov function F ( x )= | x | ,for example. See Corollary 8.7 of Robert [15] for example. (cid:3) One now extends the expression π (cid:96) for the values (cid:96) ∈S \ ∆ . This will be helpfulto describe the asymptotic dynamic of the system. See Theorem 1 further. C. FRICKER, F. GUILLEMIN, PH. ROBERT, AND G. THOMPSON Definition 3. One denotes π (cid:96) = δ −∞ , the Dirac measure at −∞ when (cid:96) ∈ ∆ − , with ∆ − def. = (cid:26) x ∈S : (cid:104) A, x (cid:105) = c , J (cid:88) j =1 ( λ j − µ j x j ) A j ≤ (cid:27) ∪ (cid:26) x ∈S : (cid:104) A, x (cid:105) For ξ > 0, denote by N ξ (d t ) a Poisson processon R + with rate ξ and ( N ξ,i (d t )) an i.i.d. sequence of such processes. All Poissonprocesses are assumed to be independent. Classically, the process ( L N ( t )) can beseen as the unique solution to the following stochastic differential equations (SDE),(8) d L N ( t ) = − L N ( t − ) (cid:88) k =1 N µ ,k (d t )+ { m N ( t − ) Proof. By using the same method as Hunt and Kurtz [12], one gets the analogueof Theorem 3 of this reference. Fix ε> c + ε 0, which implies that κ =0. Thus Relation (15) holds. Finally, Re-lations (13) and (15) give Relation (11). One concludes therefore that (cid:96) ∗ ∈ ∆ ,the associated process ( m (cid:96) ∗ ( t )) is necessarily ergodic by Proposition 2 and Rela-tions (14).To prove that the (cid:96) ∗ defined by Relations (11) and (12) is indeed an equilibriumpoint of the dynamical system defined by Equation (10), one has to show that theright-hand side of Equation (12) is indeed equal to π (cid:96) ∗ ( Z ∗− ). This is proved inProposition 5 of Section 4. (cid:3) Convergence of Invariant Distributions. In this section our main result es-tablishes the convergence of the invariant distribution of the process ( m N ( t )) as N gets large. This will give in particular the convergence with respect to N of theprobability of not downgrading a request at equilibrium. Lemma 1. If the process ( (cid:101) L Nj ( t )) is the process ( L Nj ( t )) at equilibrium then, forany ε> and T > , lim N → + ∞ P (cid:32) sup ≤ t ≤ T sup ≤ j ≤ J (cid:101) L Nj ( t ) N ≤ ρ j + ε (cid:33) = 1 . Proof. Let ( L Nj ( t )) be the process with initial state empty, then one can easilyconstruct a coupling such that the relation L Nj ( t ) ≤ (cid:101) Q Nj ( t ) , t ≥ , ≤ j ≤ J, holds almost surely, where ( Q Nj ( t )) is the M/M/ ∞ queue associated to class j requests. One deduces that, (cid:101) L Nj (0) ≤ st (cid:101) Q Nj (0)where (cid:101) Q Nj (0) is a Poisson random variable with parameter ρ j N and ≤ st is the sto-chastic ordering of random variables. One can therefore construct another couplingsuch that (cid:101) L Nj ( t ) ≤ (cid:101) Q Nj ( t ) , t ≥ , ≤ j ≤ J, LLOCATION SCHEMES OF RESOURCES WITH DOWNGRADING 11 where ( (cid:101) Q Nj ( t )) is a stationary version of the M/M/ ∞ queue associated to class j requests. The lemma is then a consequence of the following convergence in distri-bution of processes, lim N → + ∞ (cid:32) (cid:101) Q Nj ( t ) N (cid:33) = ( ρ j )for 2 ≤ j ≤ J , see Theorem 6.13 pp. 159 of Robert [15] for example. (cid:3) Definition 4. Let ( y ( t )) be the dynamical system on S satisfying (16) dd t y ( t ) = − µ y ( t ) + λ + (cid:32) J (cid:88) k =2 λ k (cid:33) A J (cid:88) k =1 A k ( λ k − µ k y k ( t )) , dd t y j ( t ) = − µ j y j ( t ) + λ j A J (cid:88) k =1 ( A k µ k y k ( t ) − λ k ) , ≤ j ≤ J, with Λ A = J (cid:88) k =1 λ k ( A k − . Lemma 2. If y (0) ∈ ∆ and if there exists an instant T > such that y ( t ) ∈ ∆ for t ∈ [0 , T ] then ( y ( t )) and ( (cid:96) ( t ) coincide on the time interval [0 , T ] , where ( (cid:96) ( t )) is thesolution of Equations (10) with (cid:96) (0)= y (0) .Proof. The proposition is a simple consequence of the representation (10) of thedifferential equations defining the dynamical system ( (cid:96) ( t )) and of the explicit ex-pression of the quantity π (cid:96) ( Z ∗− ) given by Relation (23) when (cid:96) ∈ ∆ , see Rela-tion (7). (cid:3) The next proposition investigates the stability Properties of ( y ( t )). Proposition 4. Let H be the hyperplane H = { z ∈ S : (cid:104) A, z (cid:105) = c }} if y (0) ∈ H then y ( t ) ∈ H for all t ≥ and ( y ( t )) is converging exponentially fast to (cid:96) ∗ defined in Proposition 3.Proof. It is easily checked that dd t (cid:104) A, y ( t ) (cid:105) = 0 , so that if y (0) ∈ H , then the function t (cid:55)→ (cid:104) A, y ( t ) (cid:105) is constant and equal to c ,hence y ( t ) ∈ H for all t ≥ ≤ j ≤ J , dd t y j ( t ) = λ j b − µ j y j ( t ) + λ j J (cid:88) k =2 b k y k ( t ) , with b = µ c − ΛΛ A and b j = A j ( µ j − µ )Λ A . In matrix form, if z ( t ) = ( y ( t ) , . . . , y J ( t )), it can be expressed as(17) dd t z ( t ) = e b + Bz ( t ) , with e b = b ( λ , . . . , λ J ) ∈ R J − and B = ( B jk , ≤ j, k ≤ J ) with B jk = λ j b k − µ j { k = j } . If v = ( v , . . . , v J ) is an eigenvector for the eigenvalue x of B , then( x + µ j ) v j = λ j J (cid:88) k =2 b k v k , ≤ j ≤ J, hence, x is an eigenvalue if and only if it is a solution of the equation F ( x ) def. = J (cid:88) j =2 b j λ j x + µ j = 1 . If L is the number of distinct values of µ j , 2 ≤ j ≤ J , such that µ j (cid:54) = µ , then theabove equation shows that an eigenvalue is a zero of a polynomial of degree at most L . Using Conditions ( R ), it is easy to check that the relation F (0) < B is invertible. Due to the polesof F at the − µ j , 2 ≤ j ≤ J and the relations F (0) < µ j ≥ µ for 2 ≤ j ≤ J , one hasalready L negative solutions of the equation F ( x )=1. All eigenvalues of B are thusnegative, consequently, exp( tB ) converges to 0. (See Corollary 2 of Chapter 25 ofArnol’d [2] for example.)Equation (17) can be solved as z ( t ) = e tB (cid:0) z (0)+ B − e b (cid:1) − B − e b . Therefore the function ( z ( t )) has a limit at infinity given by − B − e b which is clearly( (cid:96) ∗ j , ≤ j ≤ J ). The proposition is proved. (cid:3) One can now prove the main result of this section. Theorem 2. If (cid:96) ∗ is the quantity defined in Proposition 3, then the equilibriumdistribution of ( m N ( t )) converges to π (cid:96) ∗ when N goes to infinity.Proof. Recall that m N ( t )= (cid:10) A, L N ( t ) (cid:11) − C N and let Π N be the invariant distribu-tion of ( L N ( t )). It is assumed that the distribution of L N (0) is Π N for the rest ofthe proof. In particular ( m N ( t )) is a stationary process.One first proves that ( L N (0) /N ) converges in distribution to (cid:96) ∗ . The boundarycondition (cid:10) A, L N (0) (cid:11) ≤ C N gives that the sequence of random variables ( L N (0) /N )is tight. If ( L N k (0) /N k ) is a convergent subsequence to some random variable (cid:96) ∞ ,by Theorem 1, one gets that, for the convergence in distribution, the relationlim k → + ∞ (cid:18)(cid:18) L N k ( t ) N k (cid:19)(cid:19) = ( (cid:96) ( t ))holds, where ( (cid:96) ( t )) is a solution of Equation (10) with initial point at (cid:96) (0) = (cid:96) ∞ .Note that ( (cid:96) ( t )) is a stationary process, its distribution is invariant under any timeshift.By Lemma 1 one has that the relation (cid:96) j ( t ) ≤ ρ j , for 2 ≤ j ≤ J , holds almost surelyon any finite time interval and, by Proposition 1, (cid:104) A, (cid:96) ( t ) (cid:105) ≤ c also holds almostsurely on finite time intervals.Assume that (cid:104) A, (cid:96) (0) (cid:105) A, (cid:96) ( t ) (cid:105) A, ρ (cid:105) >c , there exists some t > (cid:104) A, (cid:96) ( t ) (cid:105) = c .Hence, by stationarity in distribution of ( (cid:96) ( t )), one can shift time at t andassume that (cid:104) A, (cid:96) (0) (cid:105) = c . On this event(18) J (cid:88) j =1 µ j (cid:96) j (0) A j ≥ µ J (cid:88) j =1 (cid:96) j (0) A j = µ c > Λ = J (cid:88) j =1 λ j . Similarly, since (cid:96) j (0) ≤ ρ j for all 2 ≤ j ≤ J , J (cid:88) j =1 A j ( λ j − µ j (cid:96) j (0))= λ − µ c + µ J (cid:88) j =2 A j (cid:96) j (0)+ J (cid:88) j =2 A j ( λ j − µ j (cid:96) j (0))(19) = − µ c + J (cid:88) j =1 A j ( λ j +( µ − µ j ) (cid:96) j (0)) ≥ − µ c + J (cid:88) j =1 A j ( λ j + ( µ − µ j ) ρ j )= − µ c + J (cid:88) j =1 A j λ j µ µ j = µ ( (cid:104) A, ρ (cid:105) − c ) > , and the last quantity is independent of (cid:96) (0). Relations (18) and (19) show that (cid:96) (0) ∈ ∆ and, by Equations (10) and (16), they also hold for t in a small neigh-borhood I of 0 independent of (cid:96) (0) so that (cid:96) ( t ) ∈ ∆ for t ∈ I . Consequently, thedynamical system ( (cid:96) ( t )) never leaves ∆ . Lemma 2 shows that the two dynamicalsystems ( (cid:96) ( t )) and ( y ( t )) (with y (0)= (cid:96) (0)) coincide. Hence, on one hand ( (cid:96) ( t )) is astationary process and, on the other hand, it is a dynamical system converging to (cid:96) ∗ , one deduces that it is constant and equal to (cid:96) ∗ . We have thus proved that thesequence ( L N (0) /N ) converges in distribution to (cid:96) ∗ .Using again Theorem 1, one gets that, for the convergence in distribution,lim N → + ∞ (cid:90) f ( m N ( u )) d u = (cid:90) Z f ( x ) π (cid:96) ∗ (d x )holds for any function f with finite support on Z . By using the stationarity of( m N ( t )) and Lebesgue’s Theorem, one obtainslim N → + ∞ E (cid:0) f ( m N (0)) (cid:1) = (cid:90) Z f ( x ) π (cid:96) ∗ (d x ) . The theorem is proved. (cid:3) Since a job arriving at time t is not downgraded if m N ( t ) < 0, one obtains thefollowing corollary. Corollary 1. As N goes to infinity, the probability that, at equilibrium, a job is notdowngraded in this allocation scheme is converging to π − defined in Proposition 11, π − = c − Λ /µ (cid:104) A, ρ (cid:105) − Λ /µ . Invariant Distribution We assume in this section that (cid:96) ∈ ∆ , as defined in Proposition 2, so that ( m (cid:96) ( t ))is an ergodic Markov process. The goal of this section is to derive an explicit expres-sion of the invariant distribution π (cid:96) on Z of ( m (cid:96) ( t )). At the same time, Proposition 5below gives the required argument to complete the proof of Proposition 3 on thecharacterization of the fixed point of the dynamical system.4.1. Functional Equation. In the following we denote by Y (cid:96) a random variablewith distribution π (cid:96) =( π (cid:96) ( n ) , n ∈ Z ).For r> 0, we will use the notation D ( r )= { z ∈ C , | z | With the notation ϕ + ( z ) = E (cid:0) z Y (cid:96) { Y (cid:96) ≥ } (cid:1) , ϕ − ( z ) = E (cid:0) z Y (cid:96) { Y (cid:96) < } (cid:1) , the random variable Y (cid:96) is such that (20) P ( z ) ϕ + ( z ) = P ( z ) ϕ − ( z ) where P and P are polynomials defined by (21) P ( z ) = J (cid:88) j =1 (cid:104) ( λ j + µ j (cid:96) j ) z A J − λ j z A J +1 − µ j (cid:96) j z A J − A j (cid:105) ,P ( z ) = J (cid:88) j =1 (cid:104) λ j z A J + A j + µ j (cid:96) j z A J − A j − ( λ j + µ j (cid:96) j ) z A J (cid:105) . Proof. For z ∈ γ (1) define f z : Z (cid:55)→ C such that f z ( x )= z x , for x ∈ Z . Equilibriumequations for ( m (cid:96) ( t )) give the identity (cid:88) x,y ∈ Z x (cid:54) = y π (cid:96) ( x ) Q (cid:96) ( x, y )( f z ( y ) − f z ( x )) = 0 , where Q (cid:96) is the Q -matrix of ( m (cid:96) ( t )) given by Equation (6). After some simplereordering, one gets the relation(22) E (cid:0) z Y (cid:96) { Y l ≥ } (cid:1) J (cid:88) j =1 (cid:0) λ j (1 − z ) + µ j (cid:96) j (cid:0) − z − A j (cid:1)(cid:1) = − E (cid:0) z Y l { Y l < } (cid:1) J (cid:88) j =1 (cid:0) λ j (cid:0) − z A j (cid:1) + µ j (cid:96) j (cid:0) − z − A j (cid:1)(cid:1) . By using the definition of ϕ + ( z ) and ϕ − ( z ), Equation (22) can be rewritten asEquation (20). (cid:3) Proposition 5. If (cid:96) ∈ ∆ then (23) π (cid:96) ( Z ∗− ) = (cid:80) Jj =1 ( A j µ j (cid:96) j − λ j ) (cid:80) Jj =1 λ j ( A j − . LLOCATION SCHEMES OF RESOURCES WITH DOWNGRADING 15 In particular if (cid:96) ∗ ∈S is given by Relation (11) then π (cid:96) ∗ ( Z ∗− ) = c − Λ /µ (cid:104) A, ρ (cid:105) − Λ /µ . Note that the right-hand side of the last relation is precisely π − of Relation (12)which is the result necessary to complete the proof of Proposition 3. Proof. With the same notations as before, from Relation (20), ϕ − ( z ) ϕ + ( z ) = P ( z ) P ( z )holds for z ∈ C , with z ∈ γ (1). By definition of ϕ − ( z ) and ϕ + ( z ),lim z → ϕ − ( z ) = π (cid:96) ( Z ∗− ) and lim z → ϕ + ( z ) = 1 − π (cid:96) ( Z ∗− ) . Since 1 is a zero of P and P , this gives the relation π (cid:96) ( Z ∗− )1 − π (cid:96) ( Z ∗− ) = P (cid:48) (1) P (cid:48) (1) = (cid:80) Jj =1 ( A j µ j (cid:96) j − λ j ) (cid:80) Jj =1 A j ( λ j − µ j (cid:96) j ) . Using the expression of ( (cid:96) ∗ j ), with some algebra, one gets π (cid:96) ∗ ( Z ∗− ) = c − J (cid:88) j =1 λ j µ (cid:44) J (cid:88) j =1 ρ j A j − J (cid:88) j =1 λ j µ = π − . The proposition is proved. (cid:3) Relation (20) is valid on the unit circle, however the function ϕ + (resp. ϕ − ) isdefined on D (resp. D c ). This can then be expressed as a Wiener-Hopf factorizationproblem analogous to the one used in the analysis of reflected random walks on N .This is used in the analysis of the GI/GI/ Z , with a drift depending on the half-space where it is located. The first (resp.second) condition in the definition of the set ∆ in Definition (7) implies that thedrift of the random walk in Z ∗− (resp. in N ) is positive (resp. negative).The first step in the analysis of Equation (20) is to determine the locations ofthe zeros of P and P . This is the purpose of the following lemma. Lemma 4. (Location of the Zeros of P and P ) Let (cid:96) be in ∆ .(i) Polynomial P has exactly two positive real roots and z ∈ ]0 , . There are A J − roots in D ( z ) and A J − roots whose modulus are strictly greaterthan .(ii) Polynomial P has exactly two positive real roots and z > . The A J − remaining roots have a modulus strictly smaller than .Proof. One first notes that P is a polynomial with the same form as the f definedby Equation (13) in Bean et al. [4] (with e j = A j , κ j = λ j and ˆ e = A J ). The roots of Q are exactly the roots of f . Lemma 2.2 of Bean et al. [4] gives assertion ( i ) of ourlemma.The proof of assertion (ii) uses an adaptation of the argument for the proof ofLemma 2.2 of Bean et al. [4]. Define the function f ( z )= z − A J P ( z ). Recall that P 16 C. FRICKER, F. GUILLEMIN, PH. ROBERT, AND G. THOMPSON is a polynomial with degree A J +1. There are exactly two real positive roots for P . Indeed, f (1) = 0 and it is easily checked that f is strictly concave with f (cid:48) (1) = J (cid:88) j =1 ( − λ j + A j µ j (cid:96) j ) > , since (cid:96) ∈ ∆ , by the second condition in Definition (7). Hence P has a real zero z greater than 1.Let r ∈ (1 , z ) be fixed, note that P ( r ) > 0. Define f ( z ) = Kz A J , with K = J (cid:88) j =1 ( λ j + A j µ j (cid:96) j ) ,f ( z ) = J (cid:88) j =1 (cid:0) λ j z A J +1 + µ j (cid:96) j z A J − A j (cid:1) , so that P = f − f .Fix some z ∈ γ ( r ). By expressing these functions in terms of real and imaginaryparts, z A J = α + iβ and f ( z ) = α + iβ , one gets(24) (cid:12)(cid:12)(cid:12) f ( z ) − f ( z ) − bz A J (cid:12)(cid:12)(cid:12) = | K ( α + iβ ) − b ( α + iβ ) − ( α + iβ ) | = ( Kα − α ) + ( Kβ − β ) + H = | f ( z ) − f ( z ) | + H, with H = ( bα ) − bα ( Kα − α ) + ( bβ ) − bβ ( Kβ − β )= b ( b − K )( α + β ) + 2 b ( α α + β β ) . Cauchy-Schwarz’s Inequality gives the relation α α + β β ≤ K | f ( z ) || f ( z ) | ≤ K f ( r ) f ( r ) , since | f i ( z ) |≤ f i ( | z | ) for i =1, 2. Thus, Hb = ( b − K )( α + β ) + 2( α α + β β ) ≤ ( b − K ) f ( r ) K + 2 f ( r ) f ( r ) K = f ( r ) K (cid:18) ( b − K ) f ( r ) + 2 Kf ( r ) (cid:19) = f ( r ) K ( bf ( r ) − KP ( r )) . Since P ( r ) > b can be chosen so that bf ( r ) < KP ( r ). From the above relationand Equation (24), one gets that for z ∈ γ ( r ), the relation (cid:12)(cid:12) f ( z ) − f ( z ) − bz A J (cid:12)(cid:12) < | f ( z ) − f ( z ) | holds. By Rouch´e’s theorem, one obtains that, for any r ∈ (1 , z ), P has exactly A J roots in D ( r ). One concludes that P has exactly A J roots in D . It is easilychecked that if z ∈ γ (1) and z (cid:54)∈ R then the real part of P ( z ) is positive, hence z cannot be a root of the polynomial P . Consequently, P has exactly A J − D . The lemma is proved. (cid:3) LLOCATION SCHEMES OF RESOURCES WITH DOWNGRADING 17 Definition 5. For U ∈ { P , P } , denote by Z U the set of the zeros of U differentfrom . Define Φ( z ) = − ϕ + ( z ) λ J − ( z − z ) (cid:89) q ∈Z P ∩ D c ( z − q ) − , z ∈ Dϕ − ( z )Λ − (cid:89) q ∈Z P ∩ D ( z − q ) (cid:89) p ∈Z P ∩ D ( z − p ) − , z ∈ D c with Λ= λ + · · · + λ J and the same notations as before. By definition, function Φis holomorphic in D and D c and, from Relation (20), is continuous on γ (1). Theanalytic continuation theorem, Theorem 16.8 of Rudin [16] for example, gives thatΦ is holomorphic on C . For z ∈ D c , | ϕ − ( z ) | ≤ E (cid:18) { Y (cid:96) < } | z | Y (cid:96) (cid:19) ≤ | z | , since the cardinality of Z P ∩ D (resp. Z P ∩ D ) is A J − A J ), the holomorphicfunction Φ is therefore bounded on C . By Liouville’s theorem, Φ is constant, equalto κ ∈ C . Therefore(25) ϕ + ( z )= − κλ J ( z − z ) − (cid:89) q ∈Z P ∩ D c ( z − q ) , z ∈ D,ϕ − ( z )= κ Λ (cid:89) q ∈Z P ∩ D ( z − q ) − (cid:89) p ∈Z P ∩ D ( z − p ) , z ∈ D c . Recall that ϕ ( z ) = ϕ + ( z ) + ϕ − ( z ) = E (cid:0) z Y (cid:96) (cid:1) is a generating function, in particular ϕ (1) = 1. Plugging the previous expressions for ϕ + and ϕ − in ϕ + (1) + ϕ − (1) = 1,one gets the relation1 = − κ (cid:89) q ∈Z P ∩ D (1 − q ) − − z ( P (cid:48) (1) + P (cid:48) (1)) , hence, using equation (21), κ = z − A (cid:89) q ∈Z P ∩ D (1 − q ) , where Λ A is introduced in Definition 4. Note that κ is positive. We can now statethe main result of this section. Proposition 6 (Invariant Measure) . If (cid:96) ∈ ∆ defined by Relation (7) , then theinvariant measure π (cid:96) can be expressed, for n ∈ Z , as π (cid:96) ( n ) = − κ (cid:88) q ∈Z P ∩ D P ( q ) q − n − ( q − z )( q − R (cid:48) D ( q ) , n< ,κ (cid:18) α n + P ( z ) z − n − ( z − R D ( z ) (cid:19) , ≤ n